Nonhomogeneous boundary condition for spectral non-local operators

TL;DR

This work establishes a unified framework for semilinear nonlocal elliptic problems driven by spectral-type operators $\psi(-L_{|D})$ in bounded $C^{1,1}$ domains with nonhomogeneous boundary data. By deriving sharp boundary estimates for Green and Poisson potentials and introducing a weak $L^1$ trace-like boundary operator, the authors characterize boundary behaviour via the renewal function $V$ and distance to the boundary, and prove existence results for broad, including sign-changing and non-monotone, nonlinearities. The analysis combines stochastic process techniques, potential theory, and Dirichlet form theory, with a spectral representation and a Poisson–Martin boundary framework that connects harmonic functions for $\psi(-L_{|D})$ to those for $L_{|D}$. A key highlight is the treatment of the interpolated fractional Laplacian as a principal example, yielding precise boundary blow-up rates and a complete nonlinear existence theory under nonhomogeneous boundary conditions. The results pave the way for a unified treatment of semilinear boundary problems in nonlocal settings and suggest sharp criteria for solvability in terms of boundary data and nonlinear growth.

Abstract

We study semilinear non-local elliptic problems driven by spectral-type operators of the form $ψ(-L_{|D})$ in a bounded $C^{1,1}$ domain $D\subset \mathbb{R}^d$ with a nonhomogeneous boundary condition. Here $ψ$ is a Bernstein function satisfying a weak scaling condition at infinity, and $L_{|D}$ is the generator of a killed Lévy 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.

Theorems & Definitions (86)

• Lemma 2.1
• Theorem 3.1
• Proposition 3.2
• proof
• Remark 3.3
• Proposition 3.4
• Proposition 3.5
• proof
• Remark 3.6
• Lemma 4.1