Existence and Uniqueness Theorems in the Inverse Problem of Recovering Surface Fluxes from Pointwise Measurements

Abstract

Inverse problems of recovering surface fluxes on the boundary of a domain from pointwise observations are considered. Sharp conditions on the data ensuring existence and uniqueness of solutions in Sobolev classes are exposed. They are smoothness conditions on the data, geometric conditions on the location of measurement points, and the boundary of a domain. The proof relies on asymptotics of fundamental solutions to the corresponding elliptic problems and the Laplace transform. The inverse problem is reduced to a linear algebraic system with a nondegerate matrix.

1. Introduction

Under consideration is the parabolic equation

(1)$Mu=u_t+Lu=f(t,x),(t,x)\in Q=(0,T)\times G,T\le \infty ,$

where $Lu=-\mathsf{\Delta }u+{\sum }_{i=1}^n{a}_i\left(x\right)u_{x_i}+a_0\left(x\right)u$, G is a domain in ${\mathbb{R}}^n$ with boundary $\Gamma \in C^2$, and $n=2,3$. The Equation (1) is furnished with the initial-boundary conditions

(2)${Bu|}_S{=g(t,x)(S=(0,T)\times \Gamma ),u|}_{t=0}=u_0\left(x\right),$

where $Bu=\frac{\partial u}{\partial \nu }+\sigma \left(x\right)u$, with $\nu$ being the outward unit normal to $\Gamma$, and with the overdetermination conditions

(3)$u(t,b_i)={\psi }_i\left(t\right)(i=1,2,\dots ,r),$

where ${\left\{b_i\right\}}_{i=1}^r$ is a collection of points lying in G. Assuming that $g(t,x)={\sum }_{j=1}^r{\alpha }_i\left(t\right){\mathsf{\Phi }}_i\left(x\right)$ for some known functions ${\mathsf{\Phi }}_j$, the problem consists in recovering both a solution to (1) under (2) and (3) and functions ${\alpha }_j$, $j=1,2,\dots ,r$, characterizing g. Note that any function can be approximated by the sums of this form for a suitable choice of basis functions ${\mathsf{\Phi }}_i$.

Inverse problems of recovering the boundary regimes are classical. They arise in many different problems of mathematical physics, in particular, in the heat and mass transfer theory, diffusion, filtration (see [1,2]), and ecology [3,4,5,6,7].

A particular attention is payed to numerical solution of the problems (1)–(3) and close to them. Most of the methods are based on reducing the problems to optimal control ones and minimization of the corresponding quadratic functionals (see, for instance, [8,9,10,11,12,13,14]). However, the problem is that these functionals can have several local minima (see Section 3.3 in [15]). First, we describe some articles, where pointwise measurements are employed as additional data. Numerical determination of constant fluxes in the case of $n=2$ is described in [9]. Similar results are presented in [16] for $n=1$. The three-dimensional problem of recovering constant fluxes of green house gases is discussed in [3], but numerical results are presented only in the one-dimensional case. In [4] (see also [5]) the method of recovering a constant surface flux relying on the approach developed in [17] is described, where special solutions to the adjoint problem are employed (see also [6,7]). The surface fluxes depending on t are recovered in [12,18,19,20] in the case of $n=1$, and in [11,21,22] in the case of $n>1$. The flux depending on time and spatial variables is reconstructed in [14,23].

In literature, there are results in the case in which additional Dirichlet data are given on a part of the boundary and the flux is reconstructed with the use of these data on another part of the boundary (see [24]). The article [13] is devoted to the recovering of the flux $h(t,x)f\left(x\right)$ (the function $f\left(x\right)$ is unknown) with the use of final or integral overdetermination data. The existence and uniqueness theorems for solutions to the inverse problems of recovering the surface flux with the use of integral data are presented in [25,26].

There is a limited number of theoretical results devoted to the problem (1)–(3). We refer the reader to the article [27] (see also [28]), where, in the case of $Mu=u_t-\mathsf{\Delta }u$, $r=1$, and $b_1\in \Gamma$, the existence and uniqueness theorems of classical solutions to the problem (1)–(3) are established. In contrast to our case, the problem is well-posed in the Hadamard sense. If the points ${\left\{b_i\right\}}_{i=1}^r$ are interior points of G then the problem becomes ill-posed and this fact was observed in many articles (see [29]). In this article we describe a new approach to the existence theory of solutions to this problem and establish the corresponding existence and uniqueness theorems. We hope that these results can be used in developing new numerical algorithms for solving the problem.

2. Preliminaries

Let E be a Banach space. By $L_p(G;E)$ (G is a domain in ${\mathbb{R}}^n$), we mean the space of E-valued measurable functions such that ${\parallel \parallel u\left(x\right)\parallel }_E{\parallel }_{L_p\left(G\right)}<\infty$ [30]. The symbols $W_p^s(G;E)$ and $W_p^s(Q;E)$ stand for the Sobolev spaces (see the definitions in [30,31]). If $E=\mathbb{R}$ or $E={\mathbb{R}}^n$ then the latter spaces is denoted by $W_p^s\left(Q\right)$. The definitions of the Hölder spaces $C^{\alpha ,\beta }\left(\overline{Q}\right),C^{\alpha ,\beta }\left(\overline{S}\right)$ can be found in [32]. By the norm of a vector, we mean the sum of the norms of coordinates. Given an interval $J=(0,T)$, put $W_p^{s,r}\left(Q\right)=W_p^s(J;L_p\left(G\right))\cap L_p(J;W_p^r\left(G\right))$ and, respectively, $W_p^{s,r}\left(S\right)=W_p^s(J;L_p(\Gamma ))\cap L_p(J;W_p^r(\Gamma ))$. Denote by ${(u,v)}_0={\int }_{G}u\left(x\right)\overline{v\left(x\right)}dx$ the inner product in $L_2\left(G\right)$. Let $\rho (Y,X)$ designate the distance between the sets $X,Y$. In this case, $\rho (x,\Gamma )$ is the distance from a point x to $\Gamma$. Denote by $B_{\eta }\left(x\right)$ the ball of radius $\eta$ centered at x.

We say that a boundary $\Gamma$ of a domain G belongs to $C^s$, $s\ge 1$ (see the definition in Chapter 1 in [32]) if, for each point $x_0\in \Gamma$, there exists a neighborhood $Y_{x_0}$ about $x_0$ and a coordinate system y (the local coordinate system) obtained from the initial one by the translation of the origin and rotation such that the axis $y_n$ is directed as the interior normal to $\Gamma$ at $x_0$ and the equation of the part $Y_{x_0}\cap \Gamma$ of the boundary is of the form $y_n=\gamma \left(y^{\prime }\right)$, $\gamma \left(0\right)=0$, $y^{\prime }=(y_1,\dots ,y_{n-1})$; moreover, $\gamma \in C^s\left(\overline{B_{\eta }^{\prime }\left(0\right)}\right)$ (where $B_{\eta }^{\prime }\left(0\right)=\{y^{\prime }:|y^{\prime }|<\eta \}$), $G\cap Y_{x_0}=\{y:|y^{\prime }|<\eta ,0<y_n-\gamma \left(y^{\prime }\right)<{\eta }_1\}$, and $({\mathbb{R}}^n\setminus G)\cap Y_{x_0}=\{y:|y^{\prime }|<\eta ,-{\eta }_1<y_n-\gamma \left(y^{\prime }\right)<0\}$. The smoothness of ${\Gamma }_0\subset \Gamma$, with ${\Gamma }_0$ an open subset of $\Gamma$, is defined similarly. The numbers $\eta ,{\eta }_1$ for a given G are fixed and we can assume without loss of generality that ${\eta }_1>(2M+1)\eta$, with M the Lipschitz constant of the function $\gamma$. We employ the straightening of the boundary, i.e., the transformation $z_n=y_n-\gamma \left(y^{\prime }\right)$, $z^{\prime }=y^{\prime }$, $y=y\left(x\right)$, with y the local coordinate system at a given point b.

Below, we assume that $G={\mathbb{R}}_{+}^n=\{x\in {\mathbb{R}}^n:x_n>0\}$ or G is a domain with compact boundary of the class $C^2$. The coefficients of the Equation (1) are assumed to be real. We consider an elliptic operator L, i.e., there exists a constant ${\eta }_0>0$ such that

$\sum _{i,j=1}^n{a}_{ij}{\xi }_i{\xi }_j\ge {\eta }_0{|\xi |}^2\forall \xi \in {\mathbb{R}}^n,\forall x\in G.$

Assign $\overrightarrow{a}=(a_1,a_2)$ for $n=2$ and $\overrightarrow{a}=(a_1,a_2,a_3)$ for $n=3$. The symbol $(·,·)$ stands for an inner product in ${\mathbb{R}}^n$. Let

(4)${\phi }_j\left(x\right)=\frac{1}{2}{\int }_0^1(\overrightarrow{a}(b_j+\tau (x-b_j)),(x-b_j))d\tau$

and assume that

(5)$a_i\in W_{\infty }^2\left(G\right)(i=1,\dots ,n),\nabla {\phi }_j,\mathsf{\Delta }{\phi }_j(j=1,\dots ,r),a_0\in L_{\infty }\left(G\right),\sigma \in C^1(\Gamma ).$

Moreover, we suppose that the functions $a_i$ admits extensions to the whole ${\mathbb{R}}^n$ such that the condition (5) is valid in $G={\mathbb{R}}^n$. If G is a domain with compact boundary of the class $C^2$ such an extension always exists (see Theorem 1 in Subsection 4.3.6 of Section Remarks in [33]). Consider the equation

(6)$L^{*}u+\overline{\lambda }u=\delta (x-b_j),x\in {\mathbb{R}}^n(n=2,3),j=1,2,\dots ,r,$

where the operator $L^{*}$ is a formal adjoint to L. Its coefficients also satisfy (5). Let $b_j=(b_j^1,\dots ,b_j^n)$. Introduce the functions ${\lambda }^{\alpha }={|\lambda |}^{\alpha }{e}^{i\alpha arg\lambda }$, $|arg\lambda |<\pi$. It follows from Theorems 3.5 and 3.1 in [34] and Theorem 3.3 in [35] that

Theorem 1.

Assume that $G={\mathbb{R}}^n$$(n=2,3)$ and the conditions (5) hold. Fix ${\eta }_0\in (0,\pi )$. Then there exists a number ${\lambda }_1\ge 0$ such that, for all λ with $|arg(\lambda -{\lambda }_1)|\le \pi -{\eta }_0$, there exists a unique solution $u_n\left(x\right)$$(n=2,3)$ to the Equation (6) decreasing at ∞ such that $u_n\in W_p^1\left(G\right)$ for all $p\in (1,n/(n-1\left)\right)$, and $u_n\in W_2^2\left(G_{\epsilon }\right)$ for all $\epsilon >0$, $G_{\epsilon }=\{x\in G:|x-b_j|>\epsilon \}$. In every domain $0<\epsilon <|x-b_j|<R$ a solution $u_n$ admits the representation

(7) $u_2\left(x\right)=\frac{1}{2\sqrt{2\pi |x-b_j|}{\lambda }^{1/4}}{e}^{-{\phi }_j\left(x\right)-\sqrt{\lambda }|x-b_j|}\left(1+O\left(\frac{1}{\sqrt{|\lambda |}}\right)\right);$

(8) $u_{2x_i}\left(x\right)=\frac{-{\lambda }^{1/4}{e}^{-{\phi }_j\left(x\right)-\sqrt{\lambda }|x-b_j|}}{2\sqrt{2\pi |x-b_j|}}\left(\frac{x_i-b_j^i}{|x-b_j|}+O\left(\frac{1}{\sqrt{|\lambda |}}\right)\right);$

(9) $u_3\left(x\right)=\frac{1}{4\pi |x-b_j|}{e}^{-{\phi }_j\left(x\right)-\sqrt{\lambda }|x-b_j|}\left(1+O\left(\frac{1}{\sqrt{|\lambda |}}\right)\right);$

(10) $u_{3x_i}\left(x\right)=\frac{-\sqrt{\lambda }{e}^{-{\phi }_j\left(x\right)-\sqrt{\lambda }|x-b_j|}}{4\pi |x-b_j|}\left(\frac{x_i-b_j^i}{|x-b_j|}+O\left(\frac{1}{\sqrt{|\lambda |}}\right)\right).$

In what follows, we denote by $v_j\left(x\right)$ a solution $u_n$ obtained in Theorem 1 for a given j.

Consider the problem

(11)${Lw+\lambda w=f\left(x\right)(x\in G),Bw|}_S=g,,$

where $G={\mathbb{R}}^n$ or $G={\mathbb{R}}_{+}^n$ or G is a domain with compact boundary of the class $C^2$.

Theorem 2.

Let $a_i\in L_{\infty }\left(G\right)$$(i=0,1,\dots ,n)$, $f\in L_p\left(G\right)$, $\sigma \in C^1(\Gamma )$, and $g\in W_p^{2-1/p}(\Gamma )$$(p>1)$. Then there exists a number ${\lambda }_0\ge 0$ such that, for all λ with $Re\lambda \ge {\lambda }_0$, there exists a unique solution $w\in W_p^2\left(G\right)$ to the problem (11).

The theorem results from Theorem 5.7 for $G={\mathbb{R}}^n$, Theorem 7.11 for $G={\mathbb{R}}_{+}^n$ and Theorem 8.2 in the case of a domain with compact boundary in [31].

The following Green formula holds.

Lemma 1.

Let the conditions (5) hold and let $Re\lambda \ge {\lambda }_0$, where ${\lambda }_0$ is chosen so that Theorem2is valid for $p=2$. If $w\in W_2^2\left(G\right)$ is a solution to the problem (11) with $f=0$ from the class specified in Theorem2then

(12)${\int }_{\Gamma }(-\frac{\partial w}{\partial \nu }-\sigma w)\overline{v_j}+w\overline{(\frac{\partial v_j}{\partial \nu }+{\sigma }^{*}{v}_j)}+w\left(b_j\right)=0,{\sigma }^{*}=\sigma +\sum _{i=1}^n{a}_i{\nu }_i.$

If $\phi \left(x\right)\in C_0^{\infty }\left({\mathbb{R}}^n\right)$ and $\phi =1$ in some neighborhood about $b_j$, then

(13) ${\int }_{\Gamma }(-\frac{\partial w}{\partial \nu }-\sigma w)\overline{\phi v_j}+w\overline{(\frac{\partial \phi v_j}{\partial \nu }+{\sigma }^{*}\phi v_j)}+w\left(b_j\right)={\int }_{G}2\nabla \phi \nabla v_j+\mathsf{\Delta }\phi v_j+\sum _{i=1}^n{a}_i{\phi }_{x_i}vdx.$

Proof. 

