Non-differentiable functionals and singular sets of minima

Kristensen, Jan; Mingione, Giuseppe

doi:10.1016/j.crma.2004.11.019

Calculus of Variations

Non-differentiable functionals and singular sets of minima
[Fonctionnelles non-différentiable et l'ensembles singulier des minima]

Présenté par : John M. Ball

Kristensen, Jan ¹ ; Mingione, Giuseppe ²

¹ Mathematical Institute, University of Oxford, St. Giles' 24-29, Oxford OX1 3LB, UK
² Dipartimento di Matematica, Università di Parma, via D'Azeglio 85/a, 43100, Parma, Italy

Comptes Rendus. Mathématique, Tome 340 (2005) no. 1, pp. 93-98.

Résumé
Abstract

Nous bornons la dimension de Hausdorff de l'ensemble singulier des minima de fonctionnelles du type $\int_{Ω} F (x, v, D v)$ oú F n'est Hölderienne que par rapport aux la variables $(x, v)$ .

We provide bounds for the Hausdorff dimension of the singular set of minima of functionals of the type $\int_{Ω} F (x, v, D v)$ , where F is only Hölder continuous with respect to the variables $(x, v)$ .

Reçu le : 2004-11-02
Accepté le : 2004-11-05
Publié le : 2004-12-21

DOI : 10.1016/j.crma.2004.11.019

Affiliations des auteurs :

Kristensen, Jan ¹ ; Mingione, Giuseppe ²

¹ Mathematical Institute, University of Oxford, St. Giles' 24-29, Oxford OX1 3LB, UK
² Dipartimento di Matematica, Università di Parma, via D'Azeglio 85/a, 43100, Parma, Italy

@article{CRMATH_2005__340_1_93_0,
     author = {Kristensen, Jan and Mingione, Giuseppe},
     title = {Non-differentiable functionals and singular sets of minima},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {93--98},
     publisher = {Elsevier},
     volume = {340},
     number = {1},
     year = {2005},
     doi = {10.1016/j.crma.2004.11.019},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2004.11.019/}
}

TY  - JOUR
AU  - Kristensen, Jan
AU  - Mingione, Giuseppe
TI  - Non-differentiable functionals and singular sets of minima
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 93
EP  - 98
VL  - 340
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2004.11.019/
DO  - 10.1016/j.crma.2004.11.019
LA  - en
ID  - CRMATH_2005__340_1_93_0
ER  -

%0 Journal Article
%A Kristensen, Jan
%A Mingione, Giuseppe
%T Non-differentiable functionals and singular sets of minima
%J Comptes Rendus. Mathématique
%D 2005
%P 93-98
%V 340
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2004.11.019/
%R 10.1016/j.crma.2004.11.019
%G en
%F CRMATH_2005__340_1_93_0

Kristensen, Jan; Mingione, Giuseppe. Non-differentiable functionals and singular sets of minima. Comptes Rendus. Mathématique, Tome 340 (2005) no. 1, pp. 93-98. doi : 10.1016/j.crma.2004.11.019. http://www.numdam.org/articles/10.1016/j.crma.2004.11.019/

Bibliographie
Cité par

[1] Acerbi, E.; Fusco, N. Regularity for minimizers of nonquadratic functionals: the case $1 < p < 2$ , J. Math. Anal. Appl., Volume 140 (1989), pp. 115-135

[2] Bojarski, B.; Iwaniec, T. Analytical foundations of the theory of quasiconformal mappings in $R^{n}$ , Ann. Acad. Sci. Fenn. Ser. A I Math., Volume 8 (1983), pp. 257-324

[3] Campanato, S. Elliptic systems with non-linearity q greater than or equal to two. Regularity of the solution to the Dirichlet problem, Ann. Mat. Pura Appl. (4), Volume 147 (1987), pp. 117-150

[4] Giaquinta, M. Introduction to Regularity Theory for Nonlinear Elliptic Systems, Lectures in Math. ETH Zürich, Birkhäuser, Basel, 1993

[5] Giaquinta, M.; Giusti, E. On the regularity of the minima of variational integrals, Acta Math., Volume 148 (1982), pp. 31-46

[6] Giaquinta, M.; Giusti, E. Differentiability of minima of nondifferentiable functionals, Invent. Math., Volume 72 (1983), pp. 285-298

[7] Giusti, E. Direct Methods in the Calculus of Variations, World Scientific, River Edge, NJ, 2003

[8] J. Kristensen, G. Mingione: The singular set of minima of integral functionals, preprint

[9] Mingione, G. The singular set of solutions to non-differentiable elliptic systems, Arch. Rational Mech. Anal., Volume 166 (2003), pp. 287-301

[10] Mingione, G. Bounds for the singular set of solutions to non linear elliptic systems, Calc. Var., Volume 18 (2003), pp. 373-400

[11] Nečas, J. Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity, Theory of Nonlinear Operators, Proc. Fourth Internat. Summer School, Acad. Sci., Berlin, 1975, pp. 197-206

[12] Rivière, T. Everywhere discontinuous harmonic maps into spheres, Acta Math., Volume 175 (1995), pp. 197-226

[13] Šverák, V.; Yan, X. Non-Lipschitz minimizers of smooth uniformly convex functionals, Proc. Natl. Acad. Sci. USA, Volume 99 (2002) no. 24, pp. 15269-15276

Cité par Sources :