The role of antisymmetric functions in nonlocal equations

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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 Ω⊂Rn must be radially symmetric if one of their level surfaces is parallel to the boundary of Ω; in turn, Ω 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.
Original languageEnglish
Pages (from-to)1671-1692
Number of pages22
JournalTransactions of the American Mathematical Society
Issue number3
Early online date10 Jan 2024
Publication statusPublished - Mar 2024


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