The role of antisymmetric functions in nonlocal equations

Abstract

We use a Hopf-type lemma for antisymmetric super-solutions to the Dirichlet problem for the fractional Laplacian with zero-th order terms, 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 ⊂ R n \Omega \subset \mathbb {R}^n must be radially symmetric if one of their level surfaces is parallel to the boundary of Ω \Omega ; in turn, Ω \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. The construction of these counter-examples relies on an approximation result that states that ‘all antisymmetric functions are locally antisymmetric and s s -harmonic up to a small error’.