Local Lipschitz regularity for degenerate elliptic systems
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 6, p. 1361-1396
We start presenting an L -gradient bound for solutions to non-homogeneous p-Laplacean type systems and equations, via suitable non-linear potentials of the right-hand side. Such a bound implies a Lorentz space characterization of Lipschitz regularity of solutions which surprisingly turns out to be independent of p, and that reveals to be the same classical one for the standard Laplacean operator. In turn, the a priori estimates derived imply the existence of locally Lipschitz regular solutions to certain degenerate systems with critical growth of the type arising when considering geometric analysis problems.
@article{AIHPC_2010__27_6_1361_0,
     author = {Duzaar, Frank and Mingione, Giuseppe},
     title = {Local Lipschitz regularity for degenerate elliptic systems},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {27},
     number = {6},
     year = {2010},
     pages = {1361-1396},
     doi = {10.1016/j.anihpc.2010.07.002},
     zbl = {1216.35063},
     mrnumber = {2738325},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2010__27_6_1361_0}
}
Duzaar, Frank; Mingione, Giuseppe. Local Lipschitz regularity for degenerate elliptic systems. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 6, pp. 1361-1396. doi : 10.1016/j.anihpc.2010.07.002. http://www.numdam.org/item/AIHPC_2010__27_6_1361_0/

[1] A. Arkhipova, Partial regularity up to the boundary of weak solutions of elliptic systems with nonlinearity q greater than two, J. Math. Sci. (N. Y.) 115 (2003), 2735-2746 | MR 1810609 | Zbl 1118.35319

[2] M.F. Betta, A. Mercaldo, F. Murat, M.M. Porzio, Existence of renormalized solutions to nonlinear elliptic equations with a lower-order term and right-hand side a measure, J. Math. Pures Appl. (9) 82 (2003), 90-124 | MR 1967494

[3] L. Boccardo, T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), 149-169 | MR 1025884 | Zbl 0707.35060

[4] L. Boccardo, T. Gallouët, Nonlinear elliptic equations with right-hand side measures, Comm. Partial Differential Equations 17 (1992), 641-655 | MR 1163440 | Zbl 0812.35043

[5] A. Cianchi, V. Maz'ya, Global Lipschitz regularity for a class of quasilinear equations, preprint.

[6] E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Acad. Sci. Torino Cl. Sci. Fis. Mat. Natur. (III) 125 no. 3 (1957), 25-43 | MR 93649 | Zbl 0084.31901

[7] E. Dibenedetto, C 1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), 827-850 | MR 709038 | Zbl 0539.35027

[8] E. Dibenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York (1993) | MR 1230384 | Zbl 0794.35090

[9] G. Dolzmann, N. Hungerbühler, S. Müller, The p-harmonic system with measure-valued right-hand side, Ann. Inst. H. Poincare Anal. Non Lineaire 14 (1997), 353-364 | Numdam | MR 1450953 | Zbl 0879.35052

[10] F. Duzaar, J. Kristensen, G. Mingione, The existence of regular boundary points for non-linear elliptic systems, J. Reine Angew. Math. (Crelles J.) 602 (2007), 17-58 | MR 2300451 | Zbl 1214.35021

[11] F. Duzaar, G. Mingione, Gradient estimates in non-linear potential theory, Rend. Lincei – Mat. Appl. 20 (2009), 179-190 | MR 2506379 | Zbl 1173.35065

[12] F. Duzaar, G. Mingione, Gradient estimates via non-linear potentials, Amer. J. Math., in press. | MR 2823872 | Zbl 1230.35028

[13] F. Duzaar, G. Mingione, Gradient continuity estimates, Calc. Var. Partial Differential Equations, doi:10.1007/s00526-010-0314-6. | MR 2729305 | Zbl 1204.35100

[14] L. Esposito, F. Leonetti, G. Mingione, Regularity results for minimizers of irregular integrals with (p,q) growth, Forum Math. 14 (2002), 245-272 | MR 1880913 | Zbl 0999.49022

[15] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ (2003) | MR 1962933 | Zbl 1028.49001

[16] C. Hamburger, Regularity of differential forms minimizing degenerate elliptic functionals, J. Reine Angew. Math. (Crelles J.) 431 (1992), 7-64 | MR 1179331 | Zbl 0776.35006

[17] F. Hélein, Harmonic Maps, Conservation Laws and Moving Frames, Cambridge Tracts in Math. vol. 150, Cambridge University Press, Cambridge (2002) | MR 1913803 | Zbl 1010.58010

