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Tuomo Kuusi; Giuseppe Mingione
Partial regularity and potentials
(Régularité partielle et potentiels)
Journal de l'École polytechnique — Mathématiques, 3 (2016), p. 309-363, doi: 10.5802/jep.35
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Class. Math.: 35B65, 31C45
Mots clés: Régularité partielle, système elliptique, théorie du potentiel non linéaire, $\varepsilon $-régularité

Résumé - Abstract

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’$\varepsilon $-régularité de solutions des systèmes elliptiques. Pour les systèmes non homogènes du type $-\mathrm{div}\, a(Du)=f$, les nouveaux critères d’$\varepsilon $-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 $-\mathrm{div}\, a(Du)=0$, 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’$\varepsilon $-régularité permettent encore d’obtenir des estimée sur la dimension de Hausdorff des ensembles singuliers.

Bibliographie

[1] E. Acerbi & N. Fusco, “A regularity theorem for minimizers of quasiconvex integrals”, Arch. Rational Mech. Anal. 99 (1987) no. 3, p. 261-281  MR 888453 |  Zbl 0627.49007
[2] D. R. Adams & L. I. Hedberg, Function spaces and potential theory, Grundlehren Math. Wiss. 314, Springer-Verlag, Berlin, 1996  MR 1411441 |  Zbl 0834.46021
[3] A. Alberico, A. Cianchi & C. Sbordone, “Continuity properties of solutions to the $p$-Laplace system”, Adv. Calc. Var. (2015), online
[4] P. Baroni, “Riesz potential estimates for a general class of quasilinear equations”, Calc. Var. Partial Differential Equations 53 (2015) no. 3-4, p. 803-846  MR 3347481
[5] L. Brasco & F. Santambrogio, “A sharp estimate à la Calderón-Zygmund for the $p$-Laplacian”, arXiv:1607.06648, 2016
[6] A. Cianchi, “Nonlinear potentials, local solutions to elliptic equations and rearrangements”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 10 (2011) no. 2, p. 335-361  MR 2856151 |  Zbl 1235.31009
[7] A. Cianchi & V. G. Maz’ya, “Global Lipschitz regularity for a class of quasilinear elliptic equations”, Comm. Partial Differential Equations 36 (2011) no. 1, p. 100-133  MR 2763349 |  Zbl 1220.35065
[8] A. Cianchi & V. G. Maz’ya, “Global boundedness of the gradient for a class of nonlinear elliptic systems”, Arch. Rational Mech. Anal. 212 (2014) no. 1, p. 129-177  MR 3162475 |  Zbl 1298.35070
[9] P. Daskalopoulos, T. Kuusi & G. Mingione, “Borderline estimates for fully nonlinear elliptic equations”, Comm. Partial Differential Equations 39 (2014) no. 3, p. 574-590  MR 3169795 |  Zbl 1290.35092
[10] E. De Giorgi, “Frontiere orientate di misura minima”, Sem. di Mat. de Scuola Norm. Sup. Pis. (1960-61), p. 1-56
[11] F. Duzaar & G. Mingione, “The $p$-harmonic approximation and the regularity of $p$-harmonic maps”, Calc. Var. Partial Differential Equations 20 (2004) no. 3, p. 235-256  MR 2062943 |  Zbl 1142.35433
[12] F. Duzaar & G. Mingione, “Regularity for degenerate elliptic problems via $p$-harmonic approximation”, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004) no. 5, p. 735-766  MR 2086757 |  Zbl 1112.35078
[13] F. Duzaar & G. Mingione, “Local Lipschitz regularity for degenerate elliptic systems”, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010) no. 6, p. 1361-1396  MR 2738325 |  Zbl 1216.35063
[14] F. Duzaar, G. Mingione & K. Steffen, Parabolic systems with polynomial growth and regularity, Mem. Amer. Math. Soc. 214, no. 1005, American Mathematical Society, 2011  MR 2866816 |  Zbl 1238.35001
[15] M. Foss & G. Mingione, “Partial continuity for elliptic problems”, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008) no. 3, p. 471-503 Numdam |  MR 2422076 |  Zbl 1153.35017
[16] N. Fusco & J. Hutchinson, “Partial regularity for minimisers of certain functionals having nonquadratic growth”, Ann. Mat. Pura Appl. (4) 155 (1989), p. 1-24  MR 1042826 |  Zbl 0698.49001
[17] M. Giaquinta, Introduction to regularity theory for nonlinear elliptic systems, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1993  MR 1239172 |  Zbl 0786.35001
[18] M. Giaquinta & G. Modica, “Remarks on the regularity of the minimizers of certain degenerate functionals”, Manuscripta Math. 57 (1986) no. 1, p. 55-99  MR 866406 |  Zbl 0607.49003
[19] E. Giusti, Direct methods in the calculus of variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003  MR 1962933 |  Zbl 1028.49001
[20] E. Giusti & M. Miranda, “Sulla regolarità delle soluzioni deboli di una classe di sistemi ellittici quasi-lineari”, Arch. Rational Mech. Anal. 31 (1968) no. 3, p. 173-184  MR 235264 |  Zbl 0167.10703
