Explicit solutions of nonconvex variational problems in dimension one (Q1968774)

The paper deals with one-dimensional variational problems of the type \[ \text{Inf}\Biggl\{\int^1_0 [W_0(x,u(x))+ W_1(x,u'(x))] dx,\;u(0)= 0,\;u(1)= \gamma,\;u\in W^{1,p}(0,1)\Biggr\} \] under the growth assumption \[ c(|\lambda|^p- 1)\leq W_1(x,\lambda)\leq C(|\lambda|^p+ 1),\quad p>1,\quad C>c>0. \] No convexity assumption of \(W_1\) on \(\lambda\) is made here. The authors propose a systematic way to analyze the above problems and to find ``generalized solutions'' when the minimizing sequences oscillate, converging in some Sobolev space to some limit that is not a minimizer. They consider a new functional involving parametrized measures supported on \(\mathbb{R}\) and solve a suitable differential inclusion motivated by necessary conditions of optimality for the generalized minimizers.