×

zbMATH — the first resource for mathematics

Regularity of minima: an invitation to the dark side of the calculus of variations. (English) Zbl 1164.49324
Summary: I am presenting a survey of regularity results for both minima of variational integrals, and solutions to non-linear elliptic, and sometimes parabolic, systems of partial differential equations. I will try to take the reader to the dark side…

MSC:
49N60 Regularity of solutions in optimal control
35J20 Variational methods for second-order elliptic equations
35J70 Degenerate elliptic equations
PDF BibTeX XML Cite
Full Text: DOI EuDML Link
References:
[1] E. Acerbi, N. Fusco: Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 86 (1984), 125–145. · Zbl 0565.49010
[2] E. Acerbi, N. Fusco: A regularity theorem for minimizers of quasiconvex integrals. Arch. Ration. Mech. Anal. 99 (1987), 261–281. · Zbl 0627.49007
[3] E. Acerbi, N. Fusco: Local regularity for minimizers of nonconvex integrals. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 16 (1989), 603–636. · Zbl 0829.49025
[4] E. Acerbi, N. Fusco: Partial regularity under anisotropic (p, q) growth conditions. J. Differ. Equations 107 (1994), 46–67. · Zbl 0807.49010
[5] E. Acerbi, G. Mingione: Regularity results for a class of functionals with non-standard growth. Arch. Ration. Mech. Anal. 156 (2001), 121–140. · Zbl 0984.49020
[6] E. Acerbi, G. Mingione: Regularity results for a class of quasiconvex functionals with nonstandard growth. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 30 (2001), 311–339. · Zbl 1027.49031
[7] E. Acerbi, G. Mingione: Regularity results for stationary electro-rheological fluids. Arch. Ration. Mech. Anal. 164 (2002), 213–259. · Zbl 1038.76058
[8] E. Acerbi, G. Mingione: Gradient estimates for the p(x)-Laplacean system. J. Reine Angew. Math. 584 (2005), 117–148. · Zbl 1093.76003
[9] E. Acerbi, G. Mingione: Gradient estimates for a class of parabolic systems. Duke Math. J. To appear.
[10] E. Acerbi, G. Mingione, and G. A. Seregin: Regularity results for parabolic systems related to a class of non Newtonian fluids. Ann. Inst. Henri Poincaré, Anal. Non Linèaire 21 (2004), 25–60. · Zbl 1052.76004
[11] Yu. A. Alkhutov: The Harnack inequality and the Hölder property of solutions of nonlinear elliptic equations with a nonstandard growth condition. Differ. Equations 33 (1997), 1653–1663. · Zbl 0949.35048
[12] F. J. Almgren: Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure. Ann. Math. 87 (1968), 321–391. · Zbl 0162.24703
[13] F. J. Almgren: Existence and Regularity Almost Everywhere of Solutions to Elliptic Variational Problems with Constraints. Mem. Am. Math. Soc. 165. Am. Math. Soc. (AMS), Providence, 1976.
[14] L. Ambrosio: Corso introduttivo alla teoria geometrica della misura ed alle superfici minime. Lecture Notes. Scuola Normale Superiore, Pisa, 1995. (In Italian.) · Zbl 0977.49028
[15] S. N. Antontsev, S. I. Shmarev: A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions. Nonlinear Anal., Theory Methods Appl. 60 (2005), 515–545. · Zbl 1066.35045
[16] S. N. Antontsev, S. I. Shmarev: Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions. · Zbl 1245.35033
[17] S. N. Antontsev, V. V. Zhikov: Higher integrability for parabolic equations of p(x; t)-Laplacian type. Adv. Differential Equations 10 (2005), 1053–1080. · Zbl 1122.35043
[18] G. Anzellotti: On the C 1,{\(\alpha\)}-regularity of {\(\omega\)}-minima of quadratic functionals. Boll. Unione Mat. Ital., VI. Ser., C, Anal. Funz. Appl. 2 (1983), 195–212. · Zbl 0522.49005
[19] A. A. Arkhipova: Partial regularity up to the boundary of weak solutions of elliptic systems with nonlinearity q greater than two. J. Math. Sci. 115 (2003), 2735–2746. · Zbl 1118.35319
[20] G. Aronsson, M. G. Crandall, and P. Juutinen: A tour of the theory of absolutely minimizing functions. Bull. Am. Math. Soc., New Ser. 41 (2004), 439–505. · Zbl 1150.35047
[21] K. Astala: Area distortion of quasiconformal mappings. Acta Math. 173 (1994), 37–60. · Zbl 0815.30015
[22] H. Attouch, C. Sbordone: Asymptotic limits for perturbed functionals of calculus of variations. Ric. Mat. 29 (1980), 85–124. · Zbl 0453.49007
[23] J. M. Ball: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63 (1977), 337–403. · Zbl 0368.73040
[24] J. M. Ball: Some open problems in elasticity. In: Geometry, Mechanics, and Dynamics (P. Newton, ed.). Springer-Verlag, New York, 2002, pp. 3–59. · Zbl 1054.74008
[25] J. M. Ball, V. J. Mizel: One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation. Arch. Ration. Mech. Anal. 90 (1985), 325–388. · Zbl 0585.49002
[26] J. M. Ball, F. Murat: W1,p -quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984), 225–253. · Zbl 0549.46019
[27] J. M. Ball, J. C. Currie, and P. J. Olver: Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Funct. Anal. 41 (1981), 135–174. · Zbl 0459.35020
[28] L. Beck: Partielle Regularität für Schwache Lösungen nichtlinearer elliptischer Systeme: der Subquadratische Fall. Diploma Thesis. Friedrich-Alexander-Universität, Erlangen-Nürnberg, 2005. (In German.)