The proof is conventional. It suffices to approximate the functions $w,v_j$ by sequences of smooth functions in the corresponding norms, to write out the above Formulas (12) and (13) for these approximations, and pass to the limit. □

Assume that $G={\mathbb{R}}_{+}^n$ or G is a domain with compact boundary of the class $C^2$. Given a collection of points $b_j\in G$$(j=1,2,\dots ,r)$, construct the points $b\in \Gamma$ such that ${\delta }_j=\rho (b_j,\Gamma )=|b-b_j|$. Denote by $K_j$ the set of these points. Let $b\in K_j$. For $n=3$, there exists a local coordinate system y such that the axes $y_1,y_2$ agree with the principal directions on the surface $\Gamma$ at $y=0$, in this case, ${\sum }_{i,j=1}^2{\gamma }_{y_i{y}_j}\left(0\right)y_i{y}_j={\gamma }_{y_1{y}_1}\left(0\right)y_1^2+{\gamma }_{y_2{y}_2}\left(0\right)y_2^2,{\gamma }_{y_1,y_2}\left(0\right)=0,$ where ${\kappa }_i={\gamma }_{y_i{y}_i}\left(0\right)$ are the principal curvatures of the surface $y_3=\gamma \left(y^{\prime }\right)$ $(y^{\prime }=(y_1,y_2))$ at 0. In the case of $n=2$, the equation of the boundary in some neighborhood about b is of the form $y_2=\gamma \left(y_1\right)$ and $\kappa ={\gamma }^{″}\left(0\right)$ is the curvature of the curve $\gamma$ at b.

Lemma 2.

Assume that, for every $j=1,2,\dots ,r$, the set $K_j$ consists of finitely many points and, for every $b\in K_j$, we have

(14) $max({\kappa }_1,{\kappa }_2)<1/{\delta }_j\mathrm{if}n=3,\mathrm{or}\kappa <1/{\delta }_j\mathrm{if}n=2,$

where ${\kappa }_i$ are principal curvatures of Γ for $n=3$ and κ is the curvature of Γ for $n=2$ at b. Then there are constants $c_0,c_1>0$, $0<{\epsilon }_1\le \eta$ such that $c_0{|x-b|}^2\le |x-b_j|-{\delta }_j\le c_1{|x-b|}^2$ for every $b\in K_j$ and all $x\in B_{{\epsilon }_1}\left(b\right)\cap \Gamma$, $j=1,2,\dots ,r$.

Remark 1.

For $n=3$, the condition (14) can be reformulated as follows. There exists a constant $q_0\in (0,1)$ such that ${\sum }_{k,l=1}^2{\gamma }_{y_k{y}_l}\left(0\right)y_k{y}_l\le q_0{|y^{\prime }|}^2/{\delta }_j\forall y^{\prime }\in {\mathbb{R}}^2,j=1,2,\dots ,r,$ where y is a local coordinate system at $b\in K_j$. The claim follows from the fact that there exists an orthogonal transformation of coordinates such that the new axes ${\tilde{y}}_1,{\tilde{y}}_2$ agree with the principal directions on the surface Γ at $y=0$.

Proof. 

Take $b\in K_j$. We prove the claim in the case of $n=3$. If $n=2$ then the proof is simpler and we omit it. Let y be a local coordinate system at b. Since $y=y\left(x\right)$ is a superposition of an orthogonal transformation and a translation, the distances between points and their images are the same. We have $b=0$, $b_j=(0,0,y_{3j})$, $x=(y^{\prime },\gamma \left(y^{\prime }\right))$$(y^{\prime }=(y_1,y_2))$, $|x-b|=\sqrt{|y^{\prime }{|}^2+{\gamma }^2\left(y^{\prime }\right)}$, $|b_j-b|=|y_{3j}|={\delta }_j$, $|x-b_j|=\sqrt{|y^{\prime }{|}^2+{|\gamma \left(y^{\prime }\right)-y_{3j}|}^2}$, and

$|x-b_j|-{\delta }_j=\frac{|x-b_j{|}^2-{\delta }_j^2}{|x-b_j|+{\delta }_j}=\frac{|y^{\prime }{|}^2+{\gamma }^2-2\gamma y_{3j}}{|x-b_j|+{\delta }_j}=J.$

Remark 1 implies that

$\gamma \left(y^{\prime }\right)=\frac{1}{2}\sum _{i,j=1}^{n-1}{\gamma }_{y_i{y}_j}\left(0\right)y_i{y}_j+o(|y^{\prime }{|}^2)\le q_0|y^{\prime }{|}^2/2{\delta }_j+o(|y^{\prime }{|}^2)$

in some neighborhood about 0. Fix a parameter ${\epsilon }_0>0$ such that ${\epsilon }_0+q_0<1$. In this case there exists ${\eta }_1\le \eta$ such that

$q_0|y^{\prime }{|}^2/2{\delta }_j+o(|y^{\prime }{|}^2)\le ({\epsilon }_0+q_0){|y^{\prime }|}^2/2{\delta }_j$

for $|y^{\prime }|\le {\eta }_1$. Therefore, we obtain

$J\ge \frac{|y^{\prime }{|}^2(1-(q_0+{\epsilon }_0))+{\gamma }^2}{|x-b_j|+{\delta }_j}\ge c_0{|x-b|}^2,c_0>0.$

The converse inequality follows directly from the definition of the quantity J.

Below, we preserve the notations of Lemma 2. Take $b\in K_j$. We can define the transformations $y=y\left(x\right)$ and $x=x\left(y\right)$. For $n=3$, put

$\begin{array}{c}{c}_j\left(b\right)=1/\sqrt{1-{\delta }_j{\kappa }_1},d_j\left(b\right)=1/\sqrt{1-{\delta }_j{\kappa }_2},I_j\left(b\right)=c_j\left(b\right)d_j\left(b\right),\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{c}_j^{*}\left(b\right)=1/\sqrt{1+{\delta }_j{\kappa }_1},d_j^{*}\left(b\right)=1/\sqrt{1+{\delta }_j{\kappa }_2},I_j^{*}\left(b\right)=c_j^{*}\left(b\right)d_j^{*}\left(b\right),\\ \hspace{1em}\hspace{1em}\hspace{1em}{B}_{j,\lambda }\left(b\right)=\{x\in \Gamma :y_1^2\left(x\right)/c_j^2\left(b\right)+y_2^2\left(x\right)/d_j^2\left(b\right)\le {|\lambda |}^{-1/2+{\epsilon }_0}\},\\ \hspace{1em}\hspace{1em}{\tilde{B}}_{j,\lambda }\left(b\right)=\{y\in {\mathbb{R}}^2:y_1^2/c_j^2\left(b\right)+y_2^2/d_j^2\left(b\right)\le {|\lambda |}^{-1/2+{\epsilon }_0}\},\\ \hspace{1em}\hspace{1em}\hspace{1em}{B}_{j,\lambda }^{*}\left(b\right)=\{x\in \Gamma :y_1^2\left(x\right)/c_j^{*2}\left(b\right)+y_2^2\left(x\right)/d_j^{*2}\left(b\right)\le {|\lambda |}^{-1/2+{\epsilon }_0}\},\\ \hfill {\tilde{B}}_{j,\lambda }^{*}\left(b\right)=\{y\in {\mathbb{R}}^2:y_1^2/c_j^{*2}\left(b\right)+y_2^2/d_j^{*2}\left(b\right)\le {|\lambda |}^{-1/2+{\epsilon }_0}\},\end{array}$

where the parameter ${\epsilon }_0\in (0,1/4)$ is chosen below. The map $y=y\left(x\right)$ takes $B_{j,\lambda }\left(b\right)$ onto ${\tilde{B}}_{j,\lambda }\left(b\right)$. Similar notations are used in the case of $n=2$, i.e.,

$\begin{array}{c}{I}_j\left(b\right)=c_j\left(b\right)=1/\sqrt{1-{\delta }_j\kappa },I_j^{*}\left(b\right)=c_j^{*}\left(b\right)=1/\sqrt{1+{\delta }_j\kappa },\hfill \\ B_{j,\lambda }\left(b\right)=\{x\in \Gamma :|y_1\left(x\right)|/c_j\left(b\right)\le {|\lambda |}^{-1/4+{\epsilon }_0/2}\},\\ {\tilde{B}}_{j,\lambda }\left(b\right)=\{y_1\in \mathbb{R}:|y_1|/c_j\left(b\right)\le {|\lambda |}^{-1/4+{\epsilon }_0/2}\},\\ \hspace{1em}{B}_{j,\lambda }^{*}\left(b\right)=\{x\in \Gamma :|y_1\left(x\right)|/c_j\left(b\right)\le {|\lambda |}^{-1/4+{\epsilon }_0/2}\},\\ \hfill {\tilde{B}}_{j,\lambda }^{*}\left(b\right)=\{y_1\in \mathbb{R}:|y_1|/c_j^{*}\left(b\right)\le {|\lambda |}^{-1/4+{\epsilon }_0/2}\}.\end{array}$

Below, we assume that, for every $j=1,2,\dots ,r$, the set $K_j$ consists of finitely many points and

(15)$\forall j=1,2,\dots ,r,\forall b\in K_j,|{\kappa }_i|{\delta }_j<1(i=1,2)\mathrm{for}n=3,|\kappa |{\delta }_j<1\mathrm{for}n=2,$

where ${\kappa }_i$ are the principal curvatures of $\Gamma$ for $n=3$ and, respectively, $\kappa$ is the curvature of $\Gamma$ for $n=2$.

Let $v_j$ be a solution to the Equation (6). Given $b\in K_j$, construct the point $b_j^b$ lying on the straight line joining $b_j$ and b and such that ${\delta }_j=|b_j-b|=|b-b_j^b|$, $|b_j-b_j^b|=2{\delta }_j$. The point $b_j^b$ is symmetric to $b_j$ with respect to the surface $\Gamma$. Let $v_j^b$ be a solution to the Equation (6), where the point $b_j$ is replaced with $b_j^b$. Denote by ${\phi }_j^b$ the functions defined by the equality (4), where $b_j$ is replaced with $b_j^b$. In what follows, we assume that the closures of coordinate neighborhoods about the points $b\in K_j$ are disjoint, otherwise, we can always reduce them. Fix a point $b\in K_j$. The quantity ${min}_{b^{\prime }\in K_j,b^{\prime }\ne b}|b^{\prime }-b_j^b|-{\delta }_j$ is positive (it depends on ${\delta }_j$ and the angles between the vectors $\overrightarrow{b{b}_j}$ and $\overrightarrow{b^{\prime }{b}_j}$). Let $X_b=\overline{Y_b\cap \Gamma }$ (where $Y_b$ is the coordinate neighborhood about b). Without loss of generality, we can also assume that the constant ${min}_{b^{\prime }\in K_j,b^{\prime }\ne b}\rho (X_{b^{\prime }},b_j^b)-{\delta }_j$ is positive for all $b\in K_j$ and all j, otherwise, we decrease the parameter $\eta$ of the coordinate neighborhoods $Y_b$. Denote by ${\epsilon }^0>0$ a constant smaller than the minimum of these constants. Theorem 1 for $b\ne b^{\prime }$ and $b\in K_j$ yields

(16)$|\sqrt{\lambda }{v}_j^b{e}^{{\delta }_j\sqrt{\lambda }}|+|e^{{\delta }_j\sqrt{\lambda }}|\nabla v_j^b||\le c_1{e}^{-{\epsilon }^0\sqrt{|\lambda |/2}}\forall x\in X_{b^{\prime }},$

where $c_1>0$ and $q_0\in (0,1)$ are constants independent of j, $b\in K_j$, and $\lambda$ such that $Re\lambda \ge {\lambda }_1$. □

Lemma 3.

Assume that the conditions (5) and (15) hold, $b\in K_j$ $(j=1,2,\dots ,r)$, and

(17) $\mathsf{\Phi }\in C^{{\alpha }_0}\left(X_b\right),X_b=\overline{Y_b\cap \Gamma }$

for some ${\alpha }_0\in (0,1]$. Then there exists a number ${\lambda }_0>0$ such that, for $Re\lambda \ge {\lambda }_0$, we have the representation

(18) ${\mathsf{\Phi }}_j=\sqrt{\lambda }{e}^{{\delta }_j\sqrt{\lambda }}{\int }_{X_b}\mathsf{\Phi }\left(x\right)\overline{v_j\left(x\right)}d\Gamma =\frac{\mathsf{\Phi }\left(b\right)e^{-{\phi }_j\left(b\right)}}{2I_j\left(b\right)}{(1+O(|\lambda |}^{-\beta }\left)\right),\beta ={\alpha }_0/4,$

(19) ${\mathsf{\Phi }}_j^{*}=\sqrt{\lambda }{e}^{{\delta }_j\sqrt{\lambda }}{\int }_{X_b}\mathsf{\Phi }\left(x\right)\overline{v_j^b\left(x\right)}d\Gamma =\frac{\mathsf{\Phi }\left(b\right)e^{-{\phi }_j^b\left(b\right)}}{2I_j^{*}\left(b\right)}{(1+O(|\lambda |}^{-\beta }),$

(20) $e^{{\delta }_j\sqrt{\lambda }}{\int }_{X_b}\overline{v_j}d\Gamma =\frac{e^{-{\phi }_j\left(b\right)}}{2I_j\left(b\right)\sqrt{\lambda }}{(1+O(|\lambda |}^{-1/4}\left)\right),e^{{\delta }_j\sqrt{\lambda }}{\int }_{X_b}\frac{\partial \overline{v_j}}{\partial \nu }d\Gamma =\frac{-e^{-{\phi }_j\left(b\right)}}{2I_j\left(b\right)}{(1+O(|\lambda |}^{-1/4}\left)\right),$

(21) $e^{{\delta }_j\sqrt{\lambda }}{\int }_{X_b}\overline{v_j^b}d\Gamma =\frac{e^{-{\phi }_j^b\left(b\right)}}{2I_j^{*}\left(b\right)\sqrt{\lambda }}{(1+O(|\lambda |}^{-1/4}\left)\right),e^{{\delta }_j\sqrt{\lambda }}{\int }_{X_b}\frac{\partial \overline{v_j^b}}{\partial \nu }d\Gamma =\frac{e^{-{\phi }_j^b\left(b\right)}}{2I_j^{*}\left(b\right)}{(1+O(|\lambda |}^{-1/4}\left)\right).$

Proof. 

