Partial regularity and potentials
[Régularité partielle et potentiels]
Journal de l’École polytechnique — Mathématiques, Tome 3 (2016), pp. 309-363.

Nous relions la théorie classique de la régularité partielle des systèmes elliptiques à la théorie du potentiel non linéaire d’équations éventuellement dégénérées. Plus précisément, nous donnons une version en théorie du potentiel des critères classiques d’ε-régularité de solutions des systèmes elliptiques. Pour les systèmes non homogènes du type -diva(Du)=f, les nouveaux critères d’ε-régularité font intervenir à la fois la fonctionnelle classique d’excès de Du et de type de Riesz optimal et les potentiels de Wolff du membre de droite f. Appliqués au cas homogène -diva(Du)=0, ces critères redonnent les critères classiques en théorie de la régularité partielle. Comme corollaire, nous montrons que les résultats classiques et précisés de régularité pour les solutions d’équations scalaires en terme d’espaces de fonctions pour f s’étendent mot pour mot aux systèmes généraux dans le cadre de la régularité partielle, à savoir la régularité partielle des solutions hors d’un ensemble singulier fermé négligeable. Enfin, ces nouveaux critères d’ε-régularité permettent encore d’obtenir des estimée sur la dimension de Hausdorff des ensembles singuliers.

We connect classical partial regularity theory for elliptic systems to Nonlinear Potential Theory of possibly degenerate equations. More precisely, we find a potential theoretic version of the classical ε-regularity criteria leading to regularity of solutions of elliptic systems. For non-homogenous systems of the type -diva(Du)=f, the new ε-regularity criteria involve both the classical excess functional of Du and optimal Riesz type and Wolff potentials of the right hand side f. When applied to the homogenous case -diva(Du)=0 such criteria recover the classical ones in partial regularity. As a corollary, we find that the classical and sharp regularity results for solutions to scalar equations in terms of function spaces for f extend verbatim to general systems in the framework of partial regularity, i.e. optimal regularity of solutions outside a negligible, closed singular set. Finally, the new ε-regularity criteria still allow to provide estimates on the Hausdorff dimension of the singular sets.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.35
Classification : 35B65, 31C45
Keywords: Partial regularity, elliptic system, nonlinear potential theory, $\varepsilon $-regularity
Mot clés : Régularité partielle, système elliptique, théorie du potentiel non linéaire, $\varepsilon $-régularité
Kuusi, Tuomo 1 ; Mingione, Giuseppe 2

1 Aalto University, Institute of Mathematics P.O. Box 11100, FI-00076 Aalto, Finland
2 Dipartimento di Matematica e Informatica, Università di Parma Parco Area delle Scienze 53/a, Campus, 43100 Parma, Italy
@article{JEP_2016__3__309_0,
     author = {Kuusi, Tuomo and Mingione, Giuseppe},
     title = {Partial regularity and potentials},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {309--363},
     publisher = {ole polytechnique},
     volume = {3},
     year = {2016},
     doi = {10.5802/jep.35},
     mrnumber = {3541851},
     zbl = {1373.35065},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jep.35/}
}
TY  - JOUR
AU  - Kuusi, Tuomo
AU  - Mingione, Giuseppe
TI  - Partial regularity and potentials
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2016
SP  - 309
EP  - 363
VL  - 3
PB  - ole polytechnique
UR  - http://archive.numdam.org/articles/10.5802/jep.35/
DO  - 10.5802/jep.35
LA  - en
ID  - JEP_2016__3__309_0
ER  - 
%0 Journal Article
%A Kuusi, Tuomo
%A Mingione, Giuseppe
%T Partial regularity and potentials
%J Journal de l’École polytechnique — Mathématiques
%D 2016
%P 309-363
%V 3
%I ole polytechnique
%U http://archive.numdam.org/articles/10.5802/jep.35/
%R 10.5802/jep.35
%G en
%F JEP_2016__3__309_0
Kuusi, Tuomo; Mingione, Giuseppe. Partial regularity and potentials. Journal de l’École polytechnique — Mathématiques, Tome 3 (2016), pp. 309-363. doi : 10.5802/jep.35. http://archive.numdam.org/articles/10.5802/jep.35/

[1] Acerbi, E.; Fusco, N. A regularity theorem for minimizers of quasiconvex integrals, Arch. Rational Mech. Anal., Volume 99 (1987) no. 3, pp. 261-281 | DOI | MR

[2] Adams, D. R.; Hedberg, L. I. Function spaces and potential theory, Grundlehren Math. Wiss., 314, Springer-Verlag, Berlin, 1996 | MR