[29] L. Beck: Boundary regularity results for systems with sub-linear growth. To appear.
[30] I. Benedetti, E. Mascolo: Regularity of minimizers for nonconvex vectorial integrals with p-q growth via relaxation methods. Abstr. Appl. Anal. (2004), 27–44. · Zbl 1067.49026
[31] A. Bensoussan, J. Frehse: Regularity Results for Nonlinear Elliptic Systems and Applications. Applied Mathematical Sciences 151. Springer-Verlag, Berlin, 2002. · Zbl 1055.35002
[32] J. J. Bevan: Polyconvexity and counterexamples to regularity in the multidimensional calculus of variations. PhD. Thesis. Oxford, 2003.
[33] J. Bevan: Singular minimizers of strictly polyconvex functionals in \(\mathbb{R}\)2{\(\times\)}2.Calc. Var. Partial Differ. Equ. 23 (2005), 347–372. · Zbl 1100.49014
[34] T. Bhattacharya, F. Leonetti: A new Poincaré inequality and its application to the regularity of minimizers of integral functionals with nonstandard growth. Nonlinear Anal., Theory Methods Appl. 17 (1991), 833–839. · Zbl 0779.49046
[35] T. Bhattacharya, F. Leonetti: W 2,2 regularity for weak solutions of elliptic systems with nonstandard growth. J. Math. Anal. Appl. 176 (1993), 224–234. · Zbl 0809.35008
[36] M. Bildhauer: Convex Variational Problems. Linear, Nearly Linear and Anisotropic Growth Conditions. Lecture Notes in Mathematics Vol. 1818. Springer-Verlag, Berlin, 2003. · Zbl 1033.49001
[37] M. Bildhauer: A priori gradient estimates for bounded generalized solutions of a class of variational problems with linear growth. J. Convex Anal. 9 (2002), 117–137. · Zbl 1011.49022
[38] M. Bildhauer, M. Fuchs: Partial regularity for variational integrals with (s, {\(\mu\)}, q)-growth. Calc. Var. Partial Differ. Equ. 13 (2001), 537–560. · Zbl 1018.49026
[39] M. Bildhauer, M. Fuchs: Partial regularity for a class of anisotropic variational integrals with convex hull property. Asymptotic Anal. 32 (2002), 293–315. · Zbl 1076.49018
[40] M. Bildhauer, M. Fuchs: C 1,{\(\alpha\)}-solutions to non-autonomous anisotropic variational problems. Calc. Var. Partial Differ. Equ. 24 (2005), 309–340. · Zbl 1101.49029
[41] M. Bildhauer, M. Fuchs, and X. Zhong: A lemma on the higher integrability of functions with applications to the regularity theory of two-dimensional generalized Newtonian fluids. Manuscr. Math. 116 (2005), 135–156. · Zbl 1116.49018
[42] S. S. Byun: Parabolic equations with BMO coe-cients in Lipschitz domains. J. Differ. Equations 209 (2005), 229–265. · Zbl 1061.35021
[43] L. Boccardo, P. Marcellini, and C. Sbordone: Lregularity for variational problems with sharp non-standard growth conditions. Boll. Unione Mat. Ital. VII. Ser. A 4 (1990), 219–225. · Zbl 0711.49058
[44] B. Bojarski, T. Iwaniec: Analytical foundations of the theory of quasiconformal mappings in \(\mathbb{R}\) n. Ann. Acad. Sci. Fenn., Ser. A I 8 (1983), 257–324. · Zbl 0548.30016
[45] B. Bojarski, C. Sbordone, and I. Wik: The Muckenhoupt class A1(R). Stud. Math. 101 (1992), 155–163. · Zbl 0808.42010
[46] E. Bombieri: Regularity theory for almost minimal currents. Arch. Ration. Mech. Anal. 78 (1982), 99–130. · Zbl 0485.49024
[47] M. Bonk, J. Heinonen: Smooth quasiregular mappings with branching. Publ. Math., Inst. Hautes É tud. Sci. 100 (2004), 153–170. · Zbl 1063.30021
[48] G. Buttazzo, M. Belloni: A survey of old and recent results about the gap phenomenon in the Calculus of Variations. In: Recent Developments in Well-posed Variational Problems 331 (R. Lucchetti et al., eds.). Kluwer Academic Publishers, Dordrecht, 1995, pp. 1–27. · Zbl 0852.49001
