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Journal of Convex Analysis, Vol. 5, No. 1, pp. 31-44 (1998)
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#
An Existence Result for a Class of Non Convex Problems of the Calculus of Variations

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Giulia Treu

Dipartimento di Matematica e Informatica, Universita di Udine, via delle Scienze 206, 33100 Udine, Italy, treu@dimi.uniud.it

**Abstract:** We consider the functional

\int_\Omega \left[h(\gamma_K(\nabla u(x)))+u(x)\right]dx \qquad u(x)\in W_0^{1,1}(\Omega)

where $\gamma_K$ is the gauge function of a convex set $K$ and $h : [0,\infty[ \rightarrow [0,\infty]$ is a possibly non convex function. In the case $K\subset\mathbb{R}^2$ is a closed polytope and $\Omega\subset\mathbb{R}^2$ is a bounded convex set we provide a sufficient condition for the existence of the minimum.

Besides, as a corollary, we give conditions on $\Omega\subset\mathbb{R}^2$ and $f:\mathbb{R}^2 \rightarrow [0,\infty]$ that are sufficient to the existence of a minimizer of

\int_\Omega \left[f(\nabla u(x))+u(x)\right]dx \qquad u(x)\in W_0^{1,1}(\Omega).

**Full text of the article:**

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