PDE Seminar Abstracts

Generalized Lax-Milgram theorem in reflexive Banach spaces and its application to elliptic boundary value problems

Hideo Kozono
Tohoku University, Japan
14 Mar 2011 2-3pm, Mills Lecture Room 209

Abstract

It is well-known that the Lax-Milgram theorem for positive definite quadratic forms on a Hilbert space is useful to show the existence of weak solutions of boundary value problems of elliptic equations. We shall generalize this theorem to the case of a one parameter family of reflexive Banach spaces. We may deal with quadratic forms which are not necessarily positive definite. Some variational inequalities play an essential role for positivity. Then we shall apply our theorem to the construction of weak solutions in ${L}^{p}$ to a boundary value problem for an elliptic system.