Geometric constraints on the domain for a class of minimum problems

Graziano Crasta; Annalisa Malusa

ESAIM: Control, Optimisation and Calculus of Variations (2003)

  • Volume: 9, page 125-133
  • ISSN: 1292-8119

Abstract

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We consider minimization problems of the form min u ∈ ϕ + W 0 1 , 1 ( Ω ) ∫ Ω [ f ( D u ( x ) ) - u ( x ) ] d x where Ω ⊆ ℝ N is a bounded convex open set, and the Borel function f : ℝ N → [ 0 , + ∞ ] 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.

How to cite

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Crasta, Graziano, and Malusa, Annalisa. "Geometric constraints on the domain for a class of minimum problems." ESAIM: Control, Optimisation and Calculus of Variations 9 (2003): 125-133. <http://eudml.org/doc/245310>.

@article{Crasta2003,
abstract = {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 $f\colon \mathbb \{R\}^N \rightarrow [0, +\infty ]$ 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.},
author = {Crasta, Graziano, Malusa, Annalisa},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {calculus of variations; existence; non-convex problems; non-coercive problems; viscosity solutions; integral functional},
language = {eng},
pages = {125-133},
publisher = {EDP-Sciences},
title = {Geometric constraints on the domain for a class of minimum problems},
url = {http://eudml.org/doc/245310},
volume = {9},
year = {2003},
}

TY - JOUR
AU - Crasta, Graziano
AU - Malusa, Annalisa
TI - Geometric constraints on the domain for a class of minimum problems
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2003
PB - EDP-Sciences
VL - 9
SP - 125
EP - 133
AB - 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 $f\colon \mathbb {R}^N \rightarrow [0, +\infty ]$ 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.
LA - eng
KW - calculus of variations; existence; non-convex problems; non-coercive problems; viscosity solutions; integral functional
UR - http://eudml.org/doc/245310
ER -

References

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