[21] C. Hamburger, “Regularity of differential forms minimizing degenerate elliptic functionals”, J. reine angew. Math. 431 (1992), p. 7-64  MR 1179331 |  Zbl 0776.35006
[22] L. I. Hedberg & Th. H. Wolff, “Thin sets in nonlinear potential theory”, Ann. Inst. Fourier (Grenoble) 33 (1983) no. 4, p. 161-187 Cedram |  MR 727526 |  Zbl 0508.31008
[23] J. Heinonen, T. Kilpeläinen & O. Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993  MR 1207810 |  Zbl 0780.31001
[24] T. Kilpeläinen & J. Malý, “Degenerate elliptic equations with measure data and nonlinear potentials”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992) no. 4, p. 591-613 Numdam |  MR 1205885 |  Zbl 0797.35052
[25] T. Kilpeläinen & J. Malý, “The Wiener test and potential estimates for quasilinear elliptic equations”, Acta Math. 172 (1994) no. 1, p. 137-161  Zbl 0820.35063
[26] R. Korte & T. Kuusi, “A note on the Wolff potential estimate for solutions to elliptic equations involving measures”, Adv. Calc. Var. 3 (2010) no. 1, p. 99-113  MR 2604619 |  Zbl 1182.35222
[27] J. Kristensen & G. Mingione, “The singular set of minima of integral functionals”, Arch. Rational Mech. Anal. 180 (2006) no. 3, p. 331-398  MR 2214961 |  Zbl 1116.49010
[28] J. Kristensen & A. Taheri, “Partial regularity of strong local minimizers in the multi-dimensional calculus of variations”, Arch. Rational Mech. Anal. 170 (2003) no. 1, p. 63-89  MR 2012647 |  Zbl 1030.49040
[29] M. Kronz, “Partial regularity results for minimizers of quasiconvex functionals of higher order”, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002) no. 1, p. 81-112 Numdam |  MR 1902546 |  Zbl 1010.49023
[30] T. Kuusi & G. Mingione, “Nonlinear vectorial potential theory”, to appear in J. Eur. Math. Soc. (JEMS)
[31] T. Kuusi & G. Mingione, “Universal potential estimates”, J. Funct. Anal. 262 (2012) no. 10, p. 4205-4269  MR 2900466 |  Zbl 1252.35097
[32] T. Kuusi & G. Mingione, “Linear potentials in nonlinear potential theory”, Arch. Rational Mech. Anal. 207 (2013) no. 1, p. 215-246  MR 3004772 |  Zbl 1266.31011
[33] T. Kuusi & G. Mingione, “Borderline gradient continuity for nonlinear parabolic systems”, Math. Ann. 360 (2014) no. 3-4, p. 937-993  MR 3273649
[34] T. Kuusi & G. Mingione, “Guide to nonlinear potential estimates”, Bull. Math. Sci. 4 (2014) no. 1, p. 1-82  MR 3174278
[35] T. Kuusi & G. Mingione, “A nonlinear Stein theorem”, Calc. Var. Partial Differential Equations 51 (2014) no. 1-2, p. 45-86  MR 3247381
[36] V. G. Maz’ja, “The continuity at a boundary point of the solutions of quasi-linear elliptic equations”, Vestnik Leningrad Univ. Math. 25 (1970) no. 13, p. 42-55  MR 274948
[37] V. G. Maz’ja & V. P. Havin, “A nonlinear potential theory”, Uspehi Mat. Nauk 27 (1972) no. 6, p. 67-138  MR 409858 |  Zbl 0247.31010
[38] G. Mingione, “The singular set of solutions to non-differentiable elliptic systems”, Arch. Rational Mech. Anal. 166 (2003) no. 4, p. 287-301  MR 1961442 |  Zbl 1142.35391
[39] G. Mingione, “Regularity of minima: an invitation to the dark side of the calculus of variations”, Appl. Math. 51 (2006) no. 4, p. 355-426  MR 2291779 |  Zbl 1164.49324
[40] G. Mingione, “Gradient potential estimates”, J. Eur. Math. Soc. (JEMS) 13 (2011) no. 2, p. 459-486  MR 2746772 |  Zbl 1217.35077
[41] C. B. Morrey, “Partial regularity results for non-linear elliptic systems”, J. Math. Mech. 17 (1967/1968), p. 649-670  MR 237947 |  Zbl 0175.11901
[42] N. C. Phuc & I. E. Verbitsky, “Quasilinear and Hessian equations of Lane-Emden type”, Ann. of Math. (2) 168 (2008) no. 3, p. 859-914  MR 2456885 |  Zbl 1175.31010
[43] N. C. Phuc & I. E. Verbitsky, “Singular quasilinear and Hessian equations and inequalities”, J. Funct. Anal. 256 (2009) no. 6, p. 1875-1906  MR 2498563 |  Zbl 1169.35026
[44] T. Schmidt, “Regularity theorems for degenerate quasiconvex energies with $(p,q)$-growth”, Adv. Calc. Var. 1 (2008) no. 3, p. 241-270  MR 2458237 |  Zbl 1151.49031
[45] J. Simon, “Régularité de solutions de problèmes nonlinéaires”, C. R. Acad. Sci. Paris Sér. A-B 282 (1976) no. 23, p. A1351-A1354  MR 454763 |  Zbl 0336.47039
[46] N. S. Trudinger & X.-J. Wang, “On the weak continuity of elliptic operators and applications to potential theory”, Amer. J. Math. 124 (2002) no. 2, p. 369-410  MR 1890997 |  Zbl 1067.35023
[47] K. Uhlenbeck, “Regularity for a class of non-linear elliptic systems”, Acta Math. 138 (1977) no. 3-4, p. 219-240  MR 474389 |  Zbl 0372.35030
[48] N. N. Ural’ceva, “Degenerate quasilinear elliptic systems”, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), p. 184-222  MR 244628