[49] G. Buttazzo, V. J. Mizel: Interpretation of the Lavrentiev Phenomenon by relaxation. J. Funct. Anal. 110 (1992), 434–460. · Zbl 0784.49006
[50] R. Caccioppoli: Limitazioni integrali per le soluzioni di un’equazione lineare ellitica a derivate parziali. Giorn. Mat. Battaglini, IV. Ser. 80 (1951), 186–212. (In Italian.) · Zbl 0043.31404
[51] L. A. Caffarelli, I. Peral: On W 1,p estimates for elliptic equations in divergence form. Commun. Pure Appl. Math. 51 (1998), 1–21. · Zbl 0906.35030
[52] S. Campanato: Hölder continuity of the solutions of some non-linear elliptic systems. Adv. Math. 48 (1983), 15–43. · Zbl 0519.35027
[53] S. Campanato: Elliptic systems with non-linearity q greater than or equal to two. Regularity of the solution of the Dirichlet problem. Ann. Mat. Pura Appl., IV. Ser. 147 (1987), 117–150. · Zbl 0635.35038
[54] S. Campanato: 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. · Zbl 0576.35041
[55] S. Campanato: Some new results on differential systems with monotonicity property. Boll. Unione Mat. Ital., VII. Ser. 2 (1988), 27–57. · Zbl 0662.35039
[56] M. Carozza, N. Fusco, and G. Mingione: Partial regularity of minimizers of quasiconvex integrals with subquadratic growth. Ann. Mat. Pura Appl., IV. Ser. 175 (1998), 141–164. · Zbl 0960.49025
[57] P. Celada, G. Cupini, and M. Guidorzi: Existence and regularity of minimizers of nonconvex integrals with p-q growth. ESAIM Control Optim. Calc. Var. To appear. · Zbl 1124.49031
[58] Y. Chen, S. Levine, and R. Rao: Functionals with p(x)-growth in image processing. Preprint. 2004.
[59] V. Chiadó Piat, A. Coscia: Hölder continuity of minimizers of functionals with variable growth exponent. Manuscr. Math. 93 (1997), 283–299. · Zbl 0878.49010
[60] F. Chiarenza, M. Frasca, and P. Longo: W 2,p -solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Trans. Am. Math. Soc. 336 (1993), 841–853. · Zbl 0818.35023
[61] M. Chipot, L. C. Evans: Linearization at infinity and Lipschitz estimates for certain problems in the calculus of variations. Proc. R. Soc. Edinb., Sect. A 102 (1986), 291–303. · Zbl 0602.49029
[62] A. Cianchi: Boundedness of solutions to variational problems under general growth conditions. Commun. Partial Differ. Equations 22 (1997), 1629–1646. · Zbl 0892.35048
[63] A. Cianchi: Local boundedness of minimizers of anisotropic functionals. Ann. Inst. Henri Poincaré, Anal. Non Linèaire 17 (2000), 147–168. · Zbl 0984.49019
[64] A. Cianchi, N. Fusco: Gradient regularity for minimizers under general growth conditions. J. Reine Angew. Math. 507 (1999), 15–36. · Zbl 0913.49024
[65] F. Colombini: Un teorema di regolarità alla frontiera per soluzioni di sistemi ellittici quasi lineari. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 25 (1971), 115–161. · Zbl 0211.13502
[66] A. Coscia, G. Mingione: Hölder continuity of the gradient of p(x)-harmonic mappings. C. R. Acad. Sci., Paris, Sér. I, Math. 328 (1999), 363–368. · Zbl 0920.49020
[67] D. Cruz-Uribe, C. J. Neugebauer: The structure of the reverse Hölder classes. Trans. Am. Math. Soc. 347 (1995), 2941–2960. · Zbl 0851.42016
[68] G. Cupini, N. Fusco, and R. Petti: Hölder continuity of local minimizers. J. Math. Anal. Appl. 235 (1999), 578–597. · Zbl 0949.49022
