On the existence and uniqueness of minimizers for a class of integral functionals

Abstract

We study the solvability of the minimization problem $$\mathop {\min }\limits_{\eta \in \mathcal{K}_\alpha } \int_0^T {\alpha (t)\left[ {f\left( {|\eta '(t)|} \right) + g\left( {\eta (t)} \right)} \right]} \,dt,$$ where \(\mathcal{K}_\alpha \) is a subset of ACloc[0, T] depending on the weight function α. Neither the convexity nor the superlinearity of f are required. The main application concerns the existence and uniqueness of minimizers to integral functionals on convex domains \(\Omega \subset \mathbb{R}^N ,\) defined in the class of functions in \(W_0 ^{1,1} \left( \Omega \right)\) depending only on the distance from the boundary of Ω. As a corollary, when Ω is a ball we obtain the existence of radially symmetric solutions to nonconvex and noncoercive functionals.