The role of the mean curvature in a mixed Hardy-Sobolev trace inequality

TL;DR

This paper investigates the existence of positive solutions to a Hardy-Sobolev trace problem with mixed boundary conditions, highlighting the influence of the boundary's mean curvature on solution existence.

Contribution

It establishes the existence of minimizers for the problem when the boundary's mean curvature is sufficiently below the potential at a point, extending previous results in the field.

Findings

Existence of minimizers under certain curvature conditions

The critical role of mean curvature in solution existence

Extension of Hardy-Sobolev trace inequalities to mixed boundary conditions

Abstract

Let $\Omega$ be a smooth bounded domain of $\mathbb{R}^{N+1}$ of boundary $\partial \Omega= \Gamma_1 \cup \Gamma_2$ and such that $\partial \Omega \cap \Gamma_2$ is a neighborhood of $0$, $h \in \mathcal{C}^0(\partial \Omega \cap \Gamma_2) $ and $s \in [0,1)$. We propose to study existence of positive solutions to the following Hardy-Sobolev trace problem with mixed boundaries conditions \begin{align} \begin{cases} \Delta u= 0& \qquad \textrm{ in } \Omega\\\ u=0 & \qquad \textrm{ on } \Gamma_1 \\\ \frac{\partial u}{\partial \nu}=h(x) u + \frac{u^{q(s)-1}}{d(x)^{s}} & \qquad \textrm{ on } \Gamma_2, \end{cases} \end{align} where $q(s):=\frac{2(N-s)}{N-1}$ is the critical Hardy-Sobolev trace exponent and $\nu$ is the outer unit normal of $\partial \Omega$. In particular, we prove existence of minimizers when $N \geq 3$ and the mean curvature is sufficiently below the potential $h$ at $0$.