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Abstract
We prove a Hopftype lemma for antisymmetric supersolutions to the Dirichlet problem for the fractional Laplacian with zeroth order terms. As an application, we use such a Hopftype lemma in combination with the method of moving planes to prove symmetry for the semilinear fractional parallel surface problem. That is, we prove that nonnegative solutions to semilinear Dirichlet problems for the fractional Laplacian in a bounded open set Ω⊂R^{n} 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 counterexamples to these theorems when only `local' assumptions are imposed on the solutions.
Original language  English 

Pages (fromto)  16711692 
Number of pages  22 
Journal  Transactions of the American Mathematical Society 
Volume  377 
Issue number  3 
Early online date  10 Jan 2024 
DOIs  
Publication status  Published  Mar 2024 
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Minimal surfaces, free boundaries and partial differential equations
ARC Australian Research Council
1/01/19 → 30/06/25
Project: Research

Partial Differential Equations, free boundaries and applications
ARC Australian Research Council
30/11/18 → 30/11/22
Project: Research