Continuity of solutions to a basic problem in the calculus of variations

Francis Clarke[1]

  • [1] Institut universitaire de France Université Claude Bernard Lyon 1, France

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2005)

  • Volume: 4, Issue: 3, page 511-530
  • ISSN: 0391-173X

Abstract

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We study the problem of minimizing ∫ Ω F ( D u ( x ) ) d x over the functions u ∈ W 1 , 1 ( Ω ) that assume given boundary values φ on Γ : = ∂ Ω . The lagrangian F and the domain Ω are assumed convex. A new type of hypothesis on the boundary function φ is introduced: thelower (or upper) bounded slope condition. This condition, which is less restrictive than the familiar bounded slope condition of Hartman, Nirenberg and Stampacchia, allows us to extend the classical Hilbert-Haar regularity theory to the case of semiconvex (or semiconcave) boundary data (instead of C 2 ). We prove in particular that the solution is locally Lipschitz in Ω . In certain cases, as when Γ is a polyhedron or else of class C 1 , 1 , we obtain in addition a global Hölder condition on Ω ¯ .

How to cite

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Clarke, Francis. "Continuity of solutions to a basic problem in the calculus of variations." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 4.3 (2005): 511-530. <http://eudml.org/doc/84569>.

@article{Clarke2005,
abstract = {We study the problem of minimizing $\int _\Omega F(Du(x))\, dx \;$ over the functions $u\in W^\{1,1\}(\Omega )$ that assume given boundary values $\phi $ on $\Gamma := \partial \Omega $. The lagrangian $F$ and the domain $\Omega $ are assumed convex. A new type of hypothesis on the boundary function $\phi $ is introduced: thelower (or upper) bounded slope condition. This condition, which is less restrictive than the familiar bounded slope condition of Hartman, Nirenberg and Stampacchia, allows us to extend the classical Hilbert-Haar regularity theory to the case of semiconvex (or semiconcave) boundary data (instead of $C^2$). We prove in particular that the solution is locally Lipschitz in $\Omega $. In certain cases, as when $\Gamma $ is a polyhedron or else of class $C^\{1,1\}$, we obtain in addition a global Hölder condition on $\,\overline\{\!\,\Omega \}$.},
affiliation = {Institut universitaire de France Université Claude Bernard Lyon 1, France},
author = {Clarke, Francis},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {3},
pages = {511-530},
publisher = {Scuola Normale Superiore, Pisa},
title = {Continuity of solutions to a basic problem in the calculus of variations},
url = {http://eudml.org/doc/84569},
volume = {4},
year = {2005},
}

TY - JOUR
AU - Clarke, Francis
TI - Continuity of solutions to a basic problem in the calculus of variations
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2005
PB - Scuola Normale Superiore, Pisa
VL - 4
IS - 3
SP - 511
EP - 530
AB - We study the problem of minimizing $\int _\Omega F(Du(x))\, dx \;$ over the functions $u\in W^{1,1}(\Omega )$ that assume given boundary values $\phi $ on $\Gamma := \partial \Omega $. The lagrangian $F$ and the domain $\Omega $ are assumed convex. A new type of hypothesis on the boundary function $\phi $ is introduced: thelower (or upper) bounded slope condition. This condition, which is less restrictive than the familiar bounded slope condition of Hartman, Nirenberg and Stampacchia, allows us to extend the classical Hilbert-Haar regularity theory to the case of semiconvex (or semiconcave) boundary data (instead of $C^2$). We prove in particular that the solution is locally Lipschitz in $\Omega $. In certain cases, as when $\Gamma $ is a polyhedron or else of class $C^{1,1}$, we obtain in addition a global Hölder condition on $\,\overline{\!\,\Omega }$.
LA - eng
UR - http://eudml.org/doc/84569
ER -

References

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