Harnack Inequalities and ABP Estimates for Nonlinear Second-Order Elliptic Equations in Unbounded Domains

Abstract

We are concerned with fully nonlinear uniformly elliptic operators with a superlinear gradient term. We look for local estimates, such as weak Harnack inequality and local maximum principle, and their extension up to the boundary. As applications, we deduce ABP-type estimates and weak maximum principles in general unbounded domains, a strong maximum principle, and a Liouville-type theorem.

(ProQuest: ... denotes non-US-ASCII text omitted.)

M. E. Amendola 1 and L. Rossi 2 and A. Vitolo 1

Recommended by Y. Giga

1, Dipartimento di Matematica e Informatica, Università di Salerno, P. Grahamstown, 84084 Fisciano (SA), Italy
2, Dipartimento di Matematica, Università di Roma "La Sapienza", P. le A. Moro 2, 00185 Roma, Italy

Received 17 July 2007; Revised 31 January 2008; Accepted 4 April 2008

1. Introduction

The qualitative theory of second-order elliptic equations received a strong effort from Harnack inequalities. Here, we will make use of this powerful technique to study continuous viscosity solutions u of fully nonlinear elliptic equations (F=f) : [figure omitted; refer to PDF] in unbounded domains Ω of ^Rn , where F is a real function of x∈Ω , t∈R , p∈ ^Rn and X... in the set ^...AE;n of n×n real symmetric matrices.

We recall that F is (degenerate) elliptic if F is nondecreasing in X and uniformly elliptic if there exist (ellipticity) constants λ and Λ such that 0<λ≤Λ and [figure omitted; refer to PDF] for Y≥0 , that is Y is semidefinite positive, where tr(Y) denotes the trace of the matrix Y .

In the class of uniformly elliptic operators, there are two extremal ones, well known as Pucci maximal and minimal operators, respectively: [figure omitted; refer to PDF] where ^X± are the positive and negative parts of X , which can be decomposed in a unique way as X=^X+ -^X- with ^X± ≥0 and ^X+X- =0 . Other examples of fully nonlinear uniformly elliptic operators can be found in [1-3].

Throughout this paper, we will consider elliptic operators with the structure conditions [figure omitted; refer to PDF] [figure omitted; refer to PDF] where ^P± are the extremal Pucci operators, b(x) is a continuous function and the exponent q∈[1,2] , so that the gradient term can have a superlinear, at most quadratic growth.

Remark 1.1.

The above structure conditions are exactly equivalent to the uniform ellipticity when F is linear in the variable X∈^...AE;n . In the nonlinear case they allow a slight generalization. Let us consider, for 0<λ<Λ and t≥0 , the function [figure omitted; refer to PDF] then the operator F=h(tr(^X+ ))-h(tr(^X- )) is elliptic and satisfies both the conditions (1.4) and (1.5), even that it is not uniformly elliptic.

However, if (1.5) (resp., (1.4)) holds, then subsolutions (resp., supersolutions) of the equation F=f are subsolutions (resp., supersolutions) of uniformly elliptic equations, and this is needed to prove our results.

We will be concerned principally with the following topics in unbounded domains; see [4-6] for classical results.

[MP] maximum principle for u.s.c. subsolutions w of F=0 in the viscosity sense (v.s.), in the form [figure omitted; refer to PDF]

[LT] Liouville theorem for continuous solutions of F=0 v.s., in the form [figure omitted; refer to PDF] Concerning MP, it is worth to note that the condition from above on the size of w can be weakened in the framework of the Phragmén-Lindelöf theory (see, e.g., [7-9]) but not omitted at all, even for classical subsolutions (see, e.g., [4, 10]). It is also well known that MP fails to hold in general in exterior domains. In fact, due to the boundedness of the fundamental solution u(x)=|x^|2-n of the Laplace equation Δu=0 , the function w=1-u provides a counterexample to MP in Ω=^Rn \_B¯1 (0) . Thus we introduce a local measure-geometric condition _Gσ in Ω at y∈^Rn , which depends on the real parameter σ∈(0,1) : there exists a ball B=B(y) such that [figure omitted; refer to PDF] where _Ωy is the connected component of B\∂Ω containing y .

If _Gσ is satisfied in Ω at all y∈Ω , we simply say that Ω is a wG domain (with parameter σ ). This is a generalization of condition G of Cabré [10], which ultimately goes back to Berestycki et al. [11].

Let R(y) denote the radius of the ball B=B(y) provided by condition wG . We will call domains of cylindrical and conical type the wG domains such that R(y)=O(1) and R(y)=O(|y|) as |y| [arrow right] +∞ , respectively. Examples of the first kind are domains with finite measure, cylinders, slabs, complements of a periodic lattice of balls, whereas cones, and complements, in the plane, of logarithmic spirals, are examples of the second kind.

In [12], it is shown that MP holds true in a wG domain for strong solutions of a linear second-order uniformly elliptic operator F=trA(x)X ; see also [13, 14] for earlier results and [15, 16] for viscosity solutions of a fully nonlinear operator with linear and quadratic growth in the gradient (i.e., in the case of the structure condition (1.5) with q=1 and q=2 ) provided that b(x)=O(1/|x|) and b(x)=O(1) as |x| [arrow right] ∞ , respectively.

With the aim to find conditions on the coefficient b(x) such that MP holds in wG domains when 1≤q≤2 , our result is the following.

Theorem 1.2 (MP).

