Diagonal non-semicontinuous variational problems

Zagatti, Sandro¹ 

¹

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1333-1343

Résumé

We study the minimum problem for non sequentially weakly lower semicontinuos fucntionals of the form$$\begin{array}{c}\hfill ℱ\left(u\right)={\int }_{I}f(x,u\left(x\right),u\text{'}\left(x\right))\mathrm{d}x,\end{array}$$ defined on Sobolev spaces, where the integrand $f:I\times {ℝ}^m\times {ℝ}^m\to ℝ$ is assumed to be non convex in the last variable. Denoting by $\overline{f}$ the lower convex envelope of $f$ with respect to the last variable, we prove the existence of minimum points of $ℱ$ assuming that the application $p\mapsto \overline{f}(·,p,·)$ is separately monotone with respect to each component $p_i$ of the vector $p$ and that the Hessian matrix of the application $\xi \mapsto \overline{f}(·,·,\xi )$ is diagonal. In the special case of functionals of sum type represented by integrands of the form $f(x,p,\xi )=g(x,\xi )+h(x,p)$, we assume that the separate monotonicity of the map $p\mapsto h(,p)$, holds true in a neighbourhood of the (unique) minimizer of the relaxed functional and not necessarily on its whole domain.

Détail

DOI : 10.1051/cocv/2017068

Classification : 46B50, 49J45
Keywords: Non semicontinuous functional, minimum problem, Γ-convergence