[69] G. Cupini, M. Guidorzi, and E. Mascolo: Regularity of minimizers of vectorial integrals with p-q growth. Nonlinear Anal., Theory Methods Appl. 54 (2003), 591–616. · Zbl 1027.49032
[70] G. Cupini, A. P. Migliorini: Hölder continuity for local minimizers of a nonconvex variational problem. J. Convex Anal. 10 (2003), 389–408. · Zbl 1084.49030
[71] B. Dacorogna: Quasiconvexity and relaxation of nonconvex problems in the calculus of variations. J. Funct. Anal. 46 (1982), 102–118. · Zbl 0547.49003
[72] B. Dacorogna: Direct Methods in the Calculus of Variations. Applied Mathematical Sciences 78. Springer-Verlag, Berlin, 1989. · Zbl 0703.49001
[73] B. Dacorogna, P. Marcellini: Implicit Partial Differential Equations. Progress in Nonlinear Differential Equations and Their Applications 37.Birkhäuser-Verlag, Boston, 1999.
[74] A. Dall’Aglio, E. Mascolo, and G. Papi: Local boundedness for minima of functionals with nonstandard growth conditions. Rend. Mat. Appl., VII Ser. 18 (1998), 305–326. · Zbl 0917.49010
[75] L. D’Apuzzo, C. Sbordone: Reverse Hölder inequalities. A sharp result. Rend. Mat. Appl., VII Ser. 10 (1990), 357–366.
[76] J. Daněček, O. John, and J. Stará: Interior C 1,{\(\gamma\)}-regularity for weak solutions of nonlinear second order elliptic systems. Math. Nachr. 276 (2004), 47–56. · Zbl 1174.35358
[77] E. De Giorgi: Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino, P. I., III. Ser. 3 (1957), 25–43. (In Italian.) · Zbl 0084.31901
[78] E. De Giorgi: Frontiere orientate di misura minima. Seminario di Matematica della Scuola Normale Superiore. Pisa, 1960–61. (In Italian.) · Zbl 0296.49031
[79] E. De Giorgi: Un esempio di estremali discontinue per un problema variazionale di tipo ellittico. Boll. Unione Mat. Ital., IV. Ser. 1 (1968), 135–137. (In Italian.) · Zbl 0155.17603
[80] E. DiBenedetto: Intrinsic Harnack type inequalities for solutions of certain degenerate parabolic equations. Arch. Ration. Mech. Anal. 100 (1988), 129–147. · Zbl 0708.35017
[81] E. DiBenedetto: Degenerate Parabolic Equations. Universitext. Springer-Verlag, New York, 1993. · Zbl 0794.35090
[82] E. DiBenedetto, A. Friedman: Hölder estimates for nonlinear degenerate parabolic systems. J. Reine Angew. Math. 357 (1985), 1–22. · Zbl 0549.35061
[83] E. DiBenedetto, J. J. Manfredi: On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems. Am. J. Math. 115 (1993), 1107–1134. · Zbl 0805.35037
[84] E. DiBenedetto, N. Trudinger: Harnack inequalities for quasiminima of variational integrals. Ann. Inst. Henri Poincaré, Anal. Non Linèaire 1 (1984), 295–308.
[85] L. Diening: Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math. 129 (2005), 657–700. · Zbl 1096.46013
[86] L. Diening, P. Hästö, and A. Nekvinda: Open problems in variable exponent Lebesgue and Sobolev spaces. In: FSDONA Proceedings, Milovy, Czech Republic, 2004 (P. Drábek, J. Rákosník, eds.). 2004, pp. 38–58.
[87] L. Diening, M. Råžička: Calderón-Zygmund operators on generalized Lebesgue spaces L p ({\(\cdot\)}) and problems related to uid dynamics. J. Reine Angew. Math. 563 (2003), 197–220. · Zbl 1072.76071
[88] G. Di Fazio: L p estimates for divergence form elliptic equations with discontinuous coefficients. Boll. Unione Mat. Ital., VII. Ser., A 10 (1996), 409–420. · Zbl 0865.35048
[89] A. Dolcini, L. Esposito, and N. Fusco: C 0,{\(\alpha\)}-regularity of {\(\omega\)}-minima. Boll. Unione Mat. Ital., VII. Ser., A 10 (1996), 113–125.