Let 0<σ<1 and 1≤q≤2 . Let Ω be a domain of ^Rn satisfying condition wG or alternatively such that, for a closed subset H of ^Rn ,

(i) MP holds in each connected component of Ω\H ;

(ii) condition _Gσ is satisfied in Ω at each y∈Ω∩H .

Suppose that w∈USC(Ω¯) is a viscosity solution of F(x,w,Dw,^D2 w)≥0 and structure condition (1.5) holds with b∈C(Ω¯) , such that b(x)=O(1/|x^|2-q ) .

If w≤0 on ∂Ω and sup...w<+∞ in Ω , then w≤0 in Ω .

This yields indeed MP in a wider class of domains than wG , for example, the cut plane and more generally the complement of continuous semi-infinite curves in ^R2 and their generalizations to hypersurfaces in ^Rn .

We also outline that the limit cases q=1 and q=2 of the above mentioned papers are obtained by continuity from the intermediate cases 1<q<2 , as it follows from Theorem 1.2. Nonetheless, there are technical improvements with respect to the previous works even in the limit cases.

Consider in particular a parabolic shaped domain Ω , satisfying condition wG with R(y)=O(|y^|α ) , 0<α<1 ; the limit cases α=0 and α=1 correspond to domains of cylindrical and conical types, respectively.

Based on an argument of [12], eventually passing to a smaller _ry ≤R(y) , we can suppose that condition _Gσ is satisfied with |B\_Ωy |=σ|B| exactly. We get the new following variant of ABP estimate.

Theorem 1.3 (ABP).

Let 0<σ, τ<1 , ^{τ[variant prime]} >1 , _R0 ,β≥0 , 1≤q≤2, and N>0 . Let Ω be a wG domain, such that condition _Gσ in Ω is fulfilled at each y∈Ω with R(y)≤_R0 +β|y^|α , 0≤α≤1 . Assume that F satisfies the structure condition (1.5), with b,f∈C(Ω¯) and _b0 :=_sup...Ω |b(x)|(1+|x^|α(2-q) )<+∞ .

If w∈USC(Ω¯) is a viscosity solution of F(x,w,Dw,^D2 w)≥f such that w≤N in Ω and w≤0 on ∂Ω , then [figure omitted; refer to PDF] where C is a positive constant depending on n , q , λ , Λ , _b0^Nq-1 , σ , τ , ^{τ[variant prime]} , _R0 , β .

Note that in the case of a domain of cylindrical type ( α=0 ), it is sufficient to have b(x)=O(1) , for all q∈[1,2] , as well as in the case of a quadratic growth in the gradient variable ( q=2 ), for all α∈[0,1] .

This result extends the previous ones contained in [10, 14] for the linear case, and [8, 16], dealing with fully nonlinear equations, in the limit situations of cylindrical/conical domains and linear/quadratic gradient terms.

Remark 1.4.

In general, unless q=1 , the above ABP type estimate is different from the so-called ABP maximum principles since C depends on the upper bound N of w if _b0 >0 and q>1 . For ABP-type estimates of this kind in bounded domains we refer to [17]. Counterexamples to the ABP maximum principle can been found in [17-19].

Consider now Ω=^Rn . The classical Liouville theorem says that harmonic functions in the entire ^Rn , which are bounded either above or below, are constant. This result continues to hold for strong solutions of quasilinear uniformly elliptic equations; see [20]. For viscosity solutions of fully nonlinear uniformly elliptic equations with an additive gradient term having linear growth, we refer to [21, 22]. Our result is the following.

Theorem 1.5 (Liouville theorem).

Let w∈C(^Rn ) be such that F(x,w,Dw,^D2 w)=0 in the viscosity sense, and assume structure conditions (1.4) and (1.5), with b∈C(^Rn ) such that b(x)=O(1/|x^|2-q ) as |x| [arrow right] +∞ . If w is bounded either above or below, then w is constant.

Remark 1.6.

Under some additional assumptions, Liouville-type results also hold in un-bounded domains of ^Rn containing balls of arbitrary large radius; see [23].

Our main tools are Krylov-Safonov Harnack inequalities and local MP; see [20] for strong solutions of quasilinear uniformly elliptic equations. For viscosity solutions and F satisfying the structure condition (1.4), they can be found in [3] if b=0 and in [24] if q=2 ; see also [25]. In the case of linear or superlinear, almost quadratic, growth in the gradient ( 1≤q<2 ), weak Harnack (wH) inequality and local MP can be deduced using arguments of [17], in which a (full) Harnack inequality has been established for ^Lp viscosity solutions; see also [26].

Nevertheless, for convenience of the reader we believed that it is worth to report systematically on this kind of inequalities in Section 3.

As the previous ones, our approach follows the lines of [3], based on the methods of [27, 28] and on the ABP maximum principle for viscosity solutions in bounded domains, due to Caffarelli [29].

Remark 1.7.

In deriving wH inequality and local MP, we only need the Alexandroff-Bakelman-Pucci (ABP) estimate with q=1 and f continuous, so [30, Proposition 2.12] and also [17, Theorem 4.1] in the case of linear growth in the gradient term, are sufficient to our purpose. But we notice that new ABP-type estimates have been established for ^Lp -viscosity solutions of equations with discontinuous coefficients by Koike and Swiech [19, 27] for q∈[1,2] and f∈^Lp .

Remark 1.8.