[3] Alberico, A.; Cianchi, A.; Sbordone, C. Continuity properties of solutions to the p-Laplace system, Adv. Calc. Var. (2015) (online)

[4] Baroni, P. Riesz potential estimates for a general class of quasilinear equations, Calc. Var. Partial Differential Equations, Volume 53 (2015) no. 3-4, pp. 803-846 | DOI | MR | Zbl

[5] Brasco, L.; Santambrogio, F. A sharp estimate à la Calderón-Zygmund for the p-Laplacian (2016) (arXiv:1607.06648)

[6] Cianchi, A. Nonlinear potentials, local solutions to elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), Volume 10 (2011) no. 2, pp. 335-361 | MR | Zbl

[7] Cianchi, A.; Maz’ya, V. G. Global Lipschitz regularity for a class of quasilinear elliptic equations, Comm. Partial Differential Equations, Volume 36 (2011) no. 1, pp. 100-133 | MR | Zbl

[8] Cianchi, A.; Maz’ya, V. G. Global boundedness of the gradient for a class of nonlinear elliptic systems, Arch. Rational Mech. Anal., Volume 212 (2014) no. 1, pp. 129-177 | DOI | MR | Zbl

[9] Daskalopoulos, P.; Kuusi, T.; Mingione, G. Borderline estimates for fully nonlinear elliptic equations, Comm. Partial Differential Equations, Volume 39 (2014) no. 3, pp. 574-590 | MR | Zbl

[10] De Giorgi, E. Frontiere orientate di misura minima, Sem. di Mat. de Scuola Norm. Sup. Pis. (1960-61), pp. 1-56

[11] Duzaar, F.; Mingione, G. The p-harmonic approximation and the regularity of p-harmonic maps, Calc. Var. Partial Differential Equations, Volume 20 (2004) no. 3, pp. 235-256 | MR | Zbl

[12] Duzaar, F.; Mingione, G. Regularity for degenerate elliptic problems via p-harmonic approximation, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 21 (2004) no. 5, pp. 735-766 | DOI | Numdam | MR | Zbl

[13] Duzaar, F.; Mingione, G. Local Lipschitz regularity for degenerate elliptic systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 27 (2010) no. 6, pp. 1361-1396 | DOI | Numdam | MR | Zbl

[14] Duzaar, F.; Mingione, G.; Steffen, K. Parabolic systems with polynomial growth and regularity, Mem. Amer. Math. Soc., 214, no. 1005, American Mathematical Society, 2011 | Zbl

[15] Foss, M.; Mingione, G. Partial continuity for elliptic problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 25 (2008) no. 3, pp. 471-503 | DOI | Numdam | MR | Zbl

[16] Fusco, N.; Hutchinson, J. Partial regularity for minimisers of certain functionals having nonquadratic growth, Ann. Mat. Pura Appl. (4), Volume 155 (1989), pp. 1-24 | DOI | MR | Zbl

[17] Giaquinta, M. Introduction to regularity theory for nonlinear elliptic systems, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1993 | Zbl

[18] Giaquinta, M.; Modica, G. Remarks on the regularity of the minimizers of certain degenerate functionals, Manuscripta Math., Volume 57 (1986) no. 1, pp. 55-99 | DOI | MR | Zbl

[19] Giusti, E. Direct methods in the calculus of variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003, viii+403 pages | DOI | Zbl

[20] Giusti, E.; Miranda, M. Sulla regolarità delle soluzioni deboli di una classe di sistemi ellittici quasi-lineari, Arch. Rational Mech. Anal., Volume 31 (1968) no. 3, pp. 173-184 | DOI | Zbl

[21] Hamburger, C. Regularity of differential forms minimizing degenerate elliptic functionals, J. reine angew. Math., Volume 431 (1992), pp. 7-64 | MR | Zbl

[22] Hedberg, L. I.; Wolff, Th. H. Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble), Volume 33 (1983) no. 4, pp. 161-187 | DOI | Numdam | MR | Zbl

[23] Heinonen, J.; Kilpeläinen, T.; Martio, O. Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993

[24] Kilpeläinen, T.; Malý, J. Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 19 (1992) no. 4, pp. 591-613 | Numdam | MR | Zbl

[25] Kilpeläinen, T.; Malý, J. The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., Volume 172 (1994) no. 1, pp. 137-161 | MR | Zbl

