$L^{\infty }$-regularity for a wide class of parabolic systems with general growth
HTML articles powered by AMS MathViewer
- by Teresa Isernia PDF
- Proc. Amer. Math. Soc. 146 (2018), 4741-4753 Request permission
Abstract:
We prove the local boundedness of weak solutions for the following non-linear second order parabolic systems: \begin{equation*} u_{t} - \mathrm {div} \left ( \frac {\mathcal {\varphi }’(|Du|)}{|Du|}Du\right )=0 \mbox { in } \Omega _{T}:=\Omega \times (-T,0), \end{equation*} where $\Omega \subset \mathbb {R}^{n}$ is a bounded domain and $\mathcal {\varphi }$ is a given $N$-function. The proof of this result is based on a Moser-type iteration argument.References
- E. Acerbi and N. Fusco, Regularity for minimizers of nonquadratic functionals: the case $1<p<2$, J. Math. Anal. Appl. 140 (1989), no. 1, 115–135. MR 997847, DOI 10.1016/0022-247X(89)90098-X
- Verena Bögelein, Frank Duzaar, and Paolo Marcellini, Existence of evolutionary variational solutions via the calculus of variations, J. Differential Equations 256 (2014), no. 12, 3912–3942. MR 3190487, DOI 10.1016/j.jde.2014.03.005
- Verena Bögelein, Frank Duzaar, Paolo Marcellini, and Stefano Signoriello, Nonlocal diffusion equations, J. Math. Anal. Appl. 432 (2015), no. 1, 398–428. MR 3371243, DOI 10.1016/j.jmaa.2015.06.053
- Dominic Breit, Bianca Stroffolini, and Anna Verde, A general regularity theorem for functionals with $\phi$-growth, J. Math. Anal. Appl. 383 (2011), no. 1, 226–233. MR 2812732, DOI 10.1016/j.jmaa.2011.05.012
- J. Burczak and P. Kaplicky, Interior regularity of space derivatives to an evolutionary, symmetric $\mathcal {\varphi }$-Laplacian, Preprint arXiv:1507.05843 .
- Hi Jun Choe, Hölder continuity for solutions of certain degenerate parabolic systems, Nonlinear Anal. 18 (1992), no. 3, 235–243. MR 1148287, DOI 10.1016/0362-546X(92)90061-I
- Andrea Dall’Aglio and Elvira Mascolo, $L^\infty$ estimates for a class of nonlinear elliptic systems with nonstandard growth, Atti Sem. Mat. Fis. Univ. Modena 50 (2002), no. 1, 65–83 (English, with English and Italian summaries). MR 1910779
- Ennio De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Un. Mat. Ital. (4) 1 (1968), 135–137 (Italian). MR 0227827
- Emmanuele DiBenedetto and Avner Friedman, Regularity of solutions of nonlinear degenerate parabolic systems, J. Reine Angew. Math. 349 (1984), 83–128. MR 743967
- Emmanuele DiBenedetto and Avner Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1–22. MR 783531, DOI 10.1515/crll.1985.357.1
- L. Diening, T. Scharle & S. Schwarzacher, Regularity for parabolic systems of Uhlenbeck type with Orlicz growth, Preprint arXiv:1603.05604.
- Lars Diening, Bianca Stroffolini, and Anna Verde, Everywhere regularity of functionals with $\phi$-growth, Manuscripta Math. 129 (2009), no. 4, 449–481. MR 2520895, DOI 10.1007/s00229-009-0277-0
- Enrico Giusti and Mario Miranda, Un esempio di soluzioni discontinue per un problema di minimo relativo ad un integrale regolare del calcolo delle variazioni, Boll. Un. Mat. Ital. (4) 1 (1968), 219–226 (Italian). MR 0232265
- Gary M. Lieberman, Boundary regularity for solutions of degenerate parabolic equations, Nonlinear Anal. 14 (1990), no. 6, 501–524. MR 1044078, DOI 10.1016/0362-546X(90)90038-I
- Paolo Marcellini, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Rational Mech. Anal. 105 (1989), no. 3, 267–284. MR 969900, DOI 10.1007/BF00251503
- Paolo Marcellini, Regularity and existence of solutions of elliptic equations with $p,q$-growth conditions, J. Differential Equations 90 (1991), no. 1, 1–30. MR 1094446, DOI 10.1016/0022-0396(91)90158-6
- Paolo Marcellini, Everywhere regularity for a class of elliptic systems without growth conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), no. 1, 1–25. MR 1401415
- Paolo Marcellini and Gloria Papi, Nonlinear elliptic systems with general growth, J. Differential Equations 221 (2006), no. 2, 412–443. MR 2196484, DOI 10.1016/j.jde.2004.11.011
- Jürgen Moser, A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1960), 457–468. MR 170091, DOI 10.1002/cpa.3160130308
- M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Marcel Dekker, Inc., New York, 1991. MR 1113700
- 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
- Michael Wiegner, On $C_\alpha$-regularity of the gradient of solutions of degenerate parabolic systems, Ann. Mat. Pura Appl. (4) 145 (1986), 385–405. MR 886719, DOI 10.1007/BF01790549
- Jiangsheng You, Regularity of solutions of certain parabolic system with nonstandard growth condition, Acta Math. Sinica (N.S.) 14 (1998), no. 2, 145–160. MR 1704826, DOI 10.1007/BF02560201
Additional Information
- Teresa Isernia
- Affiliation: Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 12, 60131 Ancona, Italy
- Email: teresa.isernia@unina.it
- Received by editor(s): July 24, 2017
- Received by editor(s) in revised form: January 16, 2018
- Published electronically: August 10, 2018
- Communicated by: Joachim Krieger
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 4741-4753
- MSC (2010): Primary 35B65, 49N60, 35K40, 46E30
- DOI: https://doi.org/10.1090/proc/14099
- MathSciNet review: 3856142