In the case of a superlinear first-order term, wH inequality and local MP are obtained by interpolation between the linear and quadratic cases, eliminating the square gradient term by means of an exponential transformation used before by Trudinger [24], see Lemmas 3.1 and 3.2 below. This kind of ideas have been also considered by Sirakov in [31].

What we definitely need are, for MP, the scaled boundary wH inequality (3.16), derived in Section 3 by means of typical viscosity methods, and, for technical reasons, its version in annular regions (3.24), and, for LT, the scaled Harnack inequality (3.11). Moreover, using the interior wH inequality (3.7) and assuming the structure condition (1.5), we also state a strong MP theorem, according to which a subsolution u of equation F=0 cannot achieve a positive maximum inside any domain (open connected set) of ^Rn unless it is constant; see Theorem 5.1 below. For a different approach, based on Hopf lemma, and more general versions see [32].

The paper is organized as follows. In Section 2, we recall some basic results of elliptic theory for viscosity solutions of second-order fully nonlinear equations with a linear gradient term; in Section 3, we extend local maximum principle and weak Harnack inequality, even up to the boundary, to the case of a superlinear gradient term; these results are applied in Section 4 to get Alexandroff-Bakelman-Pucci-type estimates and maximum principles, with the proof of Theorems 1.2 and 1.3; finally, a strong maximum principle is derived and the proof of Liouville theorem (Theorem 1.5) is given in Section 5. In the appendix, for the sake of completeness, we show the basic weak Harnack inequality and local MP for a uniformly elliptic operator with an additive first-order term having linear growth in the gradient.

2. Basic Estimates (Linear Gradient Term)

Let Ω be a domain of ^Rn , and denote by USC(Ω) and LSC(Ω) , respectively, the sets of the upper and lower semicontinuous functions in Ω . The function u∈USC(Ω) is said to be a viscosity subsolution of F=f if [figure omitted; refer to PDF] at any point x∈Ω and for all [straight phi]∈^C2 (Ω) such that [straight phi]-u has a local minimum in x . Similarly, a viscosity supersolution u∈LSC(Ω) of F=f satisfies [figure omitted; refer to PDF] at any point x∈Ω and for all [straight phi]∈^C2 (Ω) such that u-[straight phi] has a local minimum in x .

We may also assume that [straight phi](x)=u(x) in the above definition, that is the graph of the test function [straight phi] touches that one of u from above for subsolutions and from below for supersolutions [3]. Moreover, if F is continuous in the matrix-variable, as for uniformly elliptic operators, then we may assume that [straight phi](x) is a paraboloid, that is a quadratic polynomial.

We will make use of the following version of the ABP estimate, in which ^Γu+ denotes the upper contact set [figure omitted; refer to PDF] of the graph of the function u . Using [30, Proposition 2.12] or [17, Theorem 4.1], we have the following.

Lemma 2.1 (ABP estimate).

Let u∈LSC(B¯) be a viscosity supersolution of the equation [figure omitted; refer to PDF] in a ball B of unit radius, such that u≥0 on ∂B , where f∈^Ln (B)∩C(B) , for some constant _b0 ≥0 . Then [figure omitted; refer to PDF] for a positive constant C=C(n,λ,Λ,_b0 ) . Similarly, if u∈USC(B¯) is a viscosity subsolution of the equation [figure omitted; refer to PDF] such that u≤0 on ∂B , then [figure omitted; refer to PDF]

From Lemma 2.1, we obtain the following results, see the appendix, which extend [3, Theorem 4.8, (1) and (2)]; see also [15].

Here we denote by _Br a ball centered at _x0 ∈^Rn of radius r>0 .

Lemma 2.2 (wH inequality).

Let _b0 ≥0 and 0<τ<1 . Suppose that u∈LSC(_B1/τ ) is a viscosity supersolution of (2.4), with f∈C(_B¯1/τ ) , and u≥0 in _B1/τ . Then [figure omitted; refer to PDF] where C and _p0 are positive numbers, depending on n,λ,Λ,_b0 , and τ .

Lemma 2.3 (local MP).

Let _b0 ≥0 and 0<τ<1 . Suppose that u∈USC(_B1 ) is a viscosity subsolution of (2.6) with f∈C(_B¯1 ) . Then for all p>0, [figure omitted; refer to PDF] where C is a positive constant, depending on n,λ,Λ,_b0 ,τ, and p .

3. Interior and Boundary Harnack Estimates and Local MP (Superlinear Gradient Term)

Firstly, we extend interior estimates (2.8) and (2.9) to fully nonlinear operators F with a superlinear first-order term, such that, respectively, (1.4) and (1.5) hold.

Lemma 3.1 (wH inequality).

Let _b0 ≥0 , 0<τ<1, and 1≤q≤2 . Suppose that u∈LSC(_B1/τ ) is a viscosity solution of F(x,u,Du,^D2 u)≤f , under structure condition (1.4) with b≤_b0 , f∈C(_B¯1/τ ) , and 0≤u≤1 in _B1/τ . Then (2.8) holds with positive constants C and _p0 , depending on n,λ,Λ,_b0 ,τ, and q .

Lemma 3.2 (local MP).

Let _b0 ≥0 , 0<τ<1, and 1≤q≤2 . Suppose that u∈USC(_B1 ) is a viscosity solution of F(x,u,Du,^D2 u)≥f , under structure condition (1.5), with b≤_b0 , f∈C(_B¯1 ) , and u≤1 . Then (2.9) holds for all p>0 with a positive constant C , depending on n,λ,Λ,_b0 ,q,τ, and p .

