PDE Seminar Abstracts

# Minimal Solution to Some Variational Inequalities

Michel Chipot
Universität Zürich, Switzerland
Fri 18th Nov 2016, 11-12am, Carslaw Lecture Theater 157

## Abstract

We would like to present some new results regarding the existence of a minimal solution to some variational inequalities of the type

where $\Omega$ is a domain in ${ℝ}^{n}$, $K$ is a closed convex subset of ${W}_{0}^{1,p}\left(\Omega \right)$, $A$ is a nonlinear operator defined from ${W}_{0}^{1,p}\left(\Omega \right)$ into its dual by

$⟨Au,v⟩={\int }_{\Omega }a\left(x,u,\nabla u\right)\cdot \nabla v\phantom{\rule{0.3em}{0ex}}dx+{\int }_{\Omega }{a}_{0}\left(x,u,\nabla u\right)v\phantom{\rule{0.3em}{0ex}}dx$

for all $v\in {W}_{0}^{1,p}\left(\Omega \right)$ and $f\in {L}^{q}\left(\Omega \right)$ where $q$ is the conjugate of $p$.

This is joint work with S. Guesmia and S. Harkat.