Geometric constraints on the domain for a class of minimum problems
ESAIM: Control, Optimisation and Calculus of Variations, Volume 9 (2003), pp. 125-133.

We consider minimization problems of the form ${\mathrm{min}}_{u\in \varphi +{W}_{0}^{1,1}\left(\Omega \right)}{\int }_{\Omega }\left[f\left(Du\left(x\right)\right)-u\left(x\right)\right]\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x$ where $\Omega \subseteq {ℝ}^{N}$ is a bounded convex open set, and the Borel function $f:{ℝ}^{N}\to \left[0,+\infty \right]$ is assumed to be neither convex nor coercive. Under suitable assumptions involving the geometry of $\Omega$ and the zero level set of $f$, we prove that the viscosity solution of a related Hamilton-Jacobi equation provides a minimizer for the integral functional.

DOI: 10.1051/cocv:2003003
Classification: 49J10,  49L25
Keywords: calculus of variations, existence, non-convex problems, non-coercive problems, viscosity solutions
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Crasta, Graziano; Malusa, Annalisa. Geometric constraints on the domain for a class of minimum problems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 9 (2003), pp. 125-133. doi : 10.1051/cocv:2003003. http://www.numdam.org/articles/10.1051/cocv:2003003/

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