Stability of complement value problems for $p$-Lévy operators
Guy Foghem
TL;DR
This paper develops a comprehensive framework for stability of complement value problems driven by nonlinear symmetric nonlocal $p$-Lévy operators, unifying Dirichlet, Neumann and Robin data under a translation-invariant setting. It introduces tailored nonlocal Sobolev spaces and trace spaces, proves robust nonlocal Poincaré-type inequalities, and establishes well-posedness results for all three boundary conditions via variational methods. A key contribution is the rigorous analysis of nonlocal-to-local transitions, showing Gamma-convergence and BBM-type limits for a broad class of $p$-Lévy kernels, with explicit asymptotics linking $L_ u$ to the local $p$-Laplacian and a precise normalization for the fractional case. The results significantly advance the understanding of stability and approximation of nonlocal models, enabling reliable passage to local PDEs and providing a solid foundation for further study of nonlocal boundary interactions and numerical schemes.
Abstract
We set up a general framework tailor-made to solve complement value problems governed by symmetric nonlinear integrodifferential $p$-Lévy operators. A prototypical example of integrodifferential $p$-Lévy operators is the well-known fractional $p$-Laplace operator. Our main focus is on nonlinear IDEs in the presence of Dirichlet, Neumann and Robin conditions and we show well-posedness results. Several results are new even for the fractional $p$-Laplace operator but we develop the approach for general translation-invariant nonlocal operators. We also bridge a gap from nonlocal to local, by showing that solutions to the local Dirichlet and Neumann boundary value problems associated with $p$-Laplacian are strong limits of the nonlocal ones.