Proof of Lemmas 3.1 and 3.2. .

We only show the proof of Lemma 3.2, since that one of Lemma 3.1 is similar. By the structure condition (1.5), we have [figure omitted; refer to PDF] and also, in the viscosity sense, [figure omitted; refer to PDF] From this, by Young's inequality, it follows that [figure omitted; refer to PDF] with [figure omitted; refer to PDF] Using the transformation ^u+ =(λ/_b2 ) log...(1+(_b2 /λ)v) , then the USC function v=(λ/_b2 ) (exp...((_b2 /λ)^u+ )-1) satisfies the differential inequality [figure omitted; refer to PDF] in _B1/τ . Therefore, we can apply Lemma 2.3 to the subsolution v . To conclude the proof of Lemma 3.2, it is sufficient to observe that [figure omitted; refer to PDF]

Rescaling variables and functions, we highlight the dependence on geometric parameters.

Theorem 3.3 (scaled wH inequality).

Let _b0 ≥0 , 0<τ<1 , N>0, and 1≤q≤2 . Suppose that u∈LSC(_BR/τ ) is a viscosity solution of F(x,u,Du,^D2 u)≤f , under structure condition (1.4), with b≤_b0 , f∈C(_B¯R/τ ) , and 0≤u≤N in _BR/τ . Then [figure omitted; refer to PDF] with positive constants C and _p0 , depending on n,λ,Λ,q,τ, and _b0^Nq-1R2-q .

Proof.

Considering, for y∈_B1/τ , the function v(y) , defined by u(x)=Nv(x/R) , we have [figure omitted; refer to PDF] Thus, applying Lemma 3.1, we get [figure omitted; refer to PDF] with C=C(n,λ,Λ,q,τ,_b0^Nq-1R2-q ) , from which the assert follows.

Note that constants _p0 and C of the above wH inequality depend in general on the upper bound N for the supersolution and on the radius R of the ball, but in the case q=1 there is no dependence on N and in the case q=2 no dependence on R .

In the same manner as in Theorem 3.3 for wH inequality, we make the dependence on the geometric constants explicit in the following local MP.

Theorem 3.4 (scaled local MP).

Let _b0 ≥0 , 0<τ<1 , N>0, and 1≤q≤2 . Suppose that u∈USC(_BR ) is a viscosity solution of F(x,u,Du,^D2 u)≥f , under structure condition (1.5), with b≤_b0 , f∈C(_B¯R ) , and u≤N . Then for all p>0, [figure omitted; refer to PDF] with a positive constant C , depending on n,λ,Λ,q,τ,_b0^Nq-1R2-q , and p .

Combining Theorems 3.3 and 3.4, we get the full Harnack inequality for solutions.

Theorem 3.5 (Harnack inequality).

Let _b0 ≥0 , 0<τ<1 , N>0, and 1≤q≤2 . Suppose that u∈C(_BR/τ ) is a viscosity solution of F(x,u,Du,^D2 u)=f in _BR/τ , under structure conditions (1.4) and (1.5), with b≤_b0 , f∈C(_B¯R/τ ) , and 0≤u≤N . Then [figure omitted; refer to PDF] with a positive constant C=C(n,λ,Λ,q,τ,_b0^Nq-1R2-q ) .

We wish to extend the above estimates up to the boundary, that is, to balls intersecting the boundary of the domain A⊂^Rn , where the solutions are defined. For this purpose we will need suitable extensions of such solutions outside A . Precisely, take concentric balls _BτR ⊂_BR ⊂_BR/τ such that _BτR ∩A ≠ ∅ and _BR/τ \A ≠ ∅ . For a nonnegative viscosity supersolution u∈LSC(A¯) of equation F(x,u,Du,^D2 u)=f in A , we put [figure omitted; refer to PDF] where 0<τ<1 . Similarly, for a viscosity subsolution u∈USC(A¯) , we put [figure omitted; refer to PDF] Denote also by ^f0+ and ^f0- the continuations of ^f+ and ^f- vanishing outside A , respectively. Following [3, Proposition 2.8] and using the structure conditions (1.4) and (1.5), we have [figure omitted; refer to PDF] in _BR/τ for a viscosity supersolution u∈LSC(A¯) , and [figure omitted; refer to PDF] in _BR for a viscosity subsolution u∈USC(A¯) .

Observe that, if ^f+ =0 on ∂A , then ^f0+ is continuous, and then we can apply Theorem 3.3 to get a boundary wH inequality. Similarly, if ^f- =0 on ∂A , we can use Theorem 3.4 to deduce a boundary local MP.

Nevertheless, even in the general case, when ^f0+ and ^f0- are not necessarily continuous, we can get boundary estimates by means of an approximation process, as shown here below, where we use the notations defined just above.

Theorem 3.6 (boundary wH inequality).

Let _b0 ≥0 , 0<τ<1 , N>0, and 1≤q≤2 . Suppose that u∈LSC(A¯) is a viscosity solution of F(x,u,Du,^D2 u)≤f , under structure condition (1.4), with b(x)≤_b0 , f∈C(A¯) , and 0≤u≤N in A . Then [figure omitted; refer to PDF] with positive constants C and _p0 , depending on n,λ,Λ,q,τ, and _b0^Nq-1R2-q .

Proof.