Consider the case of $n=3$. We have

(22)$I={\int }_{X_b}\mathsf{\Phi }\left(x\right)\overline{v_j\left(x\right)}d\Gamma ={\int }_{B_{j,\lambda }\left(b\right)}\mathsf{\Phi }\left(x\right)\overline{v_j\left(x\right)}d\Gamma +{\int }_{X_b\setminus B_{j,\lambda }\left(b\right)}\mathsf{\Phi }\left(x\right)\overline{v_j\left(x\right)}d\Gamma .$

Theorem 1 implies that

$\overline{v_j\left(x\right)}=\frac{1}{4\pi |x-b_j|}{e}^{-{\phi }_j\left(x\right)-\sqrt{\lambda }|x-b_j|}(1+O\left(\frac{1}{\sqrt{|\lambda |}}\right)),x\in \Gamma ,$

where $Re\lambda \ge {\lambda }_1$. We can assume that $|O(\frac{1}{\sqrt{|\lambda |}})|\le 1/2$ for all such $\lambda$ and j. To estimate the second integral $J_2$ on the right-hand side of (22) from above, we derive that

(23)$|J_2{e}^{\sqrt{\lambda }{\delta }_j}|\le c{\int }_{X_b\setminus B_{j,\lambda }\left(b\right)}|e^{\sqrt{\lambda }{\delta }_j}||v_j\left(x\right)|d\Gamma \le c_1{\int }_{X_b\setminus B_{j,\lambda }\left(b\right)}{e}^{-Re\sqrt{\lambda }(|x-b_j|-{\delta }_j)}d\Gamma .$

In view of the definitions, there exists a constant ${\epsilon }_2>0$ such that $|x-b_j|-{\delta }_j\ge {\epsilon }_2{|\lambda |}^{-1/2+{\epsilon }_0}$ for all $x\in X_b\setminus B_{j,\lambda }\left(b\right)$ and, thereby,

(24)$|J_2{e}^{\sqrt{\lambda }{\delta }_j}|\le c_4{e}^{-{\epsilon }_4{|\lambda |}^{{\epsilon }_0}}$

for some constant ${\epsilon }_4>0$. For the first summand $J_1$ on the right-hand side of (22), we have

(25)$\begin{array}{c}{e}^{\sqrt{\lambda }{\delta }_j}{J}_1=e^{\sqrt{\lambda }{\delta }_j}[{\int }_{B_{j,\lambda }\left(b\right)}(e^{-{\phi }_j\left(x\right)}\mathsf{\Phi }\left(x\right)-e^{-{\phi }_j\left(b\right)}\mathsf{\Phi }\left(b\right))e^{{\phi }_j\left(x\right)}\overline{v_j\left(x\right)}d\Gamma +\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathsf{\Phi }\left(b\right)e^{-{\phi }_j\left(b\right)}{\int }_{B_{j,\lambda }\left(b\right)}{e}^{{\phi }_j\left(x\right)}\overline{v_j\left(x\right)}d\Gamma ].\end{array}$

Consider the last integral in (25) that is multiplied by $e^{\sqrt{\lambda }{\delta }_j}$. This quantity is written as

(26)$\begin{array}{c}{J}_2^{\prime }=\mathsf{\Phi }\left(b\right)e^{-{\phi }_j\left(b\right)}{\int }_{B_{j,\lambda }\left(b\right)}\frac{e^{-\sqrt{\lambda }(|x-b_j|-{\delta }_j)}}{4\pi |x-b_j|}(1+O\left(\frac{1}{\sqrt{|\lambda |}}\right))d\Gamma =\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\mathsf{\Phi }\left(b\right)e^{-{\phi }_j\left(b\right)}{\int }_{{\tilde{B}}_{j,\lambda }\left(b\right)}\frac{e^{-\sqrt{\lambda }(|y-{\tilde{b}}_j|-{\delta }_j)}}{4\pi |y-{\tilde{b}}_j|}\sqrt{1+{|\nabla \gamma \left(y\right)|}^2}(1+O\left(\frac{1}{\sqrt{|\lambda |}}\right))d{y}^{\prime }=\\ \frac{\mathsf{\Phi }\left(b\right)e^{-{\phi }_j\left(b\right)}}{4\pi {\delta }_j}{\int }_{{\tilde{B}}_{j,\lambda }\left(b\right)}{e}^{-\sqrt{\lambda }(|y-{\tilde{b}}_j|-{\delta }_j)}d{y}^{\prime }+\\ \mathsf{\Phi }\left(b\right)e^{-{\phi }_j\left(b\right)}{\int }_{{\tilde{B}}_{j,\lambda }\left(b\right)}{e}^{-\sqrt{\lambda }(|y-{\tilde{b}}_j|-{\delta }_j)}{\psi }_0\left(y\right)O\left(\frac{1}{\sqrt{|\lambda |}}\right)d{y}^{\prime }+\\ \hfill \mathsf{\Phi }\left(b\right)e^{-{\phi }_j\left(b\right)}{\int }_{{\tilde{B}}_{j,\lambda }\left(b\right)}{e}^{-\sqrt{\lambda }(|y-{\tilde{b}}_j|-{\delta }_j)}({\psi }_0\left(y\right)-{\psi }_0\left(0\right))d{y}^{\prime },{\psi }_0\left(y\right)=\frac{\sqrt{1+{|\nabla \gamma \left(y\right)|}^2}}{4\pi |y-{\tilde{b}}_j|},\end{array}$

where ${\tilde{b}}_j$ is the point $b_j$ written in the coordinate system y. Consider the integral $I_0={\int }_{{\tilde{B}}_{j,\lambda }\left(b\right)}{e}^{-\sqrt{\lambda }(|y-{\tilde{b}}_j|-{\delta }_j)}d{y}^{\prime }.$ We can assume that the axes of the local coordinate system y are directed as the principal directions on $\Gamma$ at b. In this case (see Lemma 2) we obtain that

$|y-b_j|-{\delta }_j=\frac{y_1^2(1-{\kappa }_1{\delta }_j)+y_2^2(1-{\kappa }_2{\delta }_j)+o(|y^{\prime }{|}^2)}{|x\left(y\right)-b_j|+{\delta }_j}=\frac{y_1^2(1-{\kappa }_1{\delta }_j)}{2{\delta }_j}+\frac{y_2^2(1-{\kappa }_2{\delta }_j)}{2{\delta }_j}+o(|y^{\prime }{|}^2),$

where $o(|y^{\prime }{|}^2)$ is a $C^2$-function in some neighborhood about 0. Make the change of variables $y_i={\tau }_i\sqrt{2{\delta }_j/(1-{\kappa }_i{\delta }_j)}$ in $I_0$. We obtain that

$I_0=\frac{2{\delta }_j}{c_j\left(b\right)d_j\left(b\right)}{\int }_{|\tau |\le r_0}{e}^{-\sqrt{\lambda }{(|\tau |}^2+{o(|\tau |}^2\left)\right)}d\tau ,r_0={|\lambda |}^{-1/4+{\epsilon }_0/2}/\sqrt{2{\delta }_j}.$

Introducing the polar coordinate system, we arrive at the expression

$I_0=\frac{2{\delta }_j}{c_j\left(b\right)d_j\left(b\right)}{\int }_0^{2\pi }{\int }_0^{r_0}{e}^{-\sqrt{\lambda }{\phi }_0(r,\psi )}rdrd\psi ,{\phi }_0(r,\psi )=r^2+o\left(r^2\right).$

Integrating by parts yields

$\begin{array}{c}{I}_0=\frac{-2{\delta }_j}{\sqrt{\lambda }{c}_j\left(b\right)d_j\left(b\right)}{\int }_0^{2\pi }{e}^{-\sqrt{\lambda }{\phi }_0(r,\psi )}\frac{r}{{\phi }_{0r}(r,\psi )}{|}_{r=0}^{r_0}d\psi +\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\frac{2{\delta }_j}{\sqrt{\lambda }{c}_j\left(b\right)d_j\left(b\right)}{\int }_0^{2\pi }{\int }_0^{r_0}{e}^{-\sqrt{\lambda }{\phi }_0(r,\psi )}{\left(\frac{r}{{\phi }_{0r}(r,\psi )}\right)}^{\prime }drd\psi =\\ \hspace{1em}\hspace{1em}\hspace{1em}\frac{2{\delta }_j\pi }{\sqrt{\lambda }{c}_j\left(b\right)d_j\left(b\right)}-\frac{2{\delta }_j}{\sqrt{\lambda }{c}_j\left(b\right)d_j\left(b\right)}{\int }_0^{2\pi }{e}^{-\sqrt{\lambda }{\phi }_0(r_0,\psi )}\frac{r_0}{{\phi }_{0r}(r_0,\psi )}d\psi +\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\frac{2{\delta }_j}{\sqrt{\lambda }{c}_j\left(b\right)d_j\left(b\right)}{\int }_0^{2\pi }{\int }_0^{r_0}{e}^{-\sqrt{\lambda }{\phi }_0(r,\psi )}{\left(\frac{r}{{\phi }_{0r}(r,\psi )}\right)}^{\prime }drd\psi .\hfill \end{array}$

The last integral here admits the estimate

$|{\int }_0^{2\pi }{\int }_0^{r_0}{e}^{-\sqrt{\lambda }{\phi }_0(r,\psi )}{\left(\frac{r}{{\phi }_{0r}(r,\psi )}\right)}^{\prime }drd\psi |\le c_7{\int }_0^{2\pi }{\int }_0^{r_0}{e}^{-Re\sqrt{\lambda }{c}_0{r}^2}drd\psi \le c_8{|\lambda |}^{-1/4}.$

The second integral on the right-hand side is estimated as

$\left|\frac{2{\delta }_j}{\sqrt{\lambda }{c}_j\left(b\right)d_j\left(b\right)}{\int }_0^{2\pi }{e}^{-\sqrt{\lambda }{\phi }_0(r_0,\psi )}\frac{r_0}{{\phi }_{0r}(r_0,\psi )}d\psi \right|\le c_9{e}^{-{\epsilon }_5{|\lambda |}^{{\epsilon }_0}},$

where ${\epsilon }_5$ is a positive constant. Thus, we establish the representation

(27)$I_0=\frac{2{\delta }_j\pi }{\sqrt{\lambda }{c}_j\left(b\right)d_j\left(b\right)}{(1+O(|\lambda |}^{-1/4}\left)\right).$

Consider the integral

$I_0^{\prime }={\int }_{{\tilde{B}}_{j,\lambda }\left(b\right)}|y^{\prime }{|}^{{\beta }_0}{e}^{-Re\sqrt{\lambda }(|y-{\tilde{b}}_j|-{\delta }_j)}d{y}^{\prime }\le c_0{\int }_{|\tau |\le r_0}{|\tau |}^{{\beta }_0}{e}^{-Re\sqrt{\lambda }{(|\tau |}^2+{o(|\tau |}^2\left)\right)}d\tau .$

Introducing the polar coordinate system, we infer

$I_0^{\prime }\le c_0{\int }_0^{2\pi }{\int }_0^{r_0}{e}^{-Re\sqrt{\lambda }{\phi }_0(r,\psi )}{r}^{1+{\beta }_0}drd\psi ,{\phi }_0(r,\psi )=r^2+o\left(r^2\right).$

Making the change of variables $r=t/|Re\sqrt{\lambda }{|}^{1/2}$, we obtain the estimate

$I_0^{\prime }\le c_0|Re\sqrt{\lambda }{|}^{-1-{\beta }_0/2}{\int }_0^{2\pi }{\int }_0^{r_0{|Re\sqrt{\lambda }|}^{1/2}}{e}^{-t^2(1+\frac{Re\sqrt{\lambda }}{t^2}o\left(\frac{t^2}{Re\sqrt{\lambda }}\right))}{t}^{1+{\beta }_0}dtd\psi \le c_1{|\lambda |}^{-1/2-{\beta }_0/4}.$

This inequality and (24) imply that

(28)$I_0^{\prime }\le c_1{|\lambda |}^{-1/2-{\beta }_0/4},$

where the constant $c_1$ is independent of $\lambda$. In this case the last integral $I_1$ on the right-hand side of (26) admits the estimate

$\begin{array}{c}|I_1|\le \left|{\mathsf{\Phi }}_i\left(b\right)e^{-{\phi }_j\left(b\right)}{\int }_{{\tilde{B}}_{j,\lambda }\left(b\right)}{e}^{-\sqrt{\lambda }(|y-{\tilde{b}}_j|-{\delta }_j)}({\psi }_0\left(y\right)-{\psi }_0\left(0\right))d{y}^{\prime }\right|\le \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{c}_{10}{\int }_{{\tilde{B}}_{j,\lambda }\left(b\right)}{e}^{-Re\sqrt{\lambda }(|y-{\tilde{b}}_j|-{\delta }_j)}|y^{\prime }{|}^{2}d{y}^{\prime }\le c_{11}{|\lambda |}^{-1}.\end{array}$

In view of (28), the previous integral $I_2$ in (26) (${\beta }_0=1$) is estimated as follows: $|I_2|\le c_{12}/|\lambda |.$ Finally, the second summand on the right-hand side of (25) is representable as

(29)$J_2^{\prime }=\frac{\mathsf{\Phi }\left(b\right)e^{-{\phi }_j\left(b\right)}}{2\sqrt{\lambda }{c}_j\left(b\right)d_j\left(b\right)}{(1+O(|\lambda |}^{-1/4}\left)\right).$

In view of our conditions on the coefficients, ${\phi }_j\in W_{\infty }^2\left(K\right)$ for every compact set $K\in \overline{G}$, and thereby, $|{\phi }_j\left(x\right)-{\phi }_j\left(b\right)|\le c|x-b|=c\sqrt{|y^{\prime }{|}^2+{|\gamma \left(y^{\prime }\right)|}^2}\le c_1|y^{\prime }|$. Involving the condition of the lemma and (28), we can estimate the integral $J_3=e^{\sqrt{\lambda }{\delta }_j}{\int }_{B_{j,\lambda }\left(b\right)}(\mathsf{\Phi }\left(x\right)-\mathsf{\Phi }\left(b\right))\overline{v_j\left(x\right)}+\mathsf{\Phi }\left(b\right)(1-e^{-{\phi }_j\left(b\right)+{\phi }_j\left(x\right)})\overline{v_j\left(x\right)}d\Gamma$ on the right-hand side of (25) by

(30)$|J_3|\le c_2{\int }_{{\tilde{B}}_{j,\lambda }\left(b\right)}|y^{\prime }{|}^{{\alpha }_0}{e}^{-Re\sqrt{\lambda }(|y-{\tilde{b}}_j|-{\delta }_j)}d{y}^{\prime }\le c_4{|\lambda |}^{-1/2-{\alpha }_0/4}.$

The representation (29) and the estimate (30) validate the equality (18). The equality (19) is proven analogously and the former equalities in (20) and (21) are consequences of (18) and (19). The proof in the case of $n=2$ is simpler. Display the asymptotics of the main integral