[90] G. Dolzmann, J. Kristensen: Higher integrability of minimizing Young measures. Calc. Var. Partial Differ. Equ. 22 (2005), 283–301. · Zbl 1092.49016
[91] F. Duzaar, A. Gastel: Nonlinear elliptic systems with Dini continuous coefficients. Arch. Math. 78 (2002), 58–73. · Zbl 1013.35028
[92] F. Duzaar, A. Gastel, and J. F. Grotowski: Partial regularity for almost minimizers of quasi-convex integrals. SIAM J. Math. Anal. 32 (2000), 665–687. · Zbl 0989.49026
[93] F. Duzaar, A. Gastel, and G. Mingione: Elliptic systems, singular sets and Dini continuity. Commun. Partial Differ. Equations 29 (2004), 1215–1240. · Zbl 1140.35415
[94] F. Duzaar, J. F. Grotowski: Optimal interior partial regularity for nonlinear elliptic systems: The method of A-harmonic approximation. Manuscr. Math. 103 (2000), 267–298. · Zbl 0971.35025
[95] F. Duzaar, J. F. Grotowski, and M. Kronz: Partial and full boundary regularity for minimizers of functionals with nonquadratic growth. J. Convex Anal. 11 (2004), 437–476. · Zbl 1066.49022
[96] F. Duzaar, J. F. Grotowski, and M. Kronz: Regularity of almost minimizers of quasiconvex variational integrals with subquadratic growth. Ann. Mat. Pura Appl., IV. Ser. 184 (2005), 421–448. · Zbl 1223.49040
[97] F. Duzaar, J. F. Grotowski, and K. Steffen: Optimal regularity results via A-harmonic approximation. In: Geometric Analysis and Nonlinear Partial Differential Equations (S. Hildebrandt, ed.). Springer-Verlag, Berlin, 2003, pp. 265–296. · Zbl 1290.35082
[98] F. Duzaar, J. Kristensen, and G. Mingione: The existence of regular boundary points for non-linear elliptic systems. J. Reine Angew. Math. To appear. · Zbl 1214.35021
[99] F. Duzaar, M. Kronz: Regularity of {\(\omega\)}-minimizers of quasi-convex variational integrals with polynomial growth. Differ. Geom. Appl. 17 (2002), 139–152. · Zbl 1021.49026
[100] F. Duzaar, G. Mingione: The p-harmonic approximation and the regularity of p-harmonic maps. Calc. Var. Partial Differ. Equ. 20 (2004), 235–256. · Zbl 1142.35433
[101] F. Duzaar, G. Mingione: Regularity for degenerate elliptic problems via p-harmonic approximation. Ann. Inst. Henri Poincaré, Anal. Non Linèaire 21 (2004), 735–766. · Zbl 1112.35078
[102] F. Duzaar, G. Mingione: Second order parabolic systems, optimal regularity, and singular sets of solutions. Ann. Inst. Henri Poincaré, Anal. Non Linèaire 22 (2005), 705–751. · Zbl 1099.35042
[103] F. Duzaar, G. Mingione: Non-autonomous functionals with (p, q)-growth: regularity in borderline cases. In prepration.
