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non-coercive problems

ESAIM: Control, Optimisation and Calculus of Variations (1)

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Geometric constraints on the domain for a class of minimum problems
Graziano Crasta, Annalisa Malusa
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 9 / January 2003

Published online by Cambridge University Press:

15 September 2003, pp. 125-133

Print publication:

January 2003
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We consider minimization problems of the form ${\rm min}_{u\in \varphi +W^{1,1}_0(\Omega)}\int_\Omega [f(Du(x))-u(x)]\, {\rm d}x$ where $\Omega\subseteq \mathbb{R}^N$ is a bounded convex open set, and the Borel function $f\colon \mathbb{R}^N \to [0, +\infty]$ is assumed to be neither convex nor coercive. Under suitable assumptions involving the geometry of Ω 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.