$I=\mathsf{\Phi }\left(b\right)e^{-{\phi }_j\left(b\right)}{I}_0,I_0=e^{\sqrt{\lambda }{\delta }_j}{\int }_{X_b}{e}^{{\phi }_j\left(x\right)}\overline{v_j\left(x\right)}d\Gamma ,$

where $X_b=\{x\left(y\right)\in \Gamma :|y_1|\le \eta \}$, $y=(y_1,y_2)$ is the local coordinate system at b, and $y_2=\gamma \left(y_1\right)$ is the equation of the curve $\Gamma$. To reduce arguments, we take $\eta \le {\epsilon }_1$, where the parameter ${\epsilon }_1$ is defined in Lemma 2. Theorem 1 implies that

(31)$\begin{array}{c}{I}_0=\frac{{\lambda }^{-1/4}}{2\sqrt{2\pi }}{\int }_{X_b}{e}^{-\sqrt{\lambda }(|x-b_j|-{\delta }_j)}\frac{1}{|x-b_j{|}^{1/2}}(1+O\left(\frac{1}{\sqrt{|\lambda |}}\right))d\Gamma =\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\frac{{\lambda }^{-1/4}}{2\sqrt{2\pi }}{\int }_{-\eta }^{\eta }{e}^{-\sqrt{\lambda }(|y-{\tilde{b}}_j|-{\delta }_j)}\frac{\sqrt{1+{\left({\gamma }^{\prime }\left(y_1\right)\right)}^2}}{|y-{\tilde{b}}_j{|}^{1/2}}d{y}_1+\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\frac{{\lambda }^{-1/4}}{2\sqrt{2\pi }}{\int }_{-\eta }^{\eta }{e}^{-\sqrt{\lambda }(|y-{\tilde{b}}_j|-{\delta }_j)}\frac{\sqrt{1+{\left({\gamma }^{\prime }\left(y_1\right)\right)}^2}}{|y-{\tilde{b}}_j{|}^{1/2}}O\left(\frac{1}{\sqrt{|\lambda |}}\right)d{y}_1.\end{array}$

As before, we have $|y-b_j|-{\delta }_j=\frac{y_1^2(1-\kappa {\delta }_j)-2{\delta }_j(\gamma -\kappa y_1^2/2)+{\gamma }^2}{\sqrt{y_1^2+{(\gamma -{\delta }_j)}^2}+{\delta }_j}$ $(\kappa =\kappa \left(b\right)={\gamma }^{\prime \prime }\left(0\right))$. We have the asymptotic formula (see Section 1 , Chapter 2 in [36])

${\int }_a^b{e}^{\lambda S\left(x\right)}f\left(x\right)dx=\sqrt{\frac{-2\pi }{\lambda S^{\prime \prime }\left(x_0\right)}}f\left(x_0\right){+O(1/|\lambda |}^{3/2}),$

where $x_0\in (a,b)$ is a point in which S reaches its maximum. Applying this formula to the first integral on the right-hand side of (31) and estimating the second integral by $c/{|\lambda |}^{3/4}$, we obtain

$I_0=\frac{{\lambda }^{-1/2}}{2\sqrt{1-\kappa {\delta }_j}}{+O(1/|\lambda |}^{3/4}).$

All other arguments are similar. The proof in the case of $G={\mathbb{R}}_{+}^n$ is even simpler and we omit it.

It remains to prove the latter inequalities in (19) and (20). As before, take $n=3$. The asymptotics from Theorem 1 ensure that

$e^{\sqrt{\lambda }{\delta }_j}\frac{\partial \overline{v_j}}{\partial \nu }=\frac{-\sqrt{\lambda }{e}^{{\phi }_j\left(x\left(y\right)\right)}{e}^{-\sqrt{\lambda }(|y-{\tilde{b}}_j|-{\delta }_j)}}{4\pi |y-{\tilde{b}}_j|}\left[\frac{(y-{\tilde{b}}_j,\nu )}{|y-{\tilde{b}}_j|}+{O(|\lambda |}^{-1/2})\right],$

where $\nu =\frac{1}{\sqrt{1+{|\nabla \gamma |}^2}}({\gamma }_{y_1},{\gamma }_{y_2},-1).$ If $y_n=\gamma \left(y^{\prime }\right)$ then we have

$\frac{(y-{\tilde{b}}_j,\nu )}{|y-{\tilde{b}}_j|}=\frac{y_1{\gamma }_{y_1}+y_2{\gamma }_{y_2}-\gamma \left(y\right)+{\delta }_j}{|y-{\tilde{b}}_j|}=1+O(|y^{\prime }{|}^2),{\tilde{b}}_j=(0,0,{\delta }_j).$

Thus, we obtain that

(32)$e^{\sqrt{\lambda }{\delta }_j}\frac{\partial \overline{v_j}}{\partial \nu }=-\sqrt{\lambda }{e}^{{\delta }_j\sqrt{\lambda }}\overline{v_j\left(x\left(y\right)\right)}(1+O(|y^{\prime }{|}^2{)+O(|\lambda |}^{-1/2}\left)\right).$

This equality and the previous arguments validate the claim. □

Remark 2.

Let $G=B_R\left(x_0\right)$. Then the condition (15) holds if $b_j\ne x_0$ for all j.

We consider the problem (11), where $f=0$, i.e., the problem

(33)$Lw+\lambda w=0,x\in G,$

(34)${Bw|}_S=g,$

and we obtain some estimates of its solution. Fix j and take $b\in K_j$. In Lemma 4 below, we use functions $\phi \in C_0^{\infty }\left({\mathbb{R}}^n\right)$ such that $\phi \left(y\right)=1$ on the set $U_{3\eta /4}=\{y:|y^{\prime }|\le 3\eta /4,|y_n|\le M\eta +3\eta /4\}$ and $supp\phi \subset U_{\eta }=\{y:|y^{\prime }|<\eta ,|y_n|<(M+1)\eta \}$. The condition ${\eta }_1\ge (2M+1)\eta$ ensures the inclusion $U_{\eta }\subset \overline{Y_b}$. The map $z_n=y_n-\gamma \left(y^{\prime }\right)$, $z^{\prime }=y^{\prime }$ takes a neighborhood $Y_b\cap G$ onto the set $U=\{z:|z^{\prime }|<\eta ,0<z_n<{\eta }_1\}$. Denote $B_{\eta }^{\prime }\left(0\right)=\{z^{\prime }:|z^{\prime }|<\eta \}$ and ${\Gamma }_{\eta }=\left({\cup }_{j=1}^r{\cup }_{b\in K_j}{Y}_b\right)\cap \Gamma$.

Lemma 4.

Assume that the conditions (5) hold, $b\in K_j$$(j=1,2,\dots ,r)$, and $g\in W_2^{1/2}(\Gamma )\cap W_2^1\left(X_b\right)$. Then there exists a number ${\lambda }_0>0$ such that, for $Re\lambda \ge {\lambda }_0$, there exists a unique a solution to the problem (33) and (34) in the space $W_2^2\left(G\right)$ satisfying the estimates

${\int }_G{|\nabla w|}^2+{|\lambda ||w|}^{2}dx\le c_0{\parallel g\parallel }_{L_2(\Gamma )}^2{|\lambda |}^{-1/2+2{\epsilon }_7},$

(35) ${\parallel w\parallel }_{W_2^{\alpha }(\Gamma )}\le c_1{\parallel g\parallel }_{L_2(\Gamma )}{|\lambda |}^{\alpha /2-1/2+{\epsilon }_7},\alpha \in (0,1/2).$

If $v=\phi w$, with φ from the above-described class of functions, then there exist constants $c_2,c_3>0$ such that

(36) ${\int }_U\sum _{k,l=1}^{n-1}|v_{z_k{z}_l}{|}^2+\sum _{k=1}^{n-1}|v_{z_n{z}_k}{|}^2+|\lambda ||{\nabla }_{z^{\prime }}{v|}^{2}dx\le c_2{(\parallel g\parallel }_{W_2^1\left(X_b\right)}^2+{\parallel g\parallel }_{L_2(\Gamma )}^2{)|\lambda |}^{-1/2+2{\epsilon }_7},$

(37) ${\parallel v\parallel }_{W_2^{1+\alpha }\left(B_{\eta }^{\prime }\left(0\right)\right)}\le c_3{(\parallel g\parallel }_{W_2^1\left(X_b\right)}+{\parallel g\parallel }_{L_2(\Gamma )}{)|\lambda |}^{\alpha /2-1/2+{\epsilon }_7},\alpha \in (0,1/2),$

where ${\epsilon }_7>0$ is arbitrarily small constant. If additionally $g\in W_2^2\left(X_b\right)$ and

(38) $a_0\in W_{\infty }^1\left({\cup }_{b\in K_j}(Y_b\cap G)\right),{\Gamma }_{\eta }\in C^3,\sigma \in C^{3/2+\epsilon }\left({\Gamma }_{\eta }\right)(\epsilon >0),$

then $\phi w\in W_2^3(Y_b\cap G)$ for any φ and there exist constants $c_4,c_5>0$ such that

(39) ${\int }_U\sum _{i,j,k=1}^{n-1}|v_{z_i{z}_j{z}_k}{|}^2+\sum _{k,i=1}^{n-1}|v_{z_i{z}_k{z}_n}{|}^2+|\lambda |\sum _{i,k=1}^{n-1}{v}_{z_i{z}_k}{|}^{2}dx\le c_4{(\parallel g\parallel }_{W_2^2\left(X_b\right)}^2+{\parallel g\parallel }_{L_2(\Gamma )}^2{)|\lambda |}^{\frac{-1}{2}+2{\epsilon }_7},$

(40) ${\parallel v\parallel }_{W_2^{2+\alpha }\left(B_{\eta }^{\prime }\left(0\right)\right)}\le c_5{(\parallel g\parallel }_{W_2^2\left(X_b\right)}+{\parallel g\parallel }_{L_2(\Gamma )}{)|\lambda |}^{\alpha /2-1/2+{\epsilon }_7},\alpha \in (0,1/2).$

Proof. 

Theorem 2 for $p=2$ ensures the existence and uniqueness of solutions provided that $Re\lambda \ge {\lambda }_0$ for some ${\lambda }_0>0$. Multiply the Equation (33) by a function $\overline{w}$ and integrate the result over G. Integrating by parts, we infer

${\int }_G{|\nabla w|}^2+l_0\left(w\right)\overline{w}+{\lambda |w|}^2={\int }_{\Gamma }g\overline{w}-\sigma {|w|}^{2}d\Gamma ,\mathrm{where}{l}_0\left(w\right)=\sum _{i=1}^n{a}_i{w}_{x_i}+a_{0}w.$

Separating the real and imaginary parts, we obtain

(41)${\int }_G{|\nabla w|}^2+{Re\lambda |w|}^{2}dx=Re{\int }_{\Gamma }g\overline{w}-\sigma {|w|}^{2}d\Gamma -Re{\int }_G{l}_0\left(w\right)\overline{w}dx.$

$Im\lambda {\int }_G{|w|}^{2}dx=Im{\int }_{\Gamma }g\overline{w}-\sigma {|w|}^{2}d\Gamma -Im{\int }_G{l}_0\left(w\right)\overline{w}dx.$

The last equality yields

(42)$|Im\lambda |{\int }_G{|w|}^{2}dx\le \left|Im{\int }_{\Gamma }g\overline{w}-{\sigma |w|}^{2}d\Gamma |+|Im{\int }_G{l}_{0}w\overline{w}dx\right|.$

Summing (42) and (41) and estimating the modules of the right-hand sides

(43)${\int }_G{|\nabla w|}^2+{|\lambda ||w|}^{2}dx\le c_0\left(\left|{\int }_{\Gamma }g\overline{w}-\sigma {|w|}^{2}d\Gamma \right|+\left|{\int }_G{l}_0\left(w\right)\overline{w}dx\right|\right).$

Below, we use the inequality

$|ab|\le {\epsilon |a|}^p/p+{\epsilon }^{-q/p}{|b|}^q/q,p\in (1,\infty ),q=p/(p-1),\epsilon >0.$

The last integral is estimated by

(44)$\left|{\int }_G{l}_0\left(w\right)\overline{w}dx\right|\le {\parallel \nabla w\parallel }_{L_2\left(G\right)}{\parallel w\parallel }_{L_2\left(G\right)}+{\parallel w\parallel }_{L_2\left(G\right)}^2\le \frac{1}{4}{\parallel \nabla w\parallel }_{L_2\left(G\right)}^2+c_1{\parallel w\parallel }_{L_2\left(G\right)}^2.$

Similarly, we have

$\begin{array}{c}\left|{\int }_{\Gamma }g\overline{w}-\sigma {|w|}^{2}d\Gamma \right|\le {\parallel g\parallel }_{L_2(\Gamma )}{\parallel w\parallel }_{L_2(\Gamma )}+c_2{\parallel w\parallel }_{L_2(\Gamma )}^2\le \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{c\left(\epsilon \right)\parallel g\parallel }_{L_2(\Gamma )}^2{|\lambda |}^{-1/2+2{\epsilon }_7}+{\epsilon |\lambda |}^{1/2-2{\epsilon }_7}{\parallel w\parallel }_{L_2(\Gamma )}^2+c_2{\parallel w\parallel }_{L_2(\Gamma )}^2,\end{array}$

where $\epsilon$ and ${\epsilon }_7$ are arbitrary positive constants. The embedding theorems and interpolation inequalities (see [30]) imply that

$\begin{array}{c}{|\lambda |}^{1/2-2{\epsilon }_7}{\parallel w\parallel }_{L_2(\Gamma )}^2\le c_3{|\lambda |}^{1/2-2{\epsilon }_7}{\parallel w\parallel }_{W_2^{1/2+2{\epsilon }_7}\left(G\right)}^2\le \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{c}_5{|\lambda |}^{1/2-2{\epsilon }_7}{\parallel w\parallel }_{W_2^1\left(G\right)}^{2(1/2+2{\epsilon }_7)}{\parallel w\parallel }_{L_2\left(G\right)}^{2(1/2-2{\epsilon }_7)}\le {\parallel \nabla w\parallel }^2+c_6{\parallel w\parallel }_{L_2\left(G\right)}^2|\lambda |.\end{array}$

Similarly,

(45)$c_2{\parallel w\parallel }_{L_2(\Gamma )}^2\le \frac{1}{4}{\parallel \nabla w\parallel }^2+c_7{\parallel w\parallel }_{L_2\left(G\right)}^2.$

Estimating the right-hand side of (43) with the use of (44) and (45), we arrive at the inequality

