Geometric constraints on the domain for a class of minimum problems

Crasta, Graziano; Malusa, Annalisa

ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 125-133.

We consider minimization problems of the form ${\min }_{u\in \varphi +W_0^{1,1}\left(\Omega \right)}{\int }_{\Omega }[f\left(Du\left(x\right)\right)-u\left(x\right)]\mathrm{d}x$ where $\Omega \subseteq {ℝ}^N$ is a bounded convex open set, and the Borel function $f:{ℝ}^N\to [0,+\infty ]$ 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.

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