Antisymmetric maximum principles and Hopf's lemmas for the Logarithmic Laplacian, with applications to symmetry results

TL;DR

This paper establishes maximum principles and Hopf lemmas for the Logarithmic Laplacian, leading to symmetry results and rigidity theorems for solutions of related semilinear problems.

Contribution

It introduces antisymmetric maximum principles and Hopf lemmas for the Logarithmic Laplacian, advancing the understanding of symmetry and rigidity in these nonlocal problems.

Findings

Proved antisymmetric maximum principles for the Logarithmic Laplacian.

Established Hopf-type lemmas for linear problems involving this operator.

Demonstrated symmetry of solutions in symmetric domains and a rigidity result for the parallel surface problem.

Abstract

We prove antisymmetric maximum principles and Hopf-type lemmas for linear problems described by the Logarithmic Laplacian. As application, we prove the symmetry of solutions for semilinear problems in symmetric sets, and a rigidity result for the parallel surface problem for the Logarithmic Laplacian.