$\begin{array}{c}{\int }_G{|\nabla w|}^2+{|\lambda ||w|}^{2}dx\le c_8{\parallel g\parallel }_{L_2(\Gamma )}^2{|\lambda |}^{-1/2+2{\epsilon }_7}+\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\epsilon c_6{|\lambda |\parallel w\parallel }_{L_2\left(G\right)}^2+{(\epsilon +1/2)\parallel \nabla w\parallel }_{L_2\left(G\right)}^2+c_9{\parallel w\parallel }_{L_2\left(G\right)}^2.\end{array}$

Choosing sufficiently small $\epsilon$ and increasing ${\lambda }_0$, if necessary, we derive that

(46)${\int }_G{|\nabla w|}^2+{|\lambda ||w|}^{2}dx\le c_9{\parallel g\parallel }_{L_2(\Gamma )}^2{|\lambda |}^{-1/2+2{\epsilon }_7},$

where the constant $c_9$ is independent of $\lambda$ with $Re\lambda \ge {\lambda }_0$ and ${\epsilon }_7>0$ can be taken arbitrarily small. Using (46) and interpolation inequalities we obtain that

${\parallel w\parallel }_{W_2^{\alpha }(\Gamma )}\le {c\parallel w\parallel }_{W_2^{1/2+\alpha }\left(G\right)}\le c_{10}{\parallel w\parallel }_{W_2^1\left(G\right)}^{1/2+\alpha }{\parallel w\parallel }_{L_2\left(G\right)}^{1/2-\alpha }\le c_{11}{|\lambda |}^{\alpha /2-1/2+{\epsilon }_7}{\parallel g\parallel }_{L_2(\Gamma )}$

and the estimate (35) is proven. Rewriting (33) in the coordinate system y, we obtain the problem

(47)$-\mathsf{\Delta }w+\sum _{i=1}^n{\tilde{a}}_i{w}_{y_i}+a_0{w+\lambda w=0,Bw|}_{\Gamma }=g.$

Multiply the equation (47) by $\phi \left(y\right)$. The result is the problem

(48)$-\mathsf{\Delta }v+\sum _{i=1}^n{\tilde{a}}_i{v}_{y_i}+a_{0}v+\lambda v=-2\nabla w\nabla \phi -w\mathsf{\Delta }\phi +\sum _{i=1}^n{a}_i{\phi }_{y_i}w=f_0,v=w\phi .$

(49)${Bv|}_{\Gamma }=\phi g-{\phi }_{\nu }w.$

Introduce the coordinate system z, with $z^{\prime }=y^{\prime }$, $z_n=y_n-\gamma \left(y^{\prime }\right)$. In this case, the function $v=w\phi$ is a solution to the problem

(50)$-{\mathsf{\Delta }}_{z^{\prime }}v+2\sum _{i=1}^{n-1}{\gamma }_{z_i}{v}_{z_n{z}_i}-{\sigma }_0\left(z^{\prime }\right)v_{z_n{z}_n}+\sum _{i=1}^n{c}_i{v}_{z_i}+a_{0}v+\lambda v=f_0,{\sigma }_0\left(z^{\prime }\right)=(1+|{\nabla }_{z^{\prime }}\gamma {|}^2).$

$-v_{z_n}{\sigma }_0\left(z^{\prime }\right)+\sum _{i=1}^{n-1}{v}_{z_i}{\gamma }_{z_i}+\sigma \left(x\left(z\right)\right){\sigma }_0\left(z^{\prime }\right)v{{|}_{z_n=0}=(\phi g-{\phi }_{\nu }w)|}_{z_n=0}{\sigma }_0\left(z^{\prime }\right).$

Multiplying the Equation (50) by $-{\mathsf{\Delta }}_{z^{\prime }}v$ and integrating the result over U, we obtain that

(51)$\begin{array}{c}{\int }_U{|{\mathsf{\Delta }}_{z^{\prime }}v|}^2-\sum _{i=1}^{n-1}{\gamma }_{z_i}{v}_{z_n{z}_i}\overline{{\mathsf{\Delta }}_{z^{\prime }}v}+{\partial }_{z_n}(-\sum _{i=1}^{n-1}{\gamma }_{z_i}{v}_{z_i}+{\sigma }_0\left(z^{\prime }\right)v_{z_n})\overline{{\mathsf{\Delta }}_{z^{\prime }}v}-\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(\sum _{i=1}^n{c}_i{v}_{z_i}+a_{0}v)\overline{{\mathsf{\Delta }}_{z^{\prime }}v}+\lambda {|{\nabla }_{z^{\prime }}v|}^{2}dz={(f_0,-{\mathsf{\Delta }}_{z^{\prime }}v)}_0.\end{array}$

Integrating by parts, we rewrite the first summand in the form

(52)${\int }_U|{\mathsf{\Delta }}_{z^{\prime }}{v|}^{2}dz=\sum _{k,l=1}^{n-1}{\int }_U{|v_{z_k{z}_l}|}^{2}dz.$

Note that $v\in W_2^2\left(G\right)$ and integrating by parts we obtain the integrals containing third order derivatives. However, the result of integration is easily justified if we employ smooth approximations of functions in $W_2^2\left(G\right)$. Similar arguments can be found, for instance, in the proof of Lemma 7.1 of Chapter 3 in [37]. We also have

(53)$\begin{array}{c}{\int }_U{\partial }_{z_n}(-\sum _{i=1}^{n-1}{\gamma }_{z_i}{v}_{z_i}+{\sigma }_0{v}_{z_n})\overline{{\mathsf{\Delta }}_{z^{\prime }}v}dz=-{\int }_U{\partial }_{z_n}{\nabla }_{z^{\prime }}(-\sum _{i=1}^{n-1}{\gamma }_{z_i}{v}_{z_i}+{\sigma }_0{v}_{z_n})\overline{{\nabla }_{z^{\prime }}v}dz=\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\int }_G{\nabla }_{z^{\prime }}(-\sum _{i=1}^{n-1}{\gamma }_{z_i}{v}_{z_i}+{\sigma }_0{v}_{z_n})·\overline{{\nabla }_{z^{\prime }}{v}_{z_n}}dz+{\int }_{B_{\eta }^{\prime }\left(0\right)}({\nabla }_{z^{\prime }}(-\sum _{i=1}^{n-1}{\gamma }_{z_i}{v}_{z_i}+{\sigma }_0{v}_{z_n})·\overline{{\nabla }_{z^{\prime }}v}d{z}^{\prime }=\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\int }_U(1+|{\nabla }_{z^{\prime }}{\gamma |}^2)|{\nabla }_{z^{\prime }}{v}_{z_n}{|}^{2}dz-{\int }_U\sum _{i=1}^{n-1}{\gamma }_{z_i}{\nabla }_{z^{\prime }}{v}_{z_i}·\overline{{\nabla }_{z^{\prime }}{v}_{z_n}}dz-{\int }_U\sum _{i=1}^{n-1}{v}_{z_i}{\nabla }_{z^{\prime }}{\gamma }_{z_i}·\overline{{\nabla }_{z^{\prime }}{v}_{z_n}}dz\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\int }_U{v}_{z_n}{\nabla }_{z^{\prime }}{\sigma }_0\left(z^{\prime }\right)\overline{{\nabla }_{z^{\prime }}{v}_{z_n}}dz-{\int }_{B_{\eta }^{\prime }\left(0\right)}{\nabla }_{z^{\prime }}\left((\phi g-{\phi }_{\nu }w)\sqrt{{\sigma }_0\left(z^{\prime }\right)}\right)·\overline{{\nabla }_{z^{\prime }}v}d{z}^{\prime }.\end{array}$

Consider the expression

(54)$\begin{array}{c}{\int }_U-\sum _{i=1}^{n-1}{\gamma }_{z_i}{v}_{z_n{z}_i}\overline{{\mathsf{\Delta }}_{z^{\prime }}v}dz={\int }_U\sum _{i=1}^{n-1}{\gamma }_{z_i}{v}_{z_n}\overline{{\mathsf{\Delta }}_{z^{\prime }}{v}_{z_i}}+\sum _{i=1}^{n-1}{\gamma }_{z_i{z}_i}{v}_{z_n}\overline{{\mathsf{\Delta }}_{z^{\prime }}v}dz\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}=-{\int }_U\sum _{i,k=1}^{n-1}{\gamma }_{z_i}{v}_{z_n{z}_k}\overline{v_{z_i{z}_k}}dz+{\int }_U\sum _{i=1}^{n-1}{\gamma }_{z_i{z}_i}{v}_{z_n}\overline{{\mathsf{\Delta }}_{z^{\prime }}v}dz-{\int }_U\sum _{i=1}^{n-1}{\gamma }_{z_i{z}_k}{v}_{z_n}\overline{v_{z_i{z}_k}}dz\end{array}$

Using (52)–(54) in (51), we obtain

(55)$\begin{array}{c}{\int }_U\sum _{k,l=1}^{n-1}|v_{z_k{z}_l}{|}^2+\sum _{k=1}^{n-1}{\sigma }_0|v_{z_n{z}_k}{|}^2-2Re\sum _{l,k=1}^{n-1}{\gamma }_{z_l}{v}_{z_n{z}_k}\overline{v_{z_l{z}_k}}+\lambda {|{\nabla }_{z^{\prime }}v|}^{2}dz={(f_0,-{\mathsf{\Delta }}_{z^{\prime }}v)}_0\hfill \\ -{\int }_U\sum _{i=1}^{n-1}{\gamma }_{z_i{z}_i}{v}_{z_n}\overline{{\mathsf{\Delta }}_{z^{\prime }}v}dz+{\int }_U\sum _{i=1}^{n-1}{\gamma }_{z_i{z}_k}{v}_{z_n}\overline{v_{z_i{z}_k}}dz+{\int }_U\sum _{i=1}^{n-1}{v}_{z_i}{\nabla }_{z^{\prime }}{\gamma }_{z_i}·\overline{{\nabla }_{z^{\prime }}{v}_{z_n}}dz-\\ \hfill {\int }_U{v}_{z_n}{\nabla }_{z^{\prime }}{\sigma }_0\overline{{\nabla }_{z^{\prime }}{v}_{z_n}}dz-{\int }_{B_{\eta }^{\prime }\left(0\right)}({\nabla }_{z^{\prime }}\left((\phi g-{\phi }_{\nu }w)\sqrt{{\sigma }_0}\right)·\overline{{\nabla }_{z^{\prime }}v}d{z}^{\prime }.\end{array}$

As it is seen, the inequality

$\sum _{k,l=1}^{n-1}|v_{z_k{z}_l}{|}^2+\sum _{k=1}^{n-1}{\sigma }_0|v_{z_n{z}_k}{|}^2-2Re\sum _{l,k=1}^{n-1}{\gamma }_{z_l}{v}_{z_n{z}_k}\overline{v_{z_l{z}_k}}\ge c_3(\sum _{k,l=1}^{n-1}|v_{z_k{z}_l}{|}^2+\sum _{k=1}^{n-1}|v_{z_n{z}_k}{|}^2),$

is valid for some constant $c_3>0$. Next, we infer

$\begin{array}{c}|{\int }_{B_{\eta }^{\prime }\left(0\right)}{\nabla }_{z^{\prime }}\left({\phi }_{\nu }w\sqrt{{\sigma }_0}\right)·\overline{{\nabla }_{z^{\prime }}v}|d{z}^{\prime }\le c\parallel {\nabla }_{z^{\prime }}\left({\phi }_{\nu }w\sqrt{{\sigma }_0}\right){\parallel }_{{\left(W_{2,0}^{1/2}\left(B_{\eta }^{\prime }\left(0\right)\right)\right)}^{\prime }}{\parallel {\nabla }_{z^{\prime }}v\parallel }_{W_{2,0}^{1/2}\left(B_{\eta }^{\prime }\left(0\right)\right)}\hfill \\ \hfill \le c_1{\parallel w\parallel }_{W_2^{1/2}\left(B_{\eta }^{\prime }\left(0\right)\right)}\parallel {\nabla }_{z^{\prime }}{v\parallel }_{W_2^{1/2}\left(B_{\eta }^{\prime }\left(0\right)\right)}\le \epsilon \parallel {\nabla }_{z^{\prime }}{v\parallel }_{W_2^1\left(U\right)}^2+c\left(\epsilon \right){\parallel w\parallel }_{W_2^1\left(U\right)}^2,\end{array}$

where $W_{2,0}^{1/2}\left(B_{\eta }^{\prime }\left(0\right)\right)$ is the space with the norm ${\parallel v\parallel }^2={\parallel v\parallel }_{W_2^{1/2}\left(B_{\eta }^{\prime }\left(0\right)\right)}^2+{\int }_{B_{\eta }^{\prime }\left(0\right)}{|v|}^2\frac{d{z}^{\prime }}{\rho \left(z^{\prime }\right)}$, $\rho \left(z^{\prime }\right)=\rho (z^{\prime },\partial B_{\eta }^{\prime }\left(0\right))$, $\epsilon >0$ is arbitrary, and the last summand is estimated by ${c\parallel g\parallel }_{L_2(\Gamma )}$ (see (46)). Here we rely on the conventional theorems on pointwise multipliers and Proposition 12.1 of Chapter 1 in [38]. Next, repeating the arguments of the proof of the estimate (46), we conclude that

${\int }_U\sum _{k,l=1}^{n-1}|v_{z_k{z}_l}{|}^2+\sum _{k=1}^{n-1}|v_{z_n{z}_k}{|}^2+|\lambda ||{\nabla }_{z^{\prime }}{v|}^{2}dz\le c_0{(\parallel g\parallel }_{W_2^1\left(X_b\right)}^2+{\parallel g\parallel }_{L_2(\Gamma )}^2{)|\lambda |}^{-1/2+2{\epsilon }_7}.$

To establish (37), it suffices to prove the estimate

$\parallel {\nabla }_{z^{\prime }}{v\parallel }_{W_2^{\alpha }(\Gamma )}^2+{\parallel v\parallel }_{W_2^{\alpha }(\Gamma )}^2\le c_1{(\parallel g\parallel }_{W_2^1\left(X_b\right)}+{\parallel g\parallel }_{L_2(\Gamma )}{)|\lambda |}^{\alpha /2-1/2+{\epsilon }_7},\alpha \in (0,1/2),$

which is justified by repeating of the proof of (35). To validate the second part of the claim, we first demonstrate the smoothness of a solution w. Take an arbitrary point $b\in K_j$ and the set $Y_b$. Construct a function $\phi \left(y\right)\in C_0^{\infty }\left({\mathbb{R}}^n\right)$ such that $supp\phi \subset U_{\eta }$. The function $w_0=w\phi$ is a solution to the Equation (48) from the space $W_2^2(Y_b\cap G)$ satisfying (49) on $\Gamma \cap Y_b$ and

$-\mathsf{\Delta }{w}_0=-\sum _{i=1}^n{a}_i{w}_{0x_i}-a_0{w}_0-2\nabla w\nabla \phi -\mathsf{\Delta }\phi w\in W_2^1\left(Y_b\right),$

$\frac{\partial w_0}{\partial \nu }{|}_{\Gamma }=-\sigma w_0-{\phi }_{\nu }w+g\phi \in W_2^{3/2}(\Gamma \cap Y_b).$

Using the conventional theorems on extension of boundary data inside the domain [30] and Theorem Section 3 of Chapter 4 in [39], we conclude that $w_0\in W_2^3(Y_b\cap G)$.

