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

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 $\partial D_{\mathrm{Rob}}$.