On the Minimum Problem for a Class of Noncoercive Nonconvex Functionals

Abstract

We are concerned with the problem of existence of solutions to the variational problem \[ \min\left\{\int_0^R g(t,v'(t))\,dt;\ v\in AC([0,R]),\ v(R)=0\right\}, \] with only one fixed endpoint prescribed. The map $g\colon [0,R]\times\mathbb{R}\to\overline{\mathbb{R}}$ is a normal integrand, for which neither convexity nor superlinear growth conditions are assumed. As an application, we give an existence result for the radially symmetric variational problem \[ \min_{u\in\Wuu(B_R)}\int_{B_R} [{f\pt{\mod{x},\mod{\nabla u(x)}}+a(|x|)\,u(x)}]\,dx, \] where $B_R$ is the ball of $\mathbb{R}^n$ centered at the origin and with radius $R>0$, the map $f\colon [0,R]\times[0,+\infty[\to\overline{\mathbb{R}}$ is a normal integrand, and $a\in\L(0,R)$. Again, neither convexity nor superlinear growth conditions are made on f. These kinds of problems, with nonconvex Lagrangians with respect to $\nabla u$, arise in different fields of mathematical physics such as optimal design and nonlinear elasticity.