For [straight epsilon]>0, set [figure omitted; refer to PDF] and, for x∈_B¯R/τ , [figure omitted; refer to PDF] where [figure omitted; refer to PDF] It is easy to check that ^{u_{m[straight epsilon]} -} ∈LSC(_BR/τ ) , 0≤^{u_{m[straight epsilon]} -} ≤N , _{f[straight epsilon]} ∈C(_B¯R/τ ), and [figure omitted; refer to PDF] in _BR/τ . Therefore, we can apply Theorem 3.3 with ^{u_{m[straight epsilon]} -} instead of u and _{f[straight epsilon]} instead of f to get [figure omitted; refer to PDF] Note that _inf...BR ^{u_{m[straight epsilon]} -} ≤_{inf...BR ∩A} u and 0≤_{f[straight epsilon]} ≤^f+ in A , _{f[straight epsilon]} =0 outside A . Also observing that, by lower semicontinuity, [figure omitted; refer to PDF] and therefore, by Fatou's lemma, [figure omitted; refer to PDF] from inequality (3.21) we get the assert.

In the sequel, we will make also use of a version of boundary wH inequality for annular regions _{B[straight epsilon]R,R} =_BR \_{B[straight epsilon]R} (0) , 0<[straight epsilon]<1 , which can be deduced by Theorem 3.6 reasoning as in [10, Theorem 3.1].

In this case m=_{inf...∂A∩B[straight epsilon]τR,^{τ[variant prime]} R} u , where 0<[straight epsilon]≤1/2 , 0<τ<1 , ^{τ[variant prime]} >1 .

Corollary 3.7 (boundary wH inequality).

Let 0<τ<1 , ^{τ[variant prime]} >1 , N>0, and 1≤q≤2 . Suppose that u∈LSC(A¯) is a viscosity solution of F(x,u,Du,^D2 u)≤f , under structure condition (1.4), with f∈C(A¯) , and 0≤u≤N in A . Then [figure omitted; refer to PDF] with positive constants C and _p0 , depending on n,λ,Λ,q,τ,^{τ[variant prime]} , and ^Nq-1R2-q ||b_{||^L∞ (A∩B[straight epsilon]τR,R/τ )} .

In a similar manner, we extend the local MP up to the boundary.

Theorem 3.8 (boundary local MP).

Let _b0 ≥0 , 0<τ<1 , N>0, and 1≤q≤2 . Suppose that u∈USC(A¯) is a viscosity solution of F(x,u,Du,^D2 u)≥f , under structure condition (1.5), with f∈C(A¯) and u≤N in A . Then for all p>0, [figure omitted; refer to PDF] with a positive constant C , depending on n,λ,Λ,q,τ,_b0^Nq-1R2-q , and p .

4. ABP-Type Estimates and Maximum Principles

Here we use boundary estimates of previous section to obtain MP in unbounded domains Ω of ^Rn for viscosity subsolutions u∈USC(Ω¯) , bounded above, of equation F(x,u,Du,^D2 u)=0 under structure condition (1.5).

We will make use of the measure-geometric condition _Gσ , 0<σ<1 , given in the introduction. By a continuity argument, see [12], eventually passing to a smaller R , which we will call _ry , we can assume that condition _Gσ is satisfied with |B\_Ωy |=σ|B| exactly.

We also recall that Ω is a wG domain (with parameter σ ) if each point y∈Ω satisfies condition _Gσ in Ω . In particular, if R(y) is the radius of the ball B=B(y) provided by condition _Gσ , we define domains of cylindrical and conical type as wG domains such that R(y)=O(1) and R(y)=O(|y|) , respectively as |y| [arrow right] ∞ .

4.1. Domains of Cylindrical Type

We start with the condition G of Cabré [10]. Let σ<1 , τ<1, and _R0 be positive real numbers. We say that an open connected set Ω of ^Rn is a G domain if to each y∈Ω we can associate a ball B=_BR (_xy ) of radius R≤_R0 such that [figure omitted; refer to PDF] where _Ωy is the connected component of Ω∩B containing y .

Since _Gσ ≡_Gσ,1 , then a G domain of ^Rn is of cylindrical type, like domains of finite Lebesgue n -dimensional measure, subdomains of ω×^Rn-k , where ω has finite Lebesgue k -dimensional measure, the complement of the spiral of equation r=θ in polar coordinates of ^R2 .

Given a differential operator with structure conditions, like (1.4) and (1.5), Ω will be called a narrow domain when, for given τ and _R0 , condition _Gσ,τ is satisfied for σ suitably close to 1 , depending on the structure constants and the remaining geometric constants.

A straightforward application of Theorem 3.8 yields MP in narrow domains. Indeed, assume that u≤N and F(x,u,Du,^D2 u)≥0 in Ω . Then, by (1.5), we have [figure omitted; refer to PDF] Suppose that u≤0 on ∂Ω and set M=_sup...Ω^u+ . Applying Theorem 3.8 in A=_Ωy with p=1 , we obtain [figure omitted; refer to PDF] From this, taking the supremum over y∈Ω , we get M≤0 , that is u≤0 in Ω , provided σ>1-1/C , and hence MP holds in this case.

In order to pass from narrow domains to arbitrary cylindrical domains we will use Theorem 3.6, from which the following ABP-type estimate follows.

Theorem 4.1 (ABP estimate).