Consider the equation (50). Multiply (50) by ${\mathsf{\Delta }}_{z^{\prime }}^{2}v$ and integrate the result over U. The same arguments as those of the proof of the estimate (36) can be applied to justify (37) and (39). The calculations are rather cumbersome and we omit them. □

Assume that the conditions (5) and (15) hold. In this case, for every j and $b\in K_j$, we can consruct the balls $B_j=B_{{\delta }_j}\left(b_j\right)$ and $B_j^b=B_{{\delta }_j}\left(b_j^b\right)$. Let $Y_{b,\epsilon }=\{y\in Y_b:|y^{\prime }|\le \epsilon \}$ (where $\epsilon \le \eta$).

Lemma 5.

Let the conditions (5) and (15) hold. Then, for every $j=1,2,\dots ,r$, there exists a function ${\phi }_j\in C_0^{\infty }\left({\mathbb{R}}^n\right)$ and constants ${\epsilon }_0,\rho \in (0,\eta /8)$ such that ${\phi }_j\left(x\right)=1$ for $x\in U_{\rho }=B_{{\delta }_j+\rho }\left(b_j\right)\cup {\cup }_{b\in K_j}{Y}_{b,{\epsilon }_0/2+\rho }$, ${\phi }_j\left(x\right)=0$ for $x\notin U_{3\rho }$, and $\rho (supp|\nabla {\phi }_j|\cap G,B_j^b)>0$ for all $b\in K_j$.

Proof. 

In view of (15), it is not difficult to establish that there exists a parameter ${\epsilon }_0<\eta /8$ such that $\rho (B_j\setminus {\cup }_{b\in K_j}{Y}_{b,\eta },\Gamma )={\eta }_0\left(\eta \right)>0$ for all $\eta \le {\epsilon }_0$ and $\overline{B_j^b}\cap {\cup }_{b\in K_j}(Y_{b,{\epsilon }_0}\cap \Gamma )=\left\{b\right\}$ for all $b\in K_j$. Put ${\eta }_0={min}_{\eta \in [{\epsilon }_0/2,{\epsilon }_0]}{\eta }_0\left(\eta \right)$. Obviously, ${\eta }_0>0$. Take $\rho =min({\epsilon }_0/8,{\eta }_0/8)$. Construct a nonnegative function $\omega \in C_0^{\infty }\left({\mathbb{R}}^n\right)$ such that $supp\omega \subset B_1\left(0\right)$, ${\int }_{{\mathbb{R}}^n}\omega \left(\xi \right)d\xi =1$ and the averaged function

${\phi }_j\left(x\right)=\frac{1}{{\rho }^n}{\int }_{{\mathbb{R}}^n}\omega \left(\frac{\xi -x}{\rho }\right){\chi }_{U_{2\rho }}\left(\xi \right)d\xi ,$

where ${\chi }_{U_{2\rho }}\left(\xi \right)$ is the characteristic function of the set $U_{2\rho }$. By construction, ${\phi }_j\left(x\right)=1$ for $x\in U_{\rho }$ and ${\phi }_j\left(x\right)=0$ for $x\notin U_{3\rho }$. This function satisfies our conditions. □

Let

(56)$u_0\left(x\right)\in W_2^1\left(G\right),e^{-\lambda t}f\in L_2\left(Q\right),e^{-\lambda t}{\alpha }_i\in W_2^{1/4}(0,T),{\mathsf{\Phi }}_i\left(x\right)\in W_2^{1/2}(\Gamma ).$

The following theorem results from Theorem 7.11 for $G={\mathbb{R}}_{+}^n$ and Theorems 8.2 in the case of the domain with compact boundary in [40].

Theorem 3.

Assume that $T=\infty$ and $a_i\left(x\right)\in L_{\infty }\left(G\right)$$(i=0,1,\dots ,n)$. Then there exists a constant ${\lambda }_0\ge 0$ such that if $\lambda \ge {\lambda }_0$ and the condition (56)holds then there exists a unique solution to the problem (1) and (2)such that $e^{-\lambda t}u\in W_2^{1,2}\left(Q\right)$ and

(57)$\parallel e^{-\lambda t}{u\parallel }_{W_2^{1,2}\left(Q\right)}\le C_0(\parallel u_0{\parallel }_{W_2^1\left(G\right)}+\parallel e^{-\lambda t}{f\parallel }_{L_2\left(Q\right)}+\parallel e^{-\lambda t}g{\parallel }_{W_p^{1/4,1/2}\left(S\right)}).$

Let E be a Hilbert space. Denote by ${\tilde{W}}_{2,{\gamma }_0}^s(0,\infty ;E)$ the space of functions u defined on $(0,\infty )$ whose zero extensions $\tilde{u}\left(t\right)$ to the negative semiaxis belong to $W_{2,loc}^s(\mathbb{R};E)$ and

$\parallel e^{-{\gamma }_{0}t}\tilde{u}\left(t\right){\parallel }_{W_2^s(\mathbb{R};E)}<\infty .$

The Laplace transform $\mathcal{L}$ is an isomorphism of this space ${\tilde{W}}_{2,{\gamma }_0}^s\left({\mathbb{R}}_{+};E\right)$ onto the space $E_{s,{\gamma }_0}$ of analytic functions in the domain $Rep>{\gamma }_0\ge 0$ such that

$\parallel U\left(p\right){\parallel }_{s,{\gamma }_0}^2=\underset{\gamma >{\gamma }_0}{sup}{\int }_{-\infty }^{\infty }\parallel U(\gamma +i\tau ){\parallel }_E^2\left({1+|\gamma +i\tau |}^{2s}\right)d\tau <\infty .$

If $E=\mathbb{C}$ or $E=L_2\left(G\right)$ or $E=W_2^s\left(G\right)$ (G is a domain in ${\mathbb{R}}^n$) then these properties of the Laplace transform can be found in [41] (see Theorem 7.1 and Section 8). For $T<\infty$, we similarly define the space ${\tilde{W}}_2^s(0,T)$ as the subspace of functions in $W_2^s(0,T)$ admitting the zero extensions for $t<0$ of the same class. This space coincides with $W_2^s(0,T)$ for $s<1/2$ and with the space of functions $u\in W_2^s(0,T)$ such that $u\left(0\right)=0$ for $s>1/2$. For $s=1/2$, it coincides with the space of functions in $W_2^{1/2}(0,T)$ such that $u{t}^{-1/2}\in L_2(0,T)$ [41].

3. Basic Results

We assume here that the conditions (5), (15), (17) are fulfilled. Let $\mathsf{\Psi }$ be the matrix with entries ${\mathsf{\Psi }}_{ji}={\sum }_{b\in K_j}\frac{{\mathsf{\Phi }}_i\left(b\right)e^{-{\phi }_j\left(b\right)}}{I_j\left(b\right)}$$(i,j=1,2,\dots ,r)$. We assume that

(58)$det\mathsf{\Psi }\ne 0,{\mathsf{\Phi }}_i\left(x\right)\in W_2^{1/2}(\Gamma ),$

(59)${\mathsf{\Phi }}_i\left(x\right)\in W_2^1\left(X_b\right)\mathrm{for}n=2,{\mathsf{\Phi }}_i\left(x\right)\in W_2^2\left(X_b\right)\mathrm{for}n=3,b\in {\cup }_{j=1}^r{K}_j.$

Fix a parameter ${\lambda }_0>0$ greater than the maximum of the parameters defined in Theorem 1 with ${\eta }_0=\pi /2$, Theorem 2 with $p=2$, and Theorem 3. We assume that

(60)$u_0\left(x\right)\in W_2^1\left(G\right),e^{-{\gamma }_{0}t}f\in L_2\left(Q\right).$

By Theorem 3, if the condition (60) holds for some ${\gamma }_0\ge {\lambda }_0$, then there exists a unique solution $w_0$ to the problem (1) and (2), where $g=0$, such that $e^{-{\gamma }_{0}t}{w}_0\in W_2^{1,2}\left(Q\right)$. Consider the problem (1)–(3). Changing the variables $w=u-w_0$, we obtain the simpler problem

(61)$w_t+Lw=0,Bw{{|}_S=g(t,x),w|}_{t=0}=0,$

(62)$w(b_j,t)={\psi }_j\left(t\right)-w_0(t,b_j)={\tilde{\psi }}_j\left(t\right),j=1,2,\dots ,r.$

We assume that ${\tilde{\psi }}_j\left(t\right)\in L_2(0,T)$ and

(63)${\tilde{\psi }}_j\left(t\right)={\int }_0^t{V}_{{\delta }_j}(t-\tau ){\psi }_{0j}\left(\tau \right)d\tau ,{\psi }_{0j}{e}^{-{\gamma }_{0}t}\in {\tilde{W}}_2^{n/4}(0,T)(n=2,3),$

where $V_{\gamma }\left(t\right)=\frac{e^{-{\gamma }^2/4t}}{4\pi t}$ for $n=2$ and $V_{\gamma }=\frac{\gamma e^{-{\gamma }^2/4t}}{2\sqrt{\pi }{t}^{3/2}}$ for $n=3$. For $T=\infty$, the condition (63) can be rewritten as

(64)$\underset{\sigma >{\gamma }_0}{sup}{\int }_{-\infty }^{\infty }{|\sigma +is|}^{n/2}{e}^{Re\sqrt{p}{\delta }_j}{|\mathcal{L}\left({\tilde{\psi }}_j\right)(\sigma +is)|}^{2}ds<\infty ,\mathrm{where}p=\sigma +is.$

For a finite T, the condition (63) can be stated as follows: there exists an extension of ${\tilde{\psi }}_j$ on $(0,\infty )$ satisfying (64). We have ${\widehat{V}}_{\gamma }\left(\lambda \right)=\frac{i}{4}{H}_0^{\left(1\right)}\left(i\sqrt{\lambda }\gamma \right)=\frac{1}{2\sqrt{2\pi \gamma }{\lambda }^{1/4}}{e}^{-\sqrt{\lambda }\gamma }\left(1+O\left(\frac{1}{\sqrt{|\lambda |}}\right)\right)$ for $n=2$ and ${\widehat{V}}_{\gamma }\left(\lambda \right)=e^{-\sqrt{\lambda }\gamma }$ for $n=3$. Here $H_0^{\left(1\right)}$ is the Hankel function. The latter equality is derived in Lemma 1.6.7 in [42]. The former can be easily obtained if we use the Poisson formula for a solution to the Cauchy problem for the heat equation with the right-hand side equal to the Dirac delta function.

Theorem 4.

Assume that $T=\infty$ and the conditions (5), (15), (58), (59), and (38)for $n=3$ hold. Then there exists ${\lambda }_1\ge {\lambda }_0$ such that, if $Re\lambda ={\gamma }_0\ge {\lambda }_1$ and the conditions (60), (63) are fulfilled, then there exists a unique solution to the problem (1)–(3)such that $e^{-{\gamma }_{0}t}u\in W_2^{1,2}\left(Q\right)$, $e^{-{\gamma }_{0}t}{\alpha }_i\left(t\right)\in W_2^{1/4}(0,T)$$(i=1,2,\dots ,r)$.

Proof. 

Consider the equivalent problem (61) and (62). Assuming that $w\in W_2^{1,2}\left(Q\right)$ and applying the Laplace transform to (61), we arrive at the problem

(65)$L_0\widehat{w}=\lambda \widehat{w}+L\widehat{w}=0,B\widehat{w}{|}_{\Gamma }=\sum _{i=1}^r{\widehat{\alpha }}_i{\mathsf{\Phi }}_i\left(x\right)=\widehat{g},$

(66)$\widehat{w}\left(b_j\right)={\widehat{\tilde{\psi }}}_j,j=1,2,\dots ,r.$

Next, we use the functions $v_j,v_j^b$ constructed before Lemma 3. Theorem 1 yields $v_j\in W_2^2\left(G_j\left(\epsilon \right)\right)$, $G_j\left(\epsilon \right)=\{x\in G:|x-b_j|\ge \epsilon \}$ for all $j=1,\dots ,r$, $\epsilon >0$. Construct the functions $w_j={\phi }_j(v_j+{\sum }_{b\in K_j}{v}_j^b{D}_j)$, $D_j=\frac{I_j^{*}\left(b\right)e^{{\phi }_j^b\left(b\right)-{\phi }_j\left(b\right)}}{I_j\left(b\right)}$, where the functions ${\phi }_j$ are defined in Lemma 5. The properties of the functions $v_j^b$ imply that ${\phi }_j{\sum }_{b\in K_j}{v}_j^b{D}_j\in W_2^2\left(G\right)$. Lemma 1 imply that

(67)$\begin{array}{c}\sum _{i=1}^r{\widehat{\alpha }}_i{\int }_{\Gamma }{\mathsf{\Phi }}_i\left(x\right)\overline{w_j}d\Gamma ={\int }_{\Gamma }\widehat{w}\overline{(\frac{\partial w_j}{\partial \nu }+{\sigma }^{*}{w}_j)}d\Gamma -2{\int }_G\widehat{w}\nabla {\phi }_j\overline{\nabla w_j}dx\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\int }_G-\widehat{w}\sum _{i=1}^n{a}_i{\phi }_{j{x}_i}\overline{w_j}dx-{\int }_G\widehat{w}\mathsf{\Delta }{\phi }_j\overline{w_j}dx+{\widehat{\tilde{\psi }}}_j=A_j\left(\widehat{w}\right)+{\widehat{\tilde{\psi }}}_j,\end{array}$

where the function $\widehat{w}$ is a solution to the problem (65). Consider the case of $n=3$. The case of $n=2$ is considered analogously. For the integral on the left-hand side, we have

$\begin{array}{c}\hfill {\int }_{\Gamma }{\mathsf{\Phi }}_i\left(x\right)\overline{w_j}d\Gamma ={\int }_{\Gamma }{\mathsf{\Phi }}_i\left(x\right){\phi }_j\overline{(v_j+{\sum }_{b\in K_j}{v}_j^b{D}_j)}d\Gamma =\sum _{b^{\prime }\in K_j}{\int }_{X_{b^{\prime }}}{\mathsf{\Phi }}_i\left(x\right){\phi }_j\overline{(v_j+{\sum }_{b\in K_j}{v}_j^b{D}_j)}d\Gamma .\end{array}$

