Existence of Minimizers for Nonconvex Variational Problems with Slow Growth

Abstract

Consider the minimization problem

$$(P) min\left\{ {\int_0^1 {f\left( {t,u'\left( t \right)} \right)dt;u \in W^{1.1} \left( {\left[ {0,1} \right],\mathbb{R}''} \right), u\left( 0 \right) = } u_0 ,u\left( 1 \right) = u_1 } \right\},$$

in which $f:\left[ {0,1} \right]x {\mathbb{R}}^n \to {\mathbb{R}} \cup \left\{ { + \infty } \right\}$is a normal integrand. Define the convex function $G:\mathbb{R}^n \to \mathbb{R} \cup \left\{ { + \infty } \right\}$ by $G\left( p \right)\dot = \int_0^1 {f^* \left( {t,p} \right)dt.}$ It is known that, if the essential domain H of G is open, then problem (P) has a minimizer for any pair of endpoints (u ₀, u ₁). In this paper, the same result is proved under the condition that, for every point p in H, the subgradient set ∂G(p) is either bounded or empty (when H is open, this condition holds automatically).