Local Lipschitz regularity for degenerate elliptic systems
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 6, pp. 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.

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     title = {Local {Lipschitz} regularity for degenerate elliptic systems},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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     publisher = {Elsevier},
     volume = {27},
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Duzaar, Frank; Mingione, Giuseppe. Local Lipschitz regularity for degenerate elliptic systems. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 6, pp. 1361-1396. doi : 10.1016/j.anihpc.2010.07.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2010.07.002/

[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 | Zbl

[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

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

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

[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 | Zbl

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

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

[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 | EuDML | Numdam | MR | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

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

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

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

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

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

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

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

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

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

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

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

[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 | EuDML | Numdam | MR | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | EuDML | Numdam | MR | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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