However, only two summands with $v_j$ and $v_j^{b^{\prime }}$ are essential on the set $X_{b^{\prime }}$. Indeed, in view of (16), for $b\ne b^{\prime }$ and $b\in K_j$, we infer

$|\sqrt{\lambda }{v}_j^b{e}^{{\delta }_j\sqrt{\lambda }}|+|e^{{\delta }_j\sqrt{\lambda }}|\nabla v_j^b||\le c_1{e}^{-q_0\sqrt{|\lambda |}}\forall x\in X_{b^{\prime }},$

where $q_0>0$ is a constant independent of $\lambda$. This inequality implies that the remaining integrals decay exponentially. By Lemma 3, we have

(68)$e^{{\delta }_j\sqrt{\lambda }}\sqrt{\lambda }{\int }_{\Gamma }{\mathsf{\Phi }}_i\left(x\right)\overline{w_j}d\Gamma =\sum _{b^{\prime }\in K_j}\frac{{\mathsf{\Phi }}_i\left(b^{\prime }\right)e^{-{\phi }_j\left(b^{\prime }\right)}}{I_j\left(b^{\prime }\right)}{(1+O(|\lambda |}^{-\beta }\left)\right)={\mathsf{\Psi }}_{ji}{(1+O(|\lambda |}^{-\beta }\left)\right).$

Consider the right-hand side in (67). The integrals over the domain are estimated by means of Lemma 5. On the support of $|\nabla {\phi }_j|$, Theorem 1 and Lemma 5 ensure the estimate

$|e^{{\delta }_j\sqrt{\lambda }}|(|v_j|+|\nabla v_j|+\sum _{b\in K_j}(|v_j^b|+|\nabla v_j^b|)\le c_2{e}^{-{\epsilon }_{13}\sqrt{|\lambda |}},$

where the constants $c_2,{\epsilon }_{13}>0$ are independent of $\lambda$. The Hölder inequality yields

(69)$\begin{array}{c}{e}^{{\delta }_{j}Re\sqrt{\lambda }}|-2{\int }_G\widehat{w}\nabla {\phi }_j\overline{\nabla (v_j+{\sum }_{b\in K_j}{v}_j^b{D}_j)}dx-{\int }_G\widehat{w}\sum _{i=1}^n{a}_i{\phi }_{j{x}_i}\overline{(v_j+{\sum }_{b\in K_j}{v}_j^b{D}_j)}dx\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-{\int }_G\widehat{w}\mathsf{\Delta }{\phi }_j\overline{(v_j+{\sum }_{b\in K_j}{v}_j^b{D}_j)}dx|\le c_3{\parallel \widehat{w}\parallel }_{L_2\left(G\right)}{e}^{-{\epsilon }_{13}\sqrt{|\lambda |}/2}.\end{array}$

Examine the integrals over $\Gamma$ in the right-hand side of $\left(\right)$. We have

(70)$\begin{array}{c}{\int }_{\Gamma }\widehat{w}\overline{(\frac{\partial w_j}{\partial \nu }+{\sigma }^{*}{w}_j)}d\Gamma =\sum _{b\in K_j}({\int }_{X_b}\widehat{w}{\phi }_j\frac{\partial }{\partial \nu }(\overline{v_j+v_j^b{D}_j)}d\Gamma +{\int }_{X_b}\widehat{w}\frac{\partial {\phi }_j}{\partial \nu }\overline{(v_j+v_j^b{D}_j)}d\Gamma +\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\int }_{X_b}\widehat{w}{\sigma }^{*}{\phi }_j\left(\overline{v_j+v_j^b{D}_j}\right)d\Gamma +{\int }_{X_b}\widehat{w}\sum _{b^{\prime }\in K_j,b^{\prime }\ne b}\overline{\frac{\partial \left({\phi }_j{v}_j^b{D}_j\right)}{\partial \nu }}d\Gamma +{\int }_{X_b}\widehat{w}{\sigma }^{*}{\phi }_j\overline{{\sum }_{b^{\prime }\in K_j,b^{\prime }\ne b}{v}_j^b{D}_j}d\Gamma ).\end{array}$

As in the estimate (69), the last two integrals are estimated by

$\begin{array}{c}\hfill \left|{\int }_{X_b}\widehat{w}\sum _{b^{\prime }\in K_j,b^{\prime }\ne b}\overline{\frac{\partial \left({\phi }_j{v}_j^b{D}_j\right)}{\partial \nu }}d\Gamma +{\int }_{X_b}\widehat{w}{\sigma }^{*}{\phi }_j\overline{{\sum }_{b^{\prime }\in K_j,b^{\prime }\ne b}{v}_j^b{D}_j}d\Gamma \right)|\le c_4{\parallel \widehat{w}\parallel }_{L_2(\Gamma )}{e}^{-{\epsilon }_{12}\sqrt{|\lambda |}/2}\end{array}$

in view of (16). Estimate the second and third integrals. In view of Theorem 3 and estimates of Lemma 5 (see (27)), they admit the estimates

$\begin{array}{c}\left|{\int }_{X_b}\widehat{w}\frac{\partial {\phi }_j}{\partial \nu }\overline{(v_j+v_j^b{D}_j)}d\Gamma +{\int }_{X_b}\widehat{w}{\sigma }^{*}{\phi }_j\overline{(v_j+v_j^b{D}_j)}d\Gamma \right|\le \hfill \\ c_5{\parallel w\left(x(z^{\prime },0)\right)\parallel }_{L_{\infty }\left(B_{\eta /2}^{\prime }\left(0\right)\right)}c{e}^{-{\delta }_j\sqrt{Re\lambda }}{\int }_{B_{\eta }^{\prime }\left(0\right)}{e}^{-Re\sqrt{\lambda }(|y-b_j|-{\delta }_j)}+e^{-Re\sqrt{\lambda }(|y-b_j^b|-{\delta }_j)}d{y}^{\prime }\le \\ \hfill c_6{\parallel \widehat{w}\left(x_b\left(z\right)\right)\parallel }_{L_{\infty }\left(B_{\eta /2}^{\prime }\left(0\right)\right)}{e}^{-{\delta }_{j}Re\sqrt{\lambda }}/\sqrt{|\lambda |},\end{array}$

where $x=x_b\left(z\right)$ is the straightening of the boundary in $X_b$. It remains to consider the first integral

(71)$\begin{array}{c}{I}_b={\int }_{X_b}\widehat{w}{\phi }_j\overline{\frac{\partial (v_j+v_j^b{D}_j)}{\partial \nu }}d\Gamma =\widehat{w}{\phi }_j\left(b\right){\int }_{X_b}\overline{\frac{\partial (v_j+v_j^b{D}_j)}{\partial \nu }}d\Gamma +\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\int }_{X_b}(\widehat{w}{\phi }_j\left(x\right)-\widehat{w}{\phi }_j\left(b\right))\overline{\frac{\partial (v_j+v_j^b{D}_j)}{\partial \nu }}d\Gamma .\end{array}$

Note that ${\phi }_j\left(b\right)=1$. Lemma 5 ensures the following representation for the first integral $I_1$ on the right-hand side of (71):

$\begin{array}{c}{I}_1=e^{\sqrt{\lambda }{\delta }_j}\left({\int }_{X_b}\overline{\frac{\partial v_j}{\partial \nu }}d\Gamma +{\int }_{X_b}\overline{\frac{\partial v_j^b{D}_j}{\partial \nu }}d\Gamma \right)=\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\frac{-e^{-{\phi }_j\left(b\right)}}{2I_j\left(b\right)}{(1+O(|\lambda |}^{-1/4}\left)\right)+D_j\frac{e^{-{\phi }_j^b\left(b\right)}}{2I_j^{*}\left(b\right)}{(1+O(|\lambda |}^{-1/4}\left)\right)=O\left(\frac{1}{{|\lambda |}^{1/4}}\right).\end{array}$

The second integral on the right-hand side of (71), in view of Lemma 5, (28) and (32), is estimated as follows:

(72)$\begin{array}{c}\left|{\int }_{X_b}(\widehat{w}{\phi }_j\left(y\right)-\widehat{w}{\phi }_j\left(b\right))\overline{\frac{\partial (v_j+v_j^b{D}_j)}{\partial \nu }}d\Gamma \right|\le \hfill \\ c_1\parallel \widehat{w}\left(x_b\left(z\right)\right){\parallel }_{C^1\left(B_{\eta /2}^{\prime }\left(0\right)\right)}{\int }_{{\tilde{X}}_b}|y^{\prime }|\sqrt{|\lambda |}(e^{-Re\sqrt{\lambda }(|y-b|+{\delta }_j)}+e^{-Re\sqrt{\lambda }(|y-b_j^b|+{\delta }_j)})d{y}^{\prime }\le \\ \hfill c_2\parallel \widehat{w}\left(x_b\left(z\right)\right){\parallel }_{C^1\left(B_{\eta /2}^{\prime }\left(0\right)\right)}{|\lambda |}^{-1/4}.\end{array}$

Thus, in view of (70)–(72), we have the inequality

$|e^{\sqrt{\lambda }{\delta }_j}{A}_j\left(\widehat{w}\right)|\le c_3(\sum _{b\in K_j}(\parallel \widehat{w}\left(x_b\left(z\right)\right){\parallel }_{C^1\left(B_{\eta /2}^{\prime }\left(0\right)\right)}{|\lambda |}^{-1/4}+{(\parallel w\parallel }_{L_2\left(G\right)}+{\parallel w\parallel }_{L_2(\Gamma )}),e^{-{\epsilon }_{14}Re\sqrt{\lambda }})$

with some constant ${\epsilon }_{14}>0$. Next, we employ Lemma 4. The embedding theorems for $n=3$ and Lemma 4 imply that

$\parallel \widehat{w}\left(x_b\left(z\right)\right){\parallel }_{C^1\left(B_{\eta /2}^{\prime }\left(0\right)\right)}\le c\parallel \widehat{w}\left(x_b\left(z\right)\right){\parallel }_{W_2^2\left(B_{\eta /2}^{\prime }\left(0\right)\right)}\le {(\parallel g\parallel }_{W_2^2\left(B_{\eta }^{\prime }\right)}+{\parallel g\parallel }_{L_2(\Gamma )}{)\lambda |}^{-1/2+\epsilon },$

where $\epsilon$ is an arbitrarily small constant. Similarly, Lemma 4 ensures that

$\parallel \widehat{w}{\parallel }_{L_2\left(G\right)}+\parallel \widehat{w}{\parallel }_{L_2(\Gamma )}\le {c|\lambda |}^{-1/2+\epsilon }{\parallel g\parallel }_{L_2(\Gamma )}.$

In view of the conditions on the functions ${\mathsf{\Phi }}_j$, there exists a constant $c_2$ such that

${\parallel g\parallel }_{W_2^2\left(B_{\eta }^{\prime }\right)}+{\parallel g\parallel }_{L_2(\Gamma )}\le c_2|\overrightarrow{\widehat{\alpha }}|.$

Therefore, we have the estimate

(73)$|e^{\sqrt{\lambda }{\delta }_j}{A}_j\left(\widehat{w}\right)|\le c_3|\overrightarrow{\widehat{\alpha }}{|\lambda |}^{-1/4-1/2+\epsilon },$

where $\epsilon >0$ is an arbitrarily small constant. We can rewrite (67) in the form

$e^{\sqrt{\lambda }{\delta }_j}\sqrt{\lambda }\sum _{i=1}^r{\widehat{\alpha }}_i{\int }_{\Gamma }{\mathsf{\Phi }}_i\left(x\right)\overline{v_j}d\Gamma =e^{\sqrt{\lambda }{\delta }_j}\sqrt{\lambda }{A}_j\left(\widehat{w}\right)+{\widehat{\tilde{\psi }}}_j{e}^{\sqrt{\lambda }{\delta }_j}\sqrt{\lambda },j=1,2,\dots ,r.$

The left-hand side of this equality is written as $\tilde{\mathsf{\Psi }}\left(\lambda \right)\overrightarrow{\widehat{\alpha }}$, where the entries of the matrix $\tilde{\mathsf{\Psi }}\left(\lambda \right)$ are of the form ${\mathsf{\Psi }}_{ij}{(1+O(|\lambda |}^{-\beta }\left)\right)$. The right-hand side is written in the form

(74)$A\left(\lambda \right)\overrightarrow{\widehat{\alpha }}=\overrightarrow{\beta }+S_0\left(\overrightarrow{\widehat{\alpha }}\right),\overrightarrow{\widehat{\alpha }}=(\widehat{{\alpha }_1},\dots ,\widehat{{\alpha }_r}),$

where the coordinates of the vectors $\overrightarrow{\beta },S_0\left(\overrightarrow{\widehat{\alpha }}\right)$ are as follows:

${\beta }_j=\sqrt{\lambda }{e}^{{\delta }_j\sqrt{\lambda }}{\widehat{\tilde{\psi }}}_j,S_{0j}=\sqrt{\lambda }{e}^{{\delta }_j\sqrt{\lambda }}{A}_j\left(\widehat{w}\right),j=1,\dots ,r.$

Choose ${\lambda }_1\ge {\lambda }_0$ so that the matrix $A\left(\lambda \right)$ is invertible for $Re\lambda \ge {\lambda }_1\ge {\lambda }_0$ and the norm of the operator $A^{-1}:{\mathbb{R}}^r\to {\mathbb{R}}^r$ is bounded by a constant $c_0$ for all $Re\lambda \ge {\lambda }_1$. It is more convenient to rewrite the system (74) in the form

(75)$\overrightarrow{\widehat{\alpha }}=A{\left(\lambda \right)}^{-1}\overrightarrow{\beta }+A{\left(\lambda \right)}^{-1}{S}_0\left(\overrightarrow{\widehat{\alpha }}\right).$

Estimate the norm of the operator $A{\left(\lambda \right)}^{-1}{S}_0\left(\overrightarrow{\widehat{\alpha }}\right)$. In view of (73), we have the estimate

(76)${|A\left(\lambda \right)}^{-1}{S}_0\left(\overrightarrow{\widehat{\alpha }}\right)|\le c_0\sum _{j=1}^r|S_{0j}|\le c_1|\overrightarrow{\widehat{\alpha }}{||\lambda |}^{-1/4+\epsilon }.$

