On criticality theory for elliptic mixed boundary value problems in divergence form

TL;DR

This paper develops a comprehensive criticality theory for positive solutions of elliptic divergence form equations with mixed boundary conditions, extending previous results to less regular domains and coefficients, and analyzing eigenvalues and Green functions.

Contribution

It generalizes existing criticality results for elliptic operators to more general domains and boundary conditions, including degenerate mixed boundary value problems.

Findings

Established unique solvability of the mixed boundary value problem.

Proved existence of a principal eigenvalue and positive Green function.

Extended criticality theory to less regular domains and coefficients.

Abstract

The paper is devoted to the study of positive solutions of a second-order linear elliptic equation in divergence form in a domain $D\subseteq \mathbb{R}^n$ that satisfy an oblique boundary condition on a portion of $\partial D$. First, we study the degenerate mixed boundary value problem $$ \begin{cases} Pu=f & \text{in } D, \\ Bu = 0 & \text{on } \partial D_{\mathrm{Rob}}, \\ u=0& \text{on } \partial D_{\mathrm{Dir}}, \end{cases} $$ where $D$ is a bounded Lipschitz domain, $\partial D_{\mathrm{Rob}}$ is a relatively open portion of $\partial D$, $\partial D_{\mathrm{Dir}}$ is a closed set of $\partial D$, and $B$ is an oblique (Robin) boundary operator defined on $\partial D_{\mathrm{Rob}}$. In particular, we discuss the unique solvability of the above problem, the existence of a principal eigenvalue, and the existence of a positive minimal Green function. Then we establish a criticality theory for positive weak solutions of the operator $(P,B)$ in a general domain with no boundary condition on $\partial D_{\mathrm{Dir}}$ and no growth condition at infinity. The paper generalizes and extends results obtained by Pinchover and Saadon (2002) for classical solutions of such a problem, where stronger regularity assumptions on the coefficients of $(P,B)$, and $\partial D_{\mathrm{Rob}}$.