Remarks on the nonlocal Dirichlet problem
Tomasz Grzywny, Moritz Kassmann, {\L}ukasz Le\.zaj
TL;DR
This paper explores the properties of solutions to nonlocal Dirichlet problems involving translation-invariant integrodifferential operators, focusing on solution notions, representation formulas, and differentiability conditions.
Contribution
It provides a comprehensive analysis of solution concepts, classical representation formulas, and differentiability criteria for nonlocal Dirichlet problems with Lévy process generators.
Findings
Different notions of solutions are compared and characterized.
Classical representation formulas for distributional solutions are established.
Conditions for classical twice differentiability of solutions are identified with counterexamples.
Abstract
We study translation-invariant integrodifferential operators that generate L\'{e}vy processes. First, we investigate different notions of what a solution to a nonlocal Dirichlet problem is and we provide the classical representation formula for distributional solutions. Second, we study the question under which assumptions distributional solutions are twice differentiable in the classical sense. Sufficient conditions and counterexamples are provided.
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