An extreme value theorem on closed convex subsets of Lp spaces and two applications

Abstract

In this paper we prove that each $(R\cup \{\infty \left\}\right)$-valued bounded below lower semicontinuous function defined on E reaches its infimum, where E is a closed convex subset of $L^p(X,A,\mu )$ and each element of E is dominated by some $h\in L^p(X,A,\mu )$, $p\ge 1$. As a consequence of this extreme value theorem, a version of the minimax theorem is proved in the context of dominated p-integrable functions, as above. One more consequence is that certain Euler-Lagrange equation has a weak solution in a dominated Sobolev subspace of $W_0^{1,1}$.