[104] F. Duzaar, K. Steffen: Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals. J. Reine Angew. Math. 546 (2002), 73–138. · Zbl 0999.49024
[105] I. Ekeland: On the variational principle. J. Math. Anal. Appl. 47 (1974), 324–353. · Zbl 0286.49015
[106] A. Elcrat, N. G. Meyers: Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions. Duke Math. J. 42 (1975), 121–136. · Zbl 0347.35039
[107] M. Eleuteri: Hölder continuity results for a class of functionals with non-standard growth. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 7 (2004), 129–157. · Zbl 1178.49045
[108] L. Esposito, F. Leonetti, and G. Mingione: Regularity for minimizers of functionals with p-q growth. Nonlinear Differ. Equ. Appl. 6 (1999), 133–148. · Zbl 0928.35044
[109] L. Esposito, F. Leonetti, and G. Mingione: Higher integrability for minimizers of integral functionals with (p, q) growth. J. Differ. Equations 157 (1999), 414–438. · Zbl 0939.49021
[110] L. Esposito, F. Leonetti, and G. Mingione: Regularity results for minimizers of irregular integrals with (p, q) growth. Forum Math. 14 (2002), 245–272. · Zbl 0999.49022
[111] L. Esposito, F. Leonetti, and G. Mingione: Sharp regularity for functionals with (p, q) growth. J. Differ. Equations 204 (2004), 5–55. · Zbl 1072.49024
[112] L. Esposito, G. Mingione: A regularity theorem for {\(\omega\)}-minimizers of integral functionals. Rend. Mat. Appl., VII. Ser. 19 (1999), 17–44. · Zbl 0949.49023
[113] L. Esposito, G. Mingione: Partial regularity for minimizers of convex integrals with L log L-growth. Nonlinear Differ. Equ. Appl. 7 (2000), 107–125. · Zbl 0954.49026
[114] L. Esposito, G. Mingione: Partial regularity for minimizers of degenerate polyconvex energies. J. Convex Anal. 8 (2001), 1–38. · Zbl 0977.49024
[115] L. C. Evans: Quasiconvexity and partial regularity in the calculus of variations. Arch. Ration. Mech. Anal. 95 (1986), 227–252. · Zbl 0627.49006
[116] L. C. Evans, R. F. Gariepy: On the partial regularity of energy-minimizing, area-preserving maps. Calc. Var. Partial Differ. Equ. 9 (1999), 357–372. · Zbl 0954.49024
[117] E. B. Fabes, D.W. Stroock: A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash. Arch. Ration. Mech. Anal. 96 (1986), 327–338. · Zbl 0652.35052
[118] X. Fan, D. Zhao: A class of De Giorgi type and Hölder continuity. Nonlinear Anal., Theory Methods Appl. 36 (1999), 295–318. · Zbl 0927.46022
[119] X. Fan, D. Zhao: The quasi-minimizer of integral functionals with m(x) growth conditions. Nonlinear Anal., Theory Methods Appl. 39 (2000), 807–816. · Zbl 0943.49029
[120] D. Faraco, P. Koskela, and X. Zhong: Mappings of finite distortion: the degree of regularity. Adv. Math. 190 (2005), 300–318. · Zbl 1075.30012
[121] V. Ferone, N. Fusco: Continuity properties of minimizers of integral functionals in a limit case. J. Math. Anal. Appl. 202 (1996), 27–52. · Zbl 0865.49008
[122] I. Fonseca, N. Fusco: Regularity results for anisotropic image segmentation models. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 24 (1997), 463–499. · Zbl 0899.49018
[123] I. Fonseca, N. Fusco, P. Marcellini: An existence result for a nonconvex variational problem via regularity. ESAIM, Control Optim. Calc. Var. 7 (2002), 69–95. · Zbl 1044.49011
[124] I. Fonseca, G. Leoni, and J. Malý: Weak continuity and lower semicontinuity results for determinants. Arch. Ration. Mech. Anal. 178 (2005), 411–448. · Zbl 1081.49013
[125] I. Fonseca, J. Malý: Relaxation of multiple integrals below the growth exponent. Ann. Inst. Henri Poincaré, Anal. Non Linèaire 14 (1997), 309–338. · Zbl 0868.49011
[126] I. Fonseca, J. Malý, and G. Mingione: Scalar minimizers with fractal singular sets. Arch. Ration. Mech. Anal. 172 (2004), 295–307. · Zbl 1049.49015
[127] M. Foss: Examples of the Lavrentiev Phenomenon with continuous Sobolev exponent dependence. J. Convex Anal. 10 (2003), 445–464. · Zbl 1084.49002
[128] M. Foss: A condition sufficient for the partial regularity of minimizers in two-dimensional nonlinear elasticity. In: The p-harmonic Equation and Recent Advances in Analysis. Contemp. Math. 370 (P. Poggi-Corradini, ed.). Amer. Math. Soc., Providence, 2005. · Zbl 1082.35149
[129] M. Foss: Global regularity for almost minimizers of nonconvex variational problems. · Zbl 1223.49041
[130] M. Foss, W. J. Hrusa, and V. J. Mizel: The Lavrentiev gap phenomenon in nonlinear elasticity. Arch. Ration. Mech. Anal. 167 (2003), 337–365. · Zbl 1090.74010
[131] M. Foss, W. J. Hrusa, and V. J. Mizel: The Lavrentiev phenomenon in nonlinear elasticity. J. Elasticity 72 (2003), 173–181. · Zbl 1085.74006
[132] J. Frehse: A note on the Hölder continuity of solutions of variational problems. Abh. Math. Semin. Univ. Hamb. 43 (1975), 59–63. · Zbl 0316.49008
[133] J. Frehse, G. A. Seregin: Regularity of Solutions to Variational Problems of the Deformation Theory of Plasticity with Logarithmic Hardening. Amer. Math. Soc. Transl. Ser. 2, 193.Amer. Math. Soc., Providence, 1999.
