Geometric constraints on the domain for a class of minimum problems

Graziano Crasta; Annalisa Malusa

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • 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 (2010): 125-133. <http://eudml.org/doc/90684>.

@article{Crasta2010,
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 $f\colon \mathbb\{R\}^N \to [0, +\infty]$ 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. },
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; non-coercive problems; viscosity solutions},
language = {eng},
month = {3},
pages = {125-133},
publisher = {EDP Sciences},
title = {Geometric constraints on the domain for a class of minimum problems},
url = {http://eudml.org/doc/90684},
volume = {9},
year = {2010},
}

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
DA - 2010/3//
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 $f\colon \mathbb{R}^N \to [0, +\infty]$ 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.
LA - eng
KW - Calculus of Variations; existence; non-convex problems; non-coercive problems; viscosity solutions.; integral functional; non-coercive problems; viscosity solutions
UR - http://eudml.org/doc/90684
ER -

References

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