Partial Differential Equations, Convexity and Weak Lower Semi-Continuity

Abstract

This thesis is concerned with the calculus of variations on bounded domains. The critical points of a functional I corresponding to a Lagragian function L are the solutions of the Euler-Lagrange equation. This equation is a partial differential equation. I will prove in the main theorem that there exists a minimizer to the functional I under certain conditions on L. These conditions are partial convexity and coercivity. Partial convexity is convexity in a part of the variable of L and coercivity is a bound from below of L with respect to another function. In the last subsection I will provide a motivation for the hypothesis of this theorem.