Thus, for $\epsilon <1/4$, increasing the parameter ${\lambda }_1$ if necessary, we can assume that $c_1{|\lambda |}^{-1/4+\epsilon }\le 1/2\mathrm{for}Re\lambda \ge {\lambda }_1.$ The norm of the operator $A{\left(\lambda \right)}^{-1}{S}_0\left(\overrightarrow{\widehat{\alpha }}\right):{\mathbb{C}}^r\to {\mathbb{C}}^r$ is less than 1/2 in this case and, thereby, the Equation (75) has a unique solution. Constructing a solution $\overrightarrow{\widehat{\alpha }}$ to the Equation (75), we can find a solution $\widehat{w}\in W_2^2\left(G\right)$ to the problem (61), where $Re\lambda \ge {\lambda }_1$. In view of our conditions, the estimates of Lemma 4 holds. In view of the Equation (75), a solution $\overrightarrow{\widehat{\alpha }}$ meets the estimates $|\overrightarrow{\widehat{\alpha }}|\le 2c_0|\overrightarrow{\beta }|.$ Hence, we infer

$\sum _{j=1}^r|{\widehat{\alpha }}_i{|}^2\le c_2\sum _{i=1}^r|\lambda ||e^{2{\delta }_j\sqrt{\lambda }}{|{\widehat{\tilde{\psi }}}_j|}^2,$

where the constant $c_2$ is independent of $\lambda$. The properties of Laplace transform validate the equality ${\widehat{\tilde{\psi }}}_j={\widehat{V}}_{{\delta }_j}\left(\lambda \right){\widehat{\tilde{\psi }}}_{0j}=e^{-\sqrt{\lambda }{\delta }_j}{\widehat{\psi }}_{0j}$ and the previous inequality yields

(77)$\begin{array}{c}\underset{\gamma >\lambda }{sup}{\int }_{-\infty }^{\infty }\sum _{i=1}^r{|\gamma +i\xi |}^{1/2}{|{\widehat{\alpha }}_i(\gamma +i\xi )|}^{2}d\xi \le \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}C\underset{\gamma >\lambda }{sup}{\int }_{-\infty }^{\infty }\sum _{i=1}^r{|\gamma +i\xi |}^{3/2}\parallel {\widehat{\psi }}_{0j}{(\gamma +i\xi )\parallel }^{2}d\xi \le C\sum _{j=1}^r{\parallel e^{-\lambda t}{\psi }_{0j}\parallel }_{W_2^{3/4}(0,\infty )}^2<\infty .\end{array}$

This inequality ensures that the inverse Laplace transform is defined for the functions ${\widehat{\alpha }}_i$, ${\alpha }_i\left(t\right)e^{-\lambda t}\in W_2^{1/4}(0,\infty )$, and

(78)$\sum _{i=1}^r\parallel {\alpha }_i{e}^{-\lambda t}{\parallel }_{W_2^{1/4}(0,T)}^2\le C\sum _{j=1}^r{\parallel e^{-\lambda t}{\psi }_{0j}\parallel }_{W_2^{3/4}(0,\infty )}^2<\infty .$

Note that the additional smoothness of the functions ${\tilde{\psi }}_i$ ensures the additional smoothness of the functions ${\alpha }_i$. Consider the problem (61) with the above constructed functions $\overrightarrow{\alpha }$. By Theorem 3, there exists a unique solution to this problem such that $e^{-\lambda t}w\in W_2^{1,2}\left(Q\right)$. We now demonstrate that this function satisfies (62). Indeed, applying the Laplace transform, we obtain that $\widehat{w}$ is a solution to the problem (65). Multiplying the equation in (65) by $w_j$ and integrating by parts we obtain (67) with $\widehat{w}\left(b_j\right)$ rather than ${\widehat{\tilde{\psi }}}_j$. Since ${\widehat{\alpha }}_j$ satisfy (67) with the functions ${\widehat{\tilde{\psi }}}_j$ on the right-hand side, we obtain ${\widehat{\tilde{\psi }}}_j=\widehat{w}\left(b_j\right)$.

In the case of $n=2$, the arguments are the same. However, in view of another asymtotics of the function ${\widehat{V}}_{{\delta }_j}$ the inequality (78) can be rewritten as

$\sum _{i=1}^r\parallel {\alpha }_i{e}^{-\lambda t}{\parallel }_{W_2^{1/4}(0,T)}^2\le C\sum _{j=1}^r{\parallel e^{-\lambda t}{\psi }_{0j}\parallel }_{{\tilde{W}}_2^{1/2}(0,\infty )}^2<\infty .$

Uniqueness clearly follows from the above arguments. □

If we state our theorem in the case of a finite interval $(0,T)$, then the condition (60) looks as follows:

(79)$u_0\left(x\right)\in W_2^1\left(G\right),f\in L_2\left(Q\right).$

Theorem 5.

Assume that $T<\infty$ and the conditions (5), (15), (58), (59), (79), (63) and (38) for $n=3$ hold. Then there exists a unique solution to the problem (1)–(3)such that $u\in W_2^{1,2}\left(Q\right)$, ${\alpha }_i\left(t\right)\in W_2^{1/4}(0,T)$$(1=1,2,\dots ,r)$.

Proof. 

Extend the functions ${\psi }_{0j}$ on $(0,\infty )$ as compactly supported functions of the same class. The conditions (63) are fulfilled for every $\lambda$. Extend the function f by zero on $(0,\infty )$. Theorem 4 ensures existence of a solution to the problem (1)–(3). Now we prove uniqueness of solutions. Assume that there are two solutions of the problem from the class pointed out in the statement of the theorem. In this case, their difference $v(t,x)\in W_2^{1,2}\left(Q\right)$ is a solution to the problem

$v_t+Lv=0,(t,x)\in Q.$

${Bv|}_S=g(t,x)=\sum _{i=1}^r{\alpha }_i{\mathsf{\Phi }}_i{,v|}_{t=0}=0,v(b_j,t)=0,j=1,2,\dots ,r.$

Integrating the equation and the boundary condition with respect to time two times, we obtain that the function $v_0={\int }_0^t{\int }_0^{\tau }v\left(\xi \right)d\xi d\tau$ is a solution to the problem

(80)$v_{0t}+L{v}_0=0((t,x)\in Q),B{v}_0{|}_S=g_0(t,x)=\sum _{i=1}^r{\alpha }_{0i}{\mathsf{\Phi }}_i,$

(81)$v_0{|}_{t=0}=0,v_0(b_j,t)=0,{\alpha }_{0i}={\int }_0^t{\int }_0^{\tau }{\alpha }_i\left(\xi \right)d\xi d\tau ,j=1,2,\dots ,r.$

Make the change of variables $v_0=e^{\lambda t}w$ ($Re\lambda \ge {\lambda }_0$). We have

(82)$w_t+Lw+\lambda w=0,(t,x)\in Q.$

(83)${Bw|}_S=e^{-\lambda t}{g}_0{(t,x),w|}_{t=0}=0,w(b_j,t)=0,j=1,2,\dots ,r.$

Integrating (82) over $(0,T)$, we obtain that

(84)$L\tilde{w}+\lambda \tilde{w}=-w(T,x),B\tilde{w}{|}_{\Gamma }={\int }_0^T{e}^{-\lambda t}{g}_0(t,x)dt,\tilde{w}\left(b_j\right)=0,\tilde{w}={\int }_0^{T}w(\tau ,x)d\tau .$

Let ${\widehat{\alpha }}_i={\int }_0^T{e}^{-\lambda t}{\alpha }_{0i}\left(t\right)dt$. Make the change of variables $\tilde{w}=w_0+w_1$, with $w_1$ a solution to the problem $(L+\lambda )w_1=-w(T,x)=e^{-\lambda T}{v}_0(T,x)$, $B{w}_1{|}_{\Gamma }=0$, and, respectively, $w_0$ is a solution to the problem

(85)$L{w}_0+\lambda w_0=0,B{w}_0{|}_{\Gamma }=\sum _{i=1}^r{\widehat{\alpha }}_i{\mathsf{\Phi }}_i\left(x\right),w_0\left(b_j\right)=-w_1\left(b_j\right).$

Note that $w(T,x)\in W_2^1\left(G\right)$ and, thereby, $w_1=-{(L+\lambda )}^{-1}w(T,x)\in W_2^2\left(G\right)$. Since $W_2^2\left(G\right)\subset C\left(\overline{G}\right)$ [30], we have the estimate (see Theorem 7.11 for $G={\mathbb{R}}_{+}^n$ and Theorem 8.2 in the case of a domain with compact boundary in [31])

(86)$|w_1\left(b_j\right)|\le c_0{e}^{-Re\lambda T}{\parallel v_0(T,x)\parallel }_{L_2\left(G\right)}\le c_1{e}^{-Re\lambda T}.$

Multiply the Equation (85) by the function $w_j$ defined in the proof of the previous theorem and integrate over G. As in the proof of Theorem 4, we obtain the system (see (75))

(87)$\overrightarrow{\widehat{\alpha }}=A{\left(\lambda \right)}^{-1}\overrightarrow{\beta }+A{\left(\lambda \right)}^{-1}{S}_0\left(\overrightarrow{\widehat{\alpha }}\right),$

where the coordinates of $\overrightarrow{\beta }$ are written as ${\beta }_j=-\sqrt{\lambda }{e}^{\sqrt{\lambda }{\delta }_j}{w}_1\left(b_j\right)$. The system can be rewritten as follows

(88)$\overrightarrow{\widehat{\alpha }}={(I-A{\left(\lambda \right)}^{-1}{S}_0)}^{-1}A{\left(\lambda \right)}^{-1}\overrightarrow{\beta },$

where the right-hand side is analytic for $Re\lambda \ge {\lambda }_1$ and we have

$\parallel {(I-A{\left(\lambda \right)}^{-1}{S}_0)}^{-1}A{\left(\lambda \right)}^{-1}\overrightarrow{\beta }{\parallel }_{{\mathbb{C}}^r}\le c_2{\parallel \overrightarrow{\beta }\parallel }_{{\mathbb{C}}^r},$

where $c_2$ is independent of $\lambda$. Thus, every of the quantities ${\widehat{\alpha }}_i$ is estimated by

(89)$|{\widehat{\alpha }}_i|\le c_3\sum _{j=1}^r\sqrt{\lambda }{e}^{\sqrt{\lambda }{\delta }_j}{e}^{-Re\lambda T},Re\lambda \ge {\lambda }_1.$

The function $S_i\left(z\right)={\int }_0^T{\alpha }_{0i}\left(t\right)e^{-{\lambda }_{1}t}{e}^{-zt}dt$ is the Laplace transform of the function ${\tilde{s}}_i\left(t\right)={\alpha }_{0i}\left(t\right)e^{-{\lambda }_{1}t}$ for $t\le T$ and ${\tilde{s}}_i\left(t\right)=0$ for $t>T$. Fix $\epsilon >0$ and define an additional function $W\left(z\right)=z{e}^{z(T-\epsilon )}{S}_i\left(z\right)$. It is analytic in the right half-plane and is bounded by some constant $C_1$ on the real semi-axis ${\mathbb{R}}^{+}$. To estimate this function on the on the imaginary axis, we integrate by parts as follows:

$S_i\left(z\right)=\frac{-1}{{\lambda }_1+z}({\alpha }_{0i}\left(T\right)e^{-{\lambda }_{1}T}{e}^{-zT}+{\int }_0^T{\alpha }_{0i}^{\prime }\left(t\right)e^{-{\lambda }_{1}t}{e}^{-zt}dt).$

For $z=iy$, we thus have the estimate

$|W\left(z\right)|\le c_4(|{\alpha }_{0i}\left(T\right)|+\parallel {\alpha }_{0i}^{\prime }{\parallel }_{L_1(0,T)})=c_5\forall z=iy,y\in \mathbb{R}.$

In each of the sectors $0\le argz\le \pi /2$, $-\pi /2\le argz\le 0$ the function $W\left(z\right)$ admits the estimate

$|W\left(z\right)|\le e^{|z|(T-\epsilon )}{c}_6(|{\alpha }_{0i}\left(T\right)|+\parallel {\alpha }_{0i}^{\prime }{\parallel }_{L_1(0,T)})\forall Rez\ge 0.$

Applying the Fragment-Lindelef Theorem (see Theorem 5.6.1 in [43]) we obtain that in each of the sectors $0\le argz\le \pi /2$, $-\pi /2\le argz\le 0$ the function $W\left(z\right)$ admits the estimate

$|W\left(z\right)|\le max(C_1,c_5)=C_2\forall Rez\ge 0.$

Therefore, $|S_i\left(z\right)|=|L\left({\tilde{s}}_i\left(t\right)\right)\left(z\right)|\le C_2{e}^{-(T-\epsilon )Rez}/|z|\forall Rez\ge 0.$ We have equality ($\sigma \ge {\lambda }_1,p=\sigma +i\xi$)

${\tilde{s}}_j\left(t\right)=\frac{1}{2\pi i}{\int }_{\sigma -i\infty }^{\sigma +i\infty }{e}^{pt}L\left({\tilde{s}}_j\right)\left(p\right)dp=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{e}^{\sigma t}{e}^{i\xi t}L\left({\tilde{s}}_j\right)(\sigma +i\xi )d\xi .$

and, thereby,

${\tilde{s}}_j\left(t\right)e^{-\sigma (t-(T-\epsilon \left)\right)}=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{e}^{i\xi t}{e}^{\sigma (T-\epsilon )}L\left({\tilde{s}}_j\right)(\sigma +i\xi )d\xi .$

The Parseval identity yields

$\parallel {\tilde{s}}_j\left(t\right)e^{-\sigma (t-(T-\epsilon \left)\right)}{\parallel }_{L_2\left(\mathbb{R}\right)}^2=\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}{e}^{2\sigma (T-\epsilon )}{|L\left({\tilde{s}}_i\right)(\sigma +i\xi )|}^{2}d\xi \le \frac{C_2^2}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}\frac{1}{{\sigma }^2+{\xi }^2}d\xi \le \frac{C_2^2}{2\sigma }.$

Since this inequality is true for all $\sigma >{\lambda }_1$, ${\tilde{s}}_i\left(t\right)=0$ for $t\le T-\epsilon$. Since the parameter $\epsilon$ is arbitrary, ${\alpha }_{0j}\left(t\right)=0$ for $t\le T$ and ${\alpha }_j\left(t\right)=0$ for $t\le T$ and every j and, therefore, $g(t,x)=0$ which implies that $v=0$. □

4. Discussion

We consider inverse problems of recovering surface fluxes on the boundary of a domain from pointwise observations. These problems arise in many practical applications, but there are no theoretical results concerning the existence and uniqueness questions. The problems are ill-posed in the Hadamard sense. The results can be used in developing new numerical algorithms and provide new conditions of uniqueness of solutions to these problems. We consider a model case, but it is clear what changes should be made in the general case for validating similar results. The main conditions on the data are conventional. The only distinction is the conditions on the data of measurements in the reduced problem which must belong to some special class of infinitely differentiable functions. The proof relies on an asymptotics of fundamental solutions to the corresponding elliptic problems and the Laplace transform.

Author Contributions

Investigation, S.P. and D.S. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

No new data were created or analyzed in this study.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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