Fractional differentiability for solutions of the inhomogeneous $p$-Laplace system
HTML articles powered by AMS MathViewer
- by Michał Miśkiewicz PDF
- Proc. Amer. Math. Soc. 146 (2018), 3009-3017 Request permission
Abstract:
It is shown that if $p \geqslant 3$ and $u \in W^{1,p}(\Omega ,\mathbb {R}^N)$ solves the inhomogeneous $p$-Laplace system \[ \operatorname {div} (|\nabla u|^{p-2} \nabla u) = f, \qquad f \in W^{1,p’}(\Omega ,\mathbb {R}^N), \] then locally the gradient $\nabla u$ lies in the fractional Nikol’skiĭ space $\mathcal {N}^{\theta ,2/\theta }$ with any $\theta \in [ \tfrac {2}{p}, \tfrac {2}{p-1} )$. To the author’s knowledge, this result is new even in the case of $p$-harmonic functions, slightly improving known $\mathcal {N}^{2/p,p}$ estimates. The method used here is an extension of the one used by A. Cellina in the case $2 \leqslant p < 3$ to show $W^{1,2}$ regularity.References
- Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
- Benny Avelin, Tuomo Kuusi, and Giuseppe Mingione, Nonlinear Calderón-Zygmund theory in the limiting case, Arch. Ration. Mech. Anal. 227 (2018), no. 2, 663–714. MR 3740385, DOI 10.1007/s00205-017-1171-7
- B. Bojarski and T. Iwaniec, $p$-harmonic equation and quasiregular mappings, Partial differential equations (Warsaw, 1984) Banach Center Publ., vol. 19, PWN, Warsaw, 1987, pp. 25–38. MR 1055157
- L. Brasco and F. Santambrogio, A sharp estimate à la Calderón-Zygmund for the $p$-Laplacian, Commun. Contemp. Math. (2017).
- Arrigo Cellina, The regularity of solutions to some variational problems, including the $p$-Laplace equation for $2\leq p < 3$, ESAIM Control Optim. Calc. Var. 23 (2017), no. 4, 1543–1553. MR 3716932, DOI 10.1051/cocv/2016064
- E. DiBenedetto, $C^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), no. 8, 827–850. MR 709038, DOI 10.1016/0362-546X(83)90061-5
- Lawrence C. Evans, A new proof of local $C^{1,\alpha }$ regularity for solutions of certain degenerate elliptic p.d.e, J. Differential Equations 45 (1982), no. 3, 356–373. MR 672713, DOI 10.1016/0022-0396(82)90033-X
- Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
- John L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J. 32 (1983), no. 6, 849–858. MR 721568, DOI 10.1512/iumj.1983.32.32058
- Peter Lindqvist, Notes on the $p$-Laplace equation, Report. University of Jyväskylä Department of Mathematics and Statistics, vol. 102, University of Jyväskylä, Jyväskylä, 2006. MR 2242021
- Juan J. Manfredi and Allen Weitsman, On the Fatou theorem for $p$-harmonic functions, Comm. Partial Differential Equations 13 (1988), no. 6, 651–668. MR 934377, DOI 10.1080/03605308808820556
- Giuseppe Mingione, The singular set of solutions to non-differentiable elliptic systems, Arch. Ration. Mech. Anal. 166 (2003), no. 4, 287–301. MR 1961442, DOI 10.1007/s00205-002-0231-8
- S. M. Nikol′skiĭ, Approximation of functions of several variables and imbedding theorems, Die Grundlehren der mathematischen Wissenschaften, Band 205, Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by John M. Danskin, Jr. MR 0374877, DOI 10.1007/978-3-642-65711-5
- Berardino Sciunzi, Regularity and comparison principles for $p$-Laplace equations with vanishing source term, Commun. Contemp. Math. 16 (2014), no. 6, 1450013, 20. MR 3277953, DOI 10.1142/S0219199714500138
- Peter Tolksdorf, Everywhere-regularity for some quasilinear systems with a lack of ellipticity, Ann. Mat. Pura Appl. (4) 134 (1983), 241–266. MR 736742, DOI 10.1007/BF01773507
- Peter Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126–150. MR 727034, DOI 10.1016/0022-0396(84)90105-0
- K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), no. 3-4, 219–240. MR 474389, DOI 10.1007/BF02392316
- N. N. Ural′ceva, Degenerate quasilinear elliptic systems, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 184–222 (Russian). MR 0244628
Additional Information
- Michał Miśkiewicz
- Affiliation: Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
- Email: m.miskiewicz@mimuw.edu.pl
- Received by editor(s): August 2, 2017
- Received by editor(s) in revised form: October 9, 2017
- Published electronically: February 28, 2018
- Additional Notes: The author’s research was supported by the NCN grant no. 2012/05/E/ST1/03232 (years 2013-2017).
- Communicated by: Joachim Krieger
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 3009-3017
- MSC (2010): Primary 35B65, 35J92
- DOI: https://doi.org/10.1090/proc/13993
- MathSciNet review: 3787361