Full Regularity for Convex Integral Functionals with $p(x)$ Growth in Low Dimensions

Habermann, Jens

Bollettino dell'Unione Matematica Italiana, Tome 3 (2010), p. 521-541 / Harvested from Biblioteca Digitale Italiana di Matematica

Résumé

For $\Omega \subset \mathbf{R}^{n}$; $n \ge 2$, and $N \ge 1$ we consider vector valued minimizers $u \in W_{loc}^{m,p(\cdot)}(\Omega,\mathbf{R}^{N})$ of a uniformly convex integral functional of the type $$\mathcal{F} \left[ u,\Omega \right] := \int_{\Omega} f(x,D^{m}u) \, dx,$$ where $f$ is a Carathéorody function satisfying $p(x)$ growth conditions with respect to the second variable. We show that if the dimension $n$ is small enough, dependent on the structure conditions of the functional, there holds $$D^{k}u \in C_{loc}^{0,\beta}(\Omega) \,\, \text{for} \,\, k \in \{0,\cdots,m-1\},$$ for some $\beta$, also depending on the structural data, provided that the nonlinearity exponent $p$ is uniformly continuous with modulus of continuity $\omega$ satisfying $$\limsup_{\rho\downarrow 0} \omega(\rho) \log \bigg( \frac{1}{\rho} \bigg) = 0.$$

Détail

Publié le : 2010-10-01

MR 2742780 | Zbl 1217.49029

Bibliographie

[1] Acerbi, E. - Mingione, G., Regularity results for a class of functionals with nonstandard growth, Arch. Ration. Mech. Anal., 156, 2 (2001), 121-140. | MR 1814973 | Zbl 0984.49020

[2] Acerbi, E. - Mingione, G., Regularity results for a class of quasiconvex functionals with nonstandard growth, Ann. Sc. Norm. Pisa Cl. Sci. (4), 30 (2001), 311-339. | MR 1895714 | Zbl 1027.49031

[3] Acerbi, E. - Mingione, G., Regularity results for stationary electro-rheological fluids, Arch. Rational Mech. Anal., 164 (2002), 213-259. | MR 1930392 | Zbl 1038.76058

[4] Acerbi, E. - Mingione, G., Gradient estimates for the $p(x)$-laplacean system, J. Reine Angew. Math., 584 (2005), 117-148. | MR 2155087 | Zbl 1093.76003

[5] Campanato, S., Hölder continuity of the Solutions of Some Non-linear Elliptic Systems, Adv. Math., 48 (1983), 16-43. | MR 697613

[6] Chen, Y. - Levine, S. - Rao, R., Functionals with $p(x)$-growth in image processing, SIAM J. Appl. Math., 66, no. 4 (2006), 1383-1406. | MR 2246061 | Zbl 1102.49010

[7] Coscia, A. - Mingione, G., Hölder continuity of the gradient of $p(x)$ harmonic mappings, C. R. Acad. Sci. Paris, 328 (1) (1999), 363-368. | MR 1675954 | Zbl 0920.49020

[8] Cupini, G. - Fusco, N. - Petti, R., Hölder continuity of local minimizers, J. Math. Anal. Appl., 235 (1999), 578-597. | MR 1703712 | Zbl 0949.49022

[9] Diening, L., Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces $L^{p(\cdot)}$ and $W^{k,p(\cdot)}$, Math. Nachr., 268 (2004), 31-43. | MR 2054530 | Zbl 1065.46024

[10] Duzaar, F. - Gastel, A. - Grotowski, J., Optimal partial regularity for nonlinear elliptic systems of higher order, J. Math. Sci. Univ. Tokyo, 8 (2001), 463-499. | MR 1855456 | Zbl 0995.35017

[11] Edmunds, D. - Rákosník, J., Sobolev embeddings with variable exponent, Stud. Math., 143, No. 3 (2000), 267-293. | MR 1815935 | Zbl 0974.46040

[12] Edmunds, D. - Rákosník, J., Sobolev embeddings with variable exponent II, Math. Nachr., 246-27 (2002), 53-67. | MR 1944549

[13] Eleuteri, M., Hölder continuity results for a class of functionals with non standard growth, Boll. Unione Mat. Ital., 8, 7-B (2004), 129-157. | MR 2044264 | Zbl 1178.49045

[14] Eleuteri, M. - Habermann, J., Regularity results for a class of obstacle problems under non standard growth conditions, J. Math. Anal. Appl., 344 (2), (2008), 1120-1142. | MR 2426338 | Zbl 1147.49028

[15] Esposito, L. - Leonetti, F. - Mingione, G., Sharp regularity for functionals with $(p,q)$ growth, J. Differential Equations 204, no. 1 (2004), 5-55. | MR 2076158 | Zbl 1072.49024

[16] Fan, X. - Zhao, D., A class of DeGiorgi type and Hölder continuity, Nonlin. Anal., 36 (A) (1999), 295-318. | MR 1688232

[17] Giusti, E., Direct methods in the calculus of variations, Singapore: World Scientific, vii (2003). | MR 1962933 | Zbl 1028.49001

[18] Habermann, J., Partial regularity for minima of higher order functionals with $p(x)$ growth, Manuscripta Math., 126 (1) (2007), 1-40. | MR 2395246 | Zbl 1142.49017

[19] Habermann, J., Calderón-Zygmund estimates for higher order systems with $p(x)$ growth, Math. Z., 258 (2) (2008), 427-462. | MR 2357646 | Zbl 1147.42004

[20] Kovaâčik, O. - Rákosník, Ž. J., On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J., 41, no. 116 (1991).

[21] Marcellini, P., Regularity and existence of solutions of elliptic equations with $p - q$-growth conditions, J. Differ. Equations, 90 (1991), 1-30. | MR 1094446 | Zbl 0724.35043

[22] Musielak, J., Orlicz spaces and modular spaces, Springer, Berlin, 1983. | MR 724434 | Zbl 0557.46020

[23] Rajagopal, K. R. - Růžička, M., Mathematical modelling of electro-rheological fluids, Cont. Mech. Therm., 13 (2001), 59-78.

[24] Zhikov, V. V., On some variational problems, Russian J. Math. Phys., 5 (1997), 105-116. | MR 1486765 | Zbl 0917.49006