Carnegie Mellon
Department of Mathematical Sciences

Riccardo De Arcangelis, Dipartimento di Matematica e Applicazioni "Renato Caccioppoli", University of Naples Federico II

"Relaxation of Non-Convex Pointwise Gradient Constrained Energyes"

Abstract

The talk is concerned with some relaxation problems for an integral energy $F$ of the type

\begin{displaymath}F(u)=\int_{\Omega} f(\nabla u){\rm d} {\cal L}^n,\end{displaymath}

when the admissible configurations $u$ are scalar-valued, and satisfy the pointwise gradient constraint $\nabla u(x)\in E$ for a.e. $x\in \Omega$. Here $\Omega$ is a smooth bounded open subset of $\mbox{\boldmath$R$}^n$, $f:\mbox{\boldmath$R$}^n\to[0,+\infty[$ is Borel, and $E$ is a fixed subset of $\mbox{\boldmath$R$}^n$.

Some recent results on the integral representation problem for the lower semicontinuous envelope $\overline F$ of $F$ are described when no convexity assumptions on $f$ and $E$ are assumed. In this case, it is proved that $\overline F$ can be expressed on the whole $BV(\Omega)$ as an integral with density given by the convex lower semicontinuous envelope $(f+I_E)^{**}$ of $f+I_E$.

The lack of convexity of $E$ forestalls the use of the standard integral representation techniques. The novelty of the result relies in the new approach proposed in order to avoid the analysis of the measure theoretic properties of $\overline F$ as a function of the open set $\Omega$.

As corollaries, some applications to differential inclusions are provided.

Some results on the same problem in the case when $f$ depends also on the space variable, and $E$ too has a true dependence on $x$ are discussed, showing that regularization processes occur, beside convexification, in the construction of the relaxed densities.

THURSDAY, February 2, 2006
Time: 1:30 P.M.
Location: PPB 300

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