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Abstract
The classical Laplace operator is a vital tool in modeling many physical behaviors, such as elasticity, diffusion and fluid flow. Incorporated in the Laplace operator is the requirement of twice differentiability, which implies continuity that many physical processes lack. In this thesis we introduce a new nonlocal Laplace-type operator, that is capable of dealing with strong discontinuities. Motivated by the state-based peridynamic framework, this new nonlocal Laplacian exhibits double nonlocality through the use of iterated integral operators. The operator introduces additional degrees of flexibility that can allow better representation of physical phenomena at different scales and in materials with different properties. We obtain explicit rates of convergence for this doubly nonlocal operator to the classical Laplacian as the radii for the horizons of interaction kernels shrink to zero. We study mathematical properties of this state-based Laplacian, including connections with other nonlocal and local counterparts. Finally, we study the solutions of the state-based Laplacian, and use the structure of the solutions to further exhibit the connections between other nonlocal and local Laplacians.
