Hopf's lemma for viscosity solutions to a class of non-local equations with applications. (March 2021)

• Record Type: Journal Article
• Authors: Biswas, Anup; Lőrinczi, József
• Is Part Of: Nonlinear analysis. Volume 204(2021)
• Publication Date: 2021-03
• Subjects: primary 35P30 35B50 -- secondary 35S15; Bernstein functions of the Laplacian -- Non-local Dirichlet problem -- Principal eigenvalue problem -- Hopf's lemma -- Moving planes -- Overdetermined torsion equation -- Subordinate Brownian motion -- Ascending ladder height process; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2020.112194
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider a large family of integro-differential equations and establish a non-local counterpart of Hopf's lemma, directly expressed in terms of the symbol of the operator. As closely related problems, we also obtain a variety of maximum principles for viscosity solutions. In our approach we combine direct analysis with functional integration, allowing a robust control around the boundary of the domain, and make use of the related ascending ladder height-processes. We then apply these results to a study of principal eigenvalue problems, the radial symmetry of the positive solutions, and the overdetermined non-local torsion equation.