Let σ, τ<1 , let _R0 and N be positive real numbers, and 1≤q≤2 . Let Ω be a cylindrical domain such that condition _Gσ in Ω is satisfied at each y∈Ω with R(y)≤_R0 .

Suppose that w∈USC(Ω¯) is a viscosity solution of F(x,w,Dw,^D2 w)≥f , under the structure condition (1.5), with b≤_b0 and f∈C(Ω¯) .

If w≤N in Ω and w≤0 on ∂Ω , then [figure omitted; refer to PDF] where C depends on n,λ,Λ,σ,τ, and _b0^R02-qNq-1 .

Proof.

It is enough to show the result for τ [arrow right] ^1- .

Set M=_sup...Ω^w+ and u=M-w . Let y∈Ω and B=_BR of radius R , provided by condition _Gσ in y . We choose R=_ry such that |B\_Ωy |=σ|B|; see the beginning of this section. We also denote by _BτR the concentric ball of radius τR .

Now we apply Theorem 3.6 to u in A=_Ωy with _BτR (_xy ) instead of _BR and τ close enough to 1 in such a way that |_BτR (_xy )\_Ωy |≥(σ/2)|B| and |_BτR (_xy )∩_Ωy |≥((1-σ)/2)|B| . Since w≤0 on ∂Ω , then m≥M , hence we get [figure omitted; refer to PDF] from which, for x∈_BτR (_xy )∩_Ωy , we obtain the pointwise inequality [figure omitted; refer to PDF] with 0<t<1 . On the other hand, setting K=tM+R||f_{||^Ln (B∩Ω)} and _ΩK ={x∈Ω /w(x)>K} , by virtue of (4.6) we have B\_ΩK ⊃_BτR (_xy )∩_Ωy and therefore, by our choice of R and τ , [figure omitted; refer to PDF] A further application of Theorem 3.6 to u=M-w in A=_ΩK yields [figure omitted; refer to PDF] since in this case m≤M-K . From this we deduce that, for x∈B∩_ΩK [figure omitted; refer to PDF] with 0<^{t[variant prime]} <1 . From the definition of _ΩK , see also (4.6), it follows that [figure omitted; refer to PDF] with 0<^{t[variant prime][variant prime]} <1 , for all x∈Ω∩B and hence also for x=y .

Finally, passing to the supremum over y∈Ω , we get the result.

4.2. General Domains

Firstly, we consider wG domains Ω , such that condition _Gσ in Ω holds at each y∈Ω without bounds for the radii R(y) of the balls provided by _Gσ .

Note that in general the ABP-type estimate of Theorem 4.1 is useless unless _b0 =0 , see [13], since the the constant C of ABP estimate blows up when R [arrow right] +∞ . This is why we assume b(x)=O(1/|x^|2-q ) as |x| [arrow right] +∞ in the structure condition (1.5). Moreover, to take advantage from the decay of b(x) , it is convenient to use the boundary wH inequality for annular regions of Corollary 3.7 rather than Theorem 3.6.

Reasoning as in the proof of Theorem 4.1, but quite more carefully with the aid of (3.24) instead of (3.16), see [16], we get the following ABP-type estimate.

Theorem 4.2 (ABP).

Let σ and N be positive real numbers and 1≤q≤2 . Let Ω be a wG domain (with parameter σ<1 ).

Suppose that w∈USC(Ω¯) is a viscosity solution of F(x,w,Dw,^D2 w)≥f , under the structure condition (1.5), with b,f∈C(Ω¯) such that [figure omitted; refer to PDF] for all [straight epsilon]>0 small enough, all τ<1 sufficiently close to 1, and some ^{τ[variant prime]} >1 .

If w≤N in Ω and w≤0 on ∂Ω , then [figure omitted; refer to PDF] for possibly smaller [straight epsilon]>0 and larger τ<1 , depending on n and σ .

Here C and _Cy are positive constants depending on n , q , λ , Λ , _bq^Nq-1 , σ , [straight epsilon] , τ , ^{τ[variant prime]} , while _Cy also depends on ^Nq-1ry2-q ||b_{||^L∞ (Ω∩B[straight epsilon]ry )} .

Proof of Theorem 1.2.

In the case of wG domains, Theorem 1.2 follows at once letting f=0 in Theorem 4.2. Suppose now that Ω can be split by a closed set H⊂^Rn in components where MP holds and each y∈H satisfies condition _Gσ in Ω . By MP in the components, since we assume that w≤0 on ∂Ω , then for x∈Ω we have [figure omitted; refer to PDF] Reasoning as above for (4.10), but using Corollary 3.7 instead of Theorem 3.6 as before to obtain Theorem 4.2, from condition _Gσ we deduce for y∈Ω∩H that [figure omitted; refer to PDF] where t∈]0,1[ is independent of y . Inserting this inequality in the former one, and taking the supremum over Ω , we get the result.

Examples

Provided that b(x)=O(1/|x^|2-q ) as |x| [arrow right] ∞ , this last result yields MP in very general domains such as, for instance:

(i) wG domains, like a proper cone Ω such that Ω¯ ≠ ^Rn and in general a domain of conical type, like the complement Ω in ^Rn of Γ×^Rn-2 , where Γ is a logarithmic spiral of equation r=^eθ in polar coordinates, or also complement of a larger spiral of equation r=s(θ) , with s a positive increasing function.

(ii) Domains which can be split in wG subdomains by a suitable closed set H of ^Rn , like the cut plane in ^R2 or in general the complement in ^Rn of a graph {(x,y)∈^Rn-1 ×R | _xi ≥0, i=1,...,n-1,y=f(x)} such that |f(x)|≤h+k|x| for positive constants h and k .

As a further example, we show a repeated application of Theorem 1.2. Look at the complement Ω in ^R2 of a sequence of balls _Br (k) , k=(_kx ,0),_kx ∈N , with 0<r<1/3 . Consider the nonnegative x -axis as H , then _ΩH =Ω\H is connected. If K is the half-line of equations y=(1/2)x , x≥0 , then we have the following:

(i) _ΩH \K has two components, which are domains of conical type, where MP holds;

(ii) each point of _ΩH ∩K satisfies condition _G1/2 in _ΩH .

Thus MP holds in Ω\H by Theorem 1.2. Also, each point of H satisfies condition _Gσ in Ω for some σ∈ ]0,1[ depending on r . Therefore, again by Theorem 1.2, we conclude that MP holds in Ω .

4.3. Parabolic Shaped Domains

For a parabolic shaped wG domain, condition _Gσ at y∈Ω holds with R(y)=O(|y^|α ) as |y| [arrow right] ∞ , for some 0≤α≤1 , the limit cases α=0 and α=1 representing, respectively, the cylindrical and the conical cases. Hence _ry ≤R(y)≤_R0 +β|y^|α for all y∈Ω with positive constants _R0 and β . Then, choosing [straight epsilon] sufficiently small in Theorem 4.2, if |y|≤[straight epsilon]_ry , then [figure omitted; refer to PDF] so that the supremum in the second term of the right-hand side of (4.12) is taken over a bounded subset of y∈Ω , in which _ry ≤_R1 for some positive constant _R1 . Thus [figure omitted; refer to PDF]

Proof of Theorem 1.3.

Since condition wG holds with _ry =O(|y^|α ) , 0≤α≤1 , the assumption b(x)=O(1/|x^|α(2-q) ) as |x| [arrow right] +∞ implies the finiteness of _bq in (4.11). Taking account of (4.16), by continuity of f the estimate (1.10) follows letting [straight epsilon] [arrow right] 0 .

5. Strong Maximum Principle and Liouville Theorem

The weak Harnack inequality of Theorem 3.3 can be used to show the following strong MP.

Theorem 5.1 (strong MP).

Let Ω be a domain of ^Rn . Let w∈C(Ω) be such that F(x,w,Dw,^D2 w)≥0 in the viscosity sense, and assume structure condition (1.5), with b∈C(Ω¯) . If _x0 ∈Ω and M:=w(_x0 )≥w(x), for all x∈Ω , then w≡M in Ω .

Proof.

Following [33], set _Ω1 =^w-1 ({M}) and _Ω2 =Ω\_Ω1 . By assumption _Ω1 ≠ ∅ . By continuity of w , it turns out that _Ω2 =^w-1 (]-∞,M[) is an open subset of ^Rn . Moreover, plainly, Ω=_Ω1 ∪_Ω2 and _Ω2 ∩_Ω1 =∅ .

Recall that Ω is an open connected set. Thus it is sufficient to show that _Ω1 is in turn an open subset to have Ω=_Ω1 , as claimed in the statement of the theorem. Indeed, let _x1 ∈_Ω1 , that is, w(_x1 )=M , and set u=M-w , then u is a nonnegative viscosity solution of F(x,u,Du,^D2 u)≤0 . Applying (3.7) in a ball _BR :=_BR (_x1 )⊂_BR/τ (_x1 )⊂⊂Ω , we get [figure omitted; refer to PDF] from which, by continuity, u≡M in _BR (_x1 ) . This shows that _Ω1 is an open subset of ^Rn and concludes the proof.

The Liouville type result of Theorem 1.5 is instead based on Harnack inequality (3.11) of Theorem 3.5. It is convenient to consider its version in annular regions _BR,2R =_B2R (0)\_B¯R (0) to take advantage of the decay of b(x) , obtained in standard way, using inequality (3.11) in a chain of linked balls. This yields, for continuous solutions u∈C(_B¯R/2,4R ) of equation F=f , 0≤u≤N , under the structure conditions (1.4) and (1.5), with b,f∈C(_B¯R/2,4R ) , the following inequality: [figure omitted; refer to PDF] with a positive constant C=C(n,λ,Λ,q,τ,||b_{||^L∞ (BR/2,4R )}^Nq-1R2-q ) .

Proof of Theorem 1.5.

By the strong MP of Theorem 5.1, we know that w can achieve neither a maximum nor a minimum at a point of ^Rn unless it is constant, in which case we should be done.

Suppose for instance that w≤M:=sup...w<+∞ . Let _Rk be an increasing sequence of positive numbers such that _{lim...k[arrow right]∞Rk} =∞ . Set _Mk =_sup...∂BRk w and _mk =_inf...∂BRk w . By weak maximum principle, _Mk is increasing and _mk is decreasing; thus [figure omitted; refer to PDF] Then, using Harnack inequality (5.2), with u=M-w , we get [figure omitted; refer to PDF] from which [figure omitted; refer to PDF] and, letting k [arrow right] ∞ , we get M=m , as we wanted to show.

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Appendix

Appendix

Proof of Lemmas 2.2 and 2.3

Although the proof of Lemma 2.2 is already contained in previous papers also in the case of an almost-quadratic gradient term, see for instance [17], here, for the sake of completeness, we give a sketch of the simple version in the case of linear gradient term, following [3], where the fundamental case of a second-order uniformly elliptic operator is treated, with no lower-order terms.