[134] M. Fuchs: Regularity theorems for nonlinear systems of partial differential equations under natural ellipticity conditions. Analysis 7 (1987), 83–93. · Zbl 0624.35032
[135] M. Fuchs, G. Li: Global gradient bounds for relaxed variational problems. Manuscr. Math. 92 (1997), 287–302. · Zbl 0892.49027
[136] M. Fuchs, G. Mingione: Full C 1,{\(\alpha\)}-regularity for free and constrained local minimizers of elliptic variational integrals with nearly linear growth. Manuscr. Math. 102 (2000), 227–250. · Zbl 0995.49023
[137] M. Fuchs, G. A. Seregin: Some remarks on non-Newtonian uids including nonconvex perturbations of the Bingham and Powell-Eyring model for viscoplastic fluids. Math. Models Methods Appl. Sci. 7 (1997), 405–433. · Zbl 0882.76007
[138] M. Fuchs, G. A. Seregin: A regularity theory for variational integrals with L ln L-growth. Calc. Var. Partial Differ. Equ. 6 (1998), 171–187. · Zbl 0929.49022
[139] M. Fuchs, G. A. Seregin: Hölder continuity for weak extremals of some two-dimensional variational problems related to nonlinear elasticity. Adv. Math. Sci. Appl. 7 (1997), 413–425. · Zbl 0929.49021
[140] M. Fuchs, G. A. Seregin: Partial regularity of the deformation gradient for some model problems in nonlinear two-dimensional elasticity. St. Petersbg. Math. J. 6 (1995), 1229–1248.
[141] N. Fusco: Quasiconvexity and semicontinuity for higher-order multiple integrals. Ric. Mat. 29 (1980), 307–323.
[142] N. Fusco, J. E. Hutchinson: C 1,{\(\alpha\)}-partial regularity of functions minimising quasiconvex integrals. Manuscr. Math. 54 (1986), 121–143. · Zbl 0587.49005
[143] N. Fusco, J. E. Hutchinson: Partial regularity for minimisers of certain functionals having nonquadratic growth. Ann. Mat. Pura Appl., IV. Ser. 155 (1989), 1–24. · Zbl 0698.49001
[144] N. Fusco, J. E. Hutchinson: Partial regularity in problems motivated by nonlinear elasticity. SIAM J. Math. Anal. 22 (1991), 1516–1551. · Zbl 0744.35014
[145] N. Fusco, J. E. Hutchinson: Partial regularity and everywhere continuity for a model problem from non-linear elasticity. J. Aust. Math. Soc., Ser. A 57 (1994), 158–169. · Zbl 0864.35032
[146] N. Fusco, C. Sbordone: Higher integrability from reverse Jensen inequalities with different supports. In: Partial Differential Equations and the Calculus of Variations. Essays in Honor of Ennio De Giorgi. Birkhä user-Verlag, Boston, 1989, pp. 541–562.
[147] N. Fusco, C. Sbordone: Local boundedness of minimizers in a limit case. Manuscr. Math. 69 (1990), 19–25. · Zbl 0722.49012
[148] N. Fusco, C. Sbordone: Higher integrability of the gradient of minimizers of functionals with nonstandard growth conditions. Commun. Pure Appl. Math. 43 (1990), 673–683. · Zbl 0727.49021
[149] N. Fusco, C. Sbordone: Some remarks on the regularity of minima of anisotropic integrals. Commun. Partial Differ. Equations 18 (1993), 153–167. · Zbl 0795.49025
[150] F. W. Gehring: The L p -integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130 (1973), 265–277. · Zbl 0258.30021
[151] M. Giaquinta: A counter-example to the boundary regularity of solutions to quasilinear systems. Manuscr. Math. 24 (1978), 217–220. · Zbl 0373.35027
[152] M. Giaquinta: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Annals of Mathematics Studies 105. Princeton University Press, Princeton, 1983. · Zbl 0516.49003
[153] M. Giaquinta: Direct methods for regularity in the calculus of variations. In: Nonlinear Partial Differential Equations and Their Applications. Collège de France seminar, Vol. VI (Paris, 1982/1983). Res. Notes in Math. 109. Pitman, Boston, 1984, pp. 258–274.
[154] M. Giaquinta: The problem of the regularity of minimizers. In: Proceedings of the International Congress of Mathematicians, Vol. 2 (Berkeley, California, 1986). Amer. Math. Soc., Providence, 1987, pp. 1072–1083.