[26] Korte, R.; Kuusi, T. A note on the Wolff potential estimate for solutions to elliptic equations involving measures, Adv. Calc. Var., Volume 3 (2010) no. 1, pp. 99-113 | MR | Zbl

[27] Kristensen, J.; Mingione, G. The singular set of minima of integral functionals, Arch. Rational Mech. Anal., Volume 180 (2006) no. 3, pp. 331-398 | DOI | MR | Zbl

[28] Kristensen, J.; Taheri, A. Partial regularity of strong local minimizers in the multi-dimensional calculus of variations, Arch. Rational Mech. Anal., Volume 170 (2003) no. 1, pp. 63-89 | DOI | MR | Zbl

[29] Kronz, M. Partial regularity results for minimizers of quasiconvex functionals of higher order, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 19 (2002) no. 1, pp. 81-112 | DOI | Numdam | MR | Zbl

[30] Kuusi, T.; Mingione, G. Nonlinear vectorial potential theory to appear in J. Eur. Math. Soc. (JEMS) | Zbl

[31] Kuusi, T.; Mingione, G. Universal potential estimates, J. Funct. Anal., Volume 262 (2012) no. 10, pp. 4205-4269 | DOI | MR | Zbl

[32] Kuusi, T.; Mingione, G. Linear potentials in nonlinear potential theory, Arch. Rational Mech. Anal., Volume 207 (2013) no. 1, pp. 215-246 | DOI | MR | Zbl

[33] Kuusi, T.; Mingione, G. Borderline gradient continuity for nonlinear parabolic systems, Math. Ann., Volume 360 (2014) no. 3-4, pp. 937-993 | DOI | MR | Zbl

[34] Kuusi, T.; Mingione, G. Guide to nonlinear potential estimates, Bull. Math. Sci., Volume 4 (2014) no. 1, pp. 1-82 | MR | Zbl

[35] Kuusi, T.; Mingione, G. A nonlinear Stein theorem, Calc. Var. Partial Differential Equations, Volume 51 (2014) no. 1-2, pp. 45-86 | DOI | MR | Zbl

[36] Maz’ja, V. G. The continuity at a boundary point of the solutions of quasi-linear elliptic equations, Vestnik Leningrad Univ. Math., Volume 25 (1970) no. 13, pp. 42-55 | MR

[37] Maz’ja, V. G.; Havin, V. P. A nonlinear potential theory, Uspehi Mat. Nauk, Volume 27 (1972) no. 6, pp. 67-138 | MR

[38] Mingione, G. The singular set of solutions to non-differentiable elliptic systems, Arch. Rational Mech. Anal., Volume 166 (2003) no. 4, pp. 287-301 | DOI | MR | Zbl

[39] Mingione, G. Regularity of minima: an invitation to the dark side of the calculus of variations, Appl. Math., Volume 51 (2006) no. 4, pp. 355-426 | MR | Zbl

[40] Mingione, G. Gradient potential estimates, J. Eur. Math. Soc. (JEMS), Volume 13 (2011) no. 2, pp. 459-486 | MR | Zbl

[41] Morrey, C. B. Jr. Partial regularity results for non-linear elliptic systems, J. Math. Mech., Volume 17 (1967/1968), pp. 649-670 | MR

[42] Phuc, N. C.; Verbitsky, I. E. Quasilinear and Hessian equations of Lane-Emden type, Ann. of Math. (2), Volume 168 (2008) no. 3, pp. 859-914 | DOI | MR | Zbl

[43] Phuc, N. C.; Verbitsky, I. E. Singular quasilinear and Hessian equations and inequalities, J. Funct. Anal., Volume 256 (2009) no. 6, pp. 1875-1906 | DOI | MR | Zbl

[44] Schmidt, T. Regularity theorems for degenerate quasiconvex energies with (p,q)-growth, Adv. Calc. Var., Volume 1 (2008) no. 3, pp. 241-270 | DOI | MR | Zbl

[45] Simon, J. Régularité de solutions de problèmes nonlinéaires, C. R. Acad. Sci. Paris Sér. A-B, Volume 282 (1976) no. 23, p. A1351-A1354

[46] Trudinger, N. S.; Wang, X.-J. On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math., Volume 124 (2002) no. 2, pp. 369-410 | DOI | MR

[47] Uhlenbeck, K. Regularity for a class of non-linear elliptic systems, Acta Math., Volume 138 (1977) no. 3-4, pp. 219-240 | DOI | MR | Zbl

[48] Ural’ceva, N. N. Degenerate quasilinear elliptic systems, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), Volume 7 (1968), pp. 184-222 | MR

Cité par Sources :