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The role of antisymmetric functions in nonlocal equations
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by Serena Dipierro, Giorgio Poggesi, Jack Thompson and Enrico Valdinoci;

Trans. Amer. Math. Soc. 377 (2024), 1671-1692

DOI: https://doi.org/10.1090/tran/8984

Published electronically: January 10, 2024

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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 $\Omega \subset \mathbb {R}^n$ 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. The construction of these counter-examples relies on an approximation result that states that ‘all antisymmetric functions are locally antisymmetric and $s$-harmonic up to a small error’.

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