[155] M. Giaquinta: Growth conditions and regularity. A counterexample. Manuscr. Math. 59 (1987), 245–248. · Zbl 0638.49005
[156] M. Giaquinta: Introduction to Regularity Theory for Nonlinear Elliptic Systems. Lectures in Mathematics. Birkhäuser-Verlag, Basel, 1993. · Zbl 0786.35001
[157] M. Giaquinta, E. Giusti: On the regularity of the minima of variational integrals. Acta Math. 148 (1982), 31–46. · Zbl 0494.49031
[158] M. Giaquinta, E. Giusti: Differentiability of minima of nondifferentiable functionals. Invent. Math. 72 (1983), 285–298. · Zbl 0513.49003
[159] M. Giaquinta, E. Giusti: Global C 1,{\(\alpha\)}-regularity for second order quasilinear elliptic equations in divergence form. J. Reine Angew. Math. 351 (1984), 55–65. · Zbl 0528.35014
[160] M. Giaquinta, G. Modica: Almost-everywhere regularity for solutions of nonlinear elliptic systems. Manuscr. Math. 28 (1979), 109–158. · Zbl 0411.35018
[161] M. Giaquinta, G. Modica: Regularity results for some classes of higher order nonlinear elliptic systems. J. Reine Angew. Math. 311/312 (1979), 145–169.
[162] M. Giaquinta, G. Modica: Remarks on the regularity of the minimizers of certain degenerate functionals. Manuscr. Math. 57 (1986), 55–99. · Zbl 0607.49003
[163] M. Giaquinta, M. Struwe: On the partial regularity of weak solutions on nonlinear parabolic systems. Math. Z. 179 (1982), 437–451. · Zbl 0477.35028
[164] D. Gilbarg, N. S. Trudinger: Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-Heidelberg-New York, 1977. · Zbl 0361.35003
[165] E. Giusti: Direct methods in the calculus of variations. World Scientific, Singapore, 2003. · Zbl 1028.49001
[166] E. Giusti, M. Miranda: Un esempio di soluzioni discontinue per un problema di minimo relativo ad un integrale regolare del calcolo delle variazioni. Boll. Unione Mat. Ital., IV. Ser. 1 (1968), 219–226. (In Italian.) · Zbl 0155.44501
[167] E. Giusti, M. Miranda: Sulla regolarità delle soluzioni deboli di una classe di sistemi ellittici quasi-lineari. Arch. Ration. Mech. Anal. 31 (1968), 173–184. (In Italian.) · Zbl 0167.10703
[168] M. Gromov: Partial Differential Relations. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, Berlin, 1986. · Zbl 0651.53001
[169] J. F. Grotowski: Boundary regularity for nonlinear elliptic systems. Calc. Var. Partial Differ. Equ. 15 (2002), 353–388. · Zbl 1148.35315
[170] J. F. Grotowski: Boundary regularity for quasilinear elliptic systems. Commun. Partial Differ. Equations 27 (2002), 2491–2512. · Zbl 1129.35352
[171] C. Hamburger: Regularity of differential forms minimizing degenerate elliptic functionals. J. Reine Angew. Math. 431 (1992), 7–64. · Zbl 0776.35006
[172] C. Hamburger: Quasimonotonicity, regularity and duality for nonlinear systems of partial differential equations. Ann. Mat. Pura Appl., IV. Ser. 169 (1995), 321–354. · Zbl 0852.35031
[173] C. Hamburger: Partial regularity for minimizers of variational integrals with discontinuous integrands. Ann. Inst. Henri Poincaré, Anal. Non Linèaire 13 (1996), 255–282. · Zbl 0863.35022
[174] C. Hamburger: Partial boundary regularity of solutions of nonlinear superelliptic systems. To appear.
[175] C. Hamburger: Optimal regularity of minimzers of quasiconvex variational integrals. ESAIM Control Optim. Calc. Var. To appear.
[176] W. Hao, S. Leonardi, and J. Nečas: An example of irregular solution to a nonlinear Euler-Lagrange elliptic system with real analytic coefficients. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 23 (1996), 57–67. · Zbl 0864.35031
[177] R. M. Hardt: Singularities of harmonic maps. Bull. Am. Math. Soc., New Ser. 34 (1997), 15–34. · Zbl 0871.58026
[178] R. M. Hardt, F. G. Lin, and C. Y. Wang: The p-energy minimality of x/|x|. Commun. Anal. Geom. 6 (1998), 141–152. · Zