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

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.