Nonhomogeneous boundary condition for spectral non-local operators

TL;DR

This paper develops a comprehensive framework for analyzing semilinear non-local elliptic problems with nonhomogeneous boundary conditions driven by spectral operators, extending fractional Laplacian theory and introducing new boundary estimates and solution existence results.

Contribution

It introduces a novel approach to nonhomogeneous boundary conditions for spectral non-local operators, including boundary estimates and existence results for complex nonlinearities.

Findings

Established sharp boundary estimates for Green and Poisson potentials.

Introduced a weak $L^1$ trace-like boundary operator.

Proved existence of solutions for general nonlinearities, including sign-changing cases.

Abstract

We study semilinear non-local elliptic problems driven by spectral-type operators of the form $\psi(-L_{|D})$ in a bounded $C^{1,1}$ domain $D\subset \mathbb{R}^d$ with a nonhomogeneous boundary condition. Here $\psi$ is a Bernstein function satisfying a weak scaling condition at infinity, and $L_{|D}$ is the generator of a killed L\'evy process. This general framework covers and extends the theory of the interpolated fractional Laplacian. A key novelty in this setting is the analysis of the nonhomogeneous boundary condition formulated in terms of the Poisson potential with respect to the $d-1$ Hausdorff measure on $\partial D$. We establish sharp boundary estimates for Green and Poisson potentials, introduce a weak $L^1$ trace-like boundary operator, and provide existence results for solutions under quite general nonlinearities, including sign-changing and non-monotone cases. The methodology combines stochastic process techniques, potential theory, and spectral analysis, and expresses the boundary behavior of the solution in terms of the renewal function and the distance to the boundary, suggesting a possible unified treatment of semilinear boundary problems in non-local settings.