### GEORGE MASON
UNIVERSITY

DEPARTMENT OF MATHEMATICAL
SCIENCES

COLLOQUIUM

OCTOBER 4, 2019

**Speaker: **
Carlos Rautenberg, George Mason University

**Title: *** Quasi-variational inequalities in function spaces:
Applications, theory and solution algorithms
*

**Abstract: **
Quasi-variational inequalities (QVIs) were introduced by Bensoussan and
Lions in the 1970’s, and are generalizations of variational inequalities
(VIs) where the associated constraint set is not known a priori.
In general, QVIs arise in many applications involving partial differential
operators and where very nonlinear, nonconvex and nonsmooth phenomena lead
to state-dependent constraints. The fields where QVI formulations are
prevalent involve game theory, continuum mechanics, and electromagnetism,
among others. Fundamental difficulties here are of analytical as well as of
numerical nature: For example, QVIs are not first order optimality
conditions of any optimization problem and hence direct methods of calculus
of variations are not applicable. In this talk, we consider state of the
art methods to deal with existence and approximation of solutions to QVIs
with obstacle and gradient type constraints. We further provide stability
results for the solution set in cases where multiple solutions are present
and study novel optimal control problems with QVIs as constraints. Finally,
we present numerical tests for elliptic and parabolic problems involving
a variety of applications.

**Time:** Friday, October 4, 2019, 3:30-4:20 p.m.

**Place:** Exploratory Hall, room 4106

**Refreshments** will be served at 3:00 p.m.

Department of Mathematical Sciences

George Mason University

4400 University Drive, MS 3F2

Fairfax, VA 22030-4444

http://math.gmu.edu/

Tel. 703-993-1460, Fax. 703-993-1491