Partial continuity for elliptic problems

Foss, Mikil ; Mingione, Giuseppe

Annales de l'I.H.P. Analyse non linéaire, Tome 25 (2008), p. 471-503 / Harvested from Numdam

Détail

Publié le : 2008-01-01

MR 2422076 | Zbl 1153.35017

DOI : https://doi.org/10.1016/j.anihpc.2007.02.003

Bibliographie

[1] Acerbi E., Fusco N., Semicontinuity problems in the calculus of variations, Arch. Ration. Mech. Anal. 86 (1984) 125-145. | MR 751305 | Zbl 0565.49010

[2] Acerbi E., Fusco N., A regularity theorem for minimizers of quasiconvex integrals, Arch. Ration. Mech. Anal. 99 (1987) 261-281. | MR 888453 | Zbl 0627.49007

[3] Ball J., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal. 63 (1976/77) 337-403. | MR 475169 | Zbl 0368.73040

[4] Campanato S., Hölder continuity and partial Hölder continuity results for ${H}^{1,q}$-solutions of nonlinear elliptic systems with controlled growth, Rend. Sem. Mat. Fis. Milano 52 (1982) 435-472. | MR 802957 | Zbl 0576.35041

[5] Campanato S., Hölder continuity of the solutions of some non-linear elliptic systems, Adv. Math. 48 (1983) 16-43. | MR 697613 | Zbl 0519.35027

[6] Campanato S., On the nonlinear parabolic systems in divergence form. Hölder continuity and partial Hölder continuity of the solutions, Ann. Mat. Pura Appl. (4) 137 (1984) 83-122. | MR 772253 | Zbl 0704.35024

[7] Campanato S., A few recent results for differential systems under monotonicity conditions, Boll. Un. Mat. Ital. A (7) 2 (1988) 27-57. | MR 931643 | Zbl 0662.35039

[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] Duzaar F., Gastel A., Nonlinear elliptic systems with Dini continuous coefficients, Arch. Math. (Basel) 78 (2002) 58-73. | MR 1887317 | Zbl 1013.35028

[10] Duzaar F., Gastel A., Grotowski J.F., Partial regularity for almost minimizers of quasi-convex integrals, SIAM J. Math. Anal. 32 (2000) 665-687. | MR 1786163 | Zbl 0989.49026

[11] Duzaar F., Grotowski J.F., Optimal interior partial regularity for nonlinear elliptic systems: the method of A-harmonic approximation, Manuscripta Math. 103 (2000) 267-298. | MR 1802484 | Zbl 0971.35025

[12] Duzaar F., Kronz M., Regularity of ω-minimizers of quasi-convex variational integrals with polynomial growth, Differential Geom. Appl. 17 (2002) 139-152. | MR 1925762 | Zbl 1021.49026

[13] Duzaar F., Mingione G., Regularity for degenerate elliptic problems via p-harmonic approximation, Ann. Inst. H. Poincaré Anal. Non Linèaire 21 (2004) 735-766. | Numdam | MR 2086757 | Zbl 1112.35078

[14] Duzaar F., Steffen K., Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals, J. Reine Angew. Math. (Crelles J.) 546 (2002) 73-138. | MR 1900994 | Zbl 0999.49024

[15] Evans L.C., Quasiconvexity and partial regularity in the calculus of variations, Arch. Ration. Mech. Anal. 95 (1986) 227-252. | MR 853966 | Zbl 0627.49006

[16] Fonseca I., Fusco N., Regularity results for anisotropic image segmentation models, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997) 463-499. | Numdam | MR 1612389 | Zbl 0899.49018

[17] M. Foss, Global regularity for almost minimizers of nonconvex variational problems, Ann. Mat. Pura e Appl. (4), in press, doi:10.1007/s10231-007-0045-2. | Zbl pre05292632

[18] Giaquinta M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. | MR 717034 | Zbl 0516.49003

[19] Giaquinta M., Modica G., Partial regularity of minimizers of quasiconvex integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986) 185-208. | Numdam | MR 847306 | Zbl 0594.49004

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

[21] J. Habermann, A. Zatorska-Goldstein, Regularity for minimizers of functionals with nonstandard growth by A-harmonic approximation, Preprint, 2006. | MR 2408350 | Zbl pre05285410

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

[23] Kristensen J., Mingione G., The singular set of minima of integral functionals, Arch. Ration. Mech. Anal. 180 (2006) 331-398. | MR 2214961 | Zbl 1116.49010

[24] Kronz M., Partial regularity results for minimizers of quasiconvex functionals of higher order, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002) 81-112. | Numdam | MR 1902546 | Zbl 1010.49023

[25] Manfredi J.J., Regularity for minima of functionals with p-growth, J. Differential Equations 76 (1988) 203-212. | MR 969420 | Zbl 0674.35008

[26] J.J. Manfredi, Regularity of the gradient for a class of nonlinear possibly degenerate elliptic equations, Ph.D. Thesis. University of Washington, St. Louis.

[27] Mingione G., 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

[28] Morrey C.B., Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2 (1952) 25-53. | MR 54865 | Zbl 0046.10803

[29] Müller S., Variational models for microstructure and phase transitions, in: Calculus of Variations and Geometric Evolution Problems, Cetraro, 1996, Lecture Notes in Math., vol. 1713, Springer, 1999, pp. 85-210. | MR 1731640 | Zbl 0968.74050

[30] Rivière T., Everywhere discontinuous harmonic maps into spheres, Acta Math. 175 (1995) 197-226. | MR 1368247 | Zbl 0898.58011

[31] Šverák V., Yan X., Non-Lipschitz minimizers of smooth uniformly convex variational integrals, Proc. Natl. Acad. Sci. USA 99/24 (2002) 15269-15276. | MR 1946762 | Zbl 1106.49046

[32] Wolf J., Partial regularity of weak solutions to nonlinear elliptic systems satisfying a Dini condition, Z. Anal. Anwend. 20 (2001) 315-330. | MR 1846604 | Zbl 1163.35329