Video library: Moshe Marcus, Boundary value problems for elliptic semi-linear equations with measure data

VIDEO LIBRARY


International School “Singularities, Blow-up, and Non-Classical Problems in Nonlinear PDEs for youth” November 14, 2024 10:00, Moscow, RUDN University

Boundary value problems for elliptic semi-linear equations with measure data Moshe Marcus Technical University Technion, Israel
Abstract: We consider boundary value problems of the form $-(\Delta + V )u + f(u) = \tau$ in $D\subset \mathbb R^N$, $\mathrm{tr}_V u = \nu$ on $\partial D$ where $D$ is a bounded Lipschitz domain in $\mathbb R^N$ and $\mathrm{tr}_V u$ denotes the measure boundary trace associated with $V$. Regarding the non-linear term assume: $f$ is continuous, monotone increasing and $f(0) = 0$. We discuss questions of existence and uniqueness, first in the case $V = 0$ and then for potentials $V$ that blow up at the boundary not faster then $\mathrm{dist}(x; \partial D)^{-2}$. Language: English