# 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

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topClarke, 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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- [8] M. Giaquinta, “Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems”, Princeton University Press, Princeton, N.J., 1983. Zbl0516.49003MR717034
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- [11] P. Hartman, On the bounded slope condition, Pacific J. Math. 18 (1966), 495–511. Zbl0149.32001MR197640
- [12] P. Hartman and L. Nirenberg, On spherical image maps whose Jacobians do not change sign, Amer. J. Math. 81 (1959), 901–920. Zbl0094.16303MR126812
- [13] P. Marcellini, Regularity for some scalar variational problems under general growth conditions, J. Optim. Theory Appl. 90 (1996), 161–181. Zbl0901.49030MR1397651
- [14] C. Mariconda and G. Treu, Existence and Lipschitz regularity for minima, Proc. Amer. Math. Soc. 130 (2001), 395–404. Zbl0987.49020MR1862118
- [15] C. Mariconda and G. Treu, Gradient maximum principle for minima, J. Optim. Theory Appl. 112 (2002), 167–186. Zbl1019.49029MR1881695
- [16] M. Miranda, Un teorema di esistenza e unicità per il problema dell’area minima in n variabili, Ann. Scuola. Norm. Sup. Pisa Cl. Sci. (3) 19 (1965), 233–249. Zbl0137.08201MR181918
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