[18] F. Hélein, J. Wood, Harmonic maps, Handbook of Global Analysis vol. 1213, Elsevier Sci. B.V., Amsterdam (2008), 417-491 | MR 2389639 | Zbl 1236.58002

[19] S. Hildebrandt, K.O. Widman, Some regularity results for quasilinear elliptic systems of second order, Math. Z. 142 (1975), 67-86 | MR 377273 | Zbl 0317.35040

[20] L.I. Hunt, On L(p,q) spaces, L'Einsegnement Math. 12 (1966), 249-276 | MR 223874 | Zbl 0181.40301

[21] T. Iwaniec, p-Harmonic tensors and quasiregular mappings, Ann. of Math. (2) 136 (1992), 589-624 | MR 1189867 | Zbl 0785.30009

[22] T. Iwaniec, C. Sbordone, Weak minima of variational integrals, J. Reine Angew. Math. (Crelles J.) 454 (1994), 143-161 | MR 1288682 | Zbl 0802.35016

[23] T. Kilpeläinen, J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), 137-161 | MR 1264000 | Zbl 0820.35063

[24] J. Kristensen, C. Melcher, Regularity in oscillatory nonlinear elliptic systems, Math. Z. 260 (2008), 813-847 | MR 2443332 | Zbl 1158.35037

[25] T. Kuusi, G. Mingione, Potential estimates and gradient boundedness for nonlinear parabolic systems, preprint, 2010. | MR 2916967 | Zbl 1246.35115

[26] J. Leray, J.-L. Lions, Quelques résultats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty–Browder, Bull. Soc. Math. France 93 (1965), 97-107 | Numdam | MR 194733 | Zbl 0132.10502

[27] G.M. Lieberman, Sharp forms of estimates for subsolutions and supersolutions of quasilinear elliptic equations involving measures, Comm. Partial Differential Equations 18 (1993), 1191-1212 | MR 1233190 | Zbl 0802.35041

[28] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod/Gauthier–Villars, Paris (1969) | MR 259693 | Zbl 0189.40603

[29] V. Maz'Ya, The boundedness of the first derivatives of the solution of the Dirichlet problem in a region with smooth nonregular boundary, Vestnik Leningrad. Univ. 24 (1969), 72-79 | MR 247273 | Zbl 0177.14403

[30] V. Maz'Ya, Boundedness of the gradient of a solution to the Neumann–Laplace problem in a convex domain, C. R. Acad. Sci. Paris Ser. I 347 (2008), 517-520 | MR 2576900 | Zbl 1167.35013

[31] G. Mingione, Regularity of minima: an invitation to the dark side of the calculus of variations, Appl. Math. 51 (2006), 355-425 | MR 2291779 | Zbl 1164.49324

[32] G. Mingione, The Calderón–Zygmund theory for elliptic problems with measure data, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), 195-261 | Numdam | MR 2352517 | Zbl 1178.35168

[33] G. Mingione, Gradient estimates below the duality exponent, Math. Ann. 346 (2010), 571-627 | MR 2578563 | Zbl 1193.35077

[34] G. Mingione, Gradient potential estimates, J. Europ. Math. Soc., in press. | MR 2746772 | Zbl 1217.35077

[35] T. Rivière, Conservation laws for conformally invariant variational problems, Invent. Math. 168 (2007), 1-22 | MR 2285745 | Zbl 1128.58010

[36] T. Rivière, Integrability by compensation in the analysis of conformally invariant problems, lecture notes.

[37] T. Rivière, Sub-criticality of Schroedinger systems with antisymmetric potentials, preprint, 2009. | MR 2772189 | Zbl 1210.35072

[38] E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser. vol. 32, Princeton University Press, Princeton, NJ (1971) | MR 304972 | Zbl 0232.42007

[39] N.S. Trudinger, X.J. Wang, On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math. 124 (2002), 369-410 | MR 1890997 | Zbl 1067.35023

[40] N.S. Trudinger, X.J. Wang, Quasilinear elliptic equations with signed measure data, Discrete Contin. Dyn. Syst. A 23 (2009), 477-494 | MR 2449089 | Zbl 1154.35351

[41] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), 219-240 | MR 474389 | Zbl 0372.35030

[42] N.N. Ural'Tseva, Degenerate quasilinear elliptic systems, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 184-222 | MR 244628 | Zbl 0199.42502