2022/03/22 by Dipierro, Serena, Poggesi, Giorgio, Thompson, Jack +1 · 1 citation
#35B06 #35B50 #35N25 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2203.11468
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⊂ \mathbb 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.