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A sharp attainment result for nonconvex variational problems

Authors:

Email author" target="_blank">P?Celada Email author G?Cupini M?Guidorzi

Institution:

1.Dipartimento di Matematica,Universitá degli Studi di Parma,Parma,Italy

Abstract:

We consider the problem of minimizing autonomous, multiple integrals like

$\min \left\{ \int_\Omega f\left(u ,\nabla u\right) dx : u\in u_0 + W_0^{1,p}(\Omega) \right\}$

()

where $f: \ensuremath{\mathbb{R}}\times \ensuremath{\mathbb{R}} ^N\to 0 ,\infty)$ is a continuous, possibly nonconvex function of the gradient variable $\nabla u$ . Assuming that the bipolar function f^** of f is affine as a function of the gradient $\nabla u$ on each connected component of the sections of the detachment set $\mathcal{D} = \left\{f^{**} < f\right\}$ , we prove attainment for ( $\mathcal{P}$ ) under mild assumptions on f and f^**. We present examples that show that the hypotheses on f and f^** considered here for attainment are essentially sharp.Received: 12 May 2003, Accepted: 26 August 2003, Published online: 24 November 2003Mathematics Subject Classification (2000): 49J10, 49K10

Keywords:

:" target="_blank">: Nonconvex variational problems Existence of minimizers Convex integration

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