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arXiv · 2203.11468

The role of antisymmetric functions in nonlocal equations

Serena Dipierro·Giorgio Poggesi·Jack Thompson·Enrico Valdinoci
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Abstract

We prove a Hopf-type lemma for antisymmetric super-solutions to the Dirichlet problem for the fractional Laplacian with zero-th order terms. As an application, we use such a Hopf-type lemma in combination with the method of moving planes to prove symmetry for the semilinear fractional parallel surface problem. That is, we prove that non-negative solutions to semilinear Dirichlet problems for the fractional Laplacian in a bounded open set $\Omega \subset \mathbb R^n$ must be radially symmetric if one of their level surfaces is parallel to the boundary of $\Omega$; in turn, $\Omega$ must be a ball. Furthermore, we discuss maximum principles and the Harnack inequality for antisymmetric functions in the fractional setting and provide counter-examples to these theorems when only `local' assumptions are imposed on the solutions.

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Serena Dipierro, Giorgio Poggesi, Jack Thompson, Enrico Valdinoci. 2022-03-22. The role of antisymmetric functions in nonlocal equations. https://arxiv.org/abs/2203.11468

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