However, it seems useless to repeat the nice proof of [3], to which we refer for the ideas and details. We only outline the steps which are influenced by the first-order term. For this reason, we keep the same notations of [3].

Also, for the sake of brevity, we will refer to constants depending only on n,λ,Λ,_b0 as to structural constants.

Firstly, we introduce a test function; see [3, Lemma 4.1].

Lemma 5.

There exist positive structural constants M , C and a function [straight phi]∈^C2 (^Rn ) such that [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] where ξ∈C(^Rn ) , 0≤ξ≤1 , supp ξ⊂_B1/2 .

Proof.

We search for a function of type [straight phi](x)=_A1 -_A2^e-αr , for r=|x|≥1/4 , where _A1 and _A2 are positive constants to be chosen in order that [varphi]((3/2)n)=-2 and [varphi](2n)=0 . Next, we extend [straight phi] to ^Rn in such a way that [straight phi]≥-M . By calculations, choosing α=4Λ(n-1)+1/λ we have [figure omitted; refer to PDF] for r≥1/4 . Also, for r≤1/4 , [figure omitted; refer to PDF] and therefore (A.3) holds taking a cut-off function ξ∈C(^Rn ) such that ξ=1 in _B¯1/4 and ξ=0 outside _B1/2 .

Next, we get a lower bound for the size of level sets of supersolutions. Denoting by _Ql a cube of side l , consider a nonnegative viscosity solution u∈LSC(_Q4n ) of the differential inequality ^Pλ,Λ- (^D2 u)-_b0 |Du|≤f .

Setting w=u+[straight phi] and observing that [figure omitted; refer to PDF] a positive lower bound [figure omitted; refer to PDF] with a structural positive constant μ<1 , see (A.2), follows, in the same way as in [3, Lemma 4.5], using ABP estimate (2.5), provided that _inf...Q3 u≤1 and ||^f+_{||^Ln (Q4n )} ≤_{[straight epsilon]0} for a positive structural constant _{[straight epsilon]0} . Moreover, under the same assumptions, [3, Lemma 4.6] says that [figure omitted; refer to PDF] for all k∈N , which for k=1 agrees with (A.7). Then we point out that (A.8) follows by an induction process, based on the Calderón-Zygmund decomposition of the cube _Q1 , centered at the origin, supposing (A.8) to hold for k-1 .

To perform the induction step it is crucial that, for a supersolution u(x) of (2.4), the rescaled function u (y)=u(x)/^Mk-1 , where x=_x0 +^2-i y runs in the dyadic cube _{Q^2-i 4n} , centered at _x0 , is in turn a supersolution of (2.4) with a correspondingly scaled f , namely [figure omitted; refer to PDF] for y∈_Q4n . From (A.8) it follows that [figure omitted; refer to PDF] for all t>0 , with d and [straight epsilon] positive structural constants. Then, following the proof of Theorem 4.8(1) of [3], we use (A.8) in the identity [figure omitted; refer to PDF] see [1], with _p0 =[straight epsilon]/2 and, by rescaling, remove the normalization conditions _inf...Q3 u≤1 and ||^f+_{||^Ln (Q4n )} ≤_{[straight epsilon]0} to get [figure omitted; refer to PDF] From this, with a covering argument as in [10, Theorem 3.1], we obtain (2.8).

We argue in the same manner for Lemma 2.3. Suppose again that _Q1 is centered at the origin. Following the proof of Theorem 4.8(2) of [3], firstly we consider a subsolution u of (2.6) such that ||^f+_{||^Ln (Q4n )} ≤_{[straight epsilon]0} and ||^u+_{||^{L[straight epsilon]} (Q1 )} ≤^{d1/[straight epsilon]} to get, even in this case, (A.10). Then, arguing as in [3, Lemma 4.7], there exist structural constants _M0 >1 and σ>0 such that, for all j∈N large enough, [figure omitted; refer to PDF] where ν=_M0 /(_M0 -1/2) and _lj =σ^{M0-[straight epsilon]/nν-[straight epsilon]j/n} . As above, to get this result we use the invariance of equation by scale transformations, namely that the function v(y)=ν/(ν-1)-u(x)/^νj-1 (ν-1)_M0 , where x=_x0 +^(4n)-1_lj y runs in the small cube ^Qj , is in turn a supersolution of (2.4) with a correspondingly scaled f , that is, [figure omitted; refer to PDF] for y∈_Q4n , provided that j>1+log...(2-1/_M0 )/log...ν .

On the base of (A.13), reasoning as in the proof of Lemma 4.4 of [3], we infer that _sup...Q1/4 u≤C , from which, by rescaling to remove normalization conditions ||^f+_{||^Ln (Q4n )} ≤_{[straight epsilon]0} and ||^u+_{||^{L[straight epsilon]} (Q1 )} ≤^{d1/[straight epsilon]} , we get [figure omitted; refer to PDF] as in the proof of Theorem 4.8 (2) of [3]. By a covering argument, as above for supersolutions, we get (2.9) for p=[straight epsilon] . Note that (A.10) a fortiori holds replacing [straight epsilon] with p<[straight epsilon] . Thus (2.9) follows for all 0<p<[straight epsilon] . Finally, by Hölder inequality, we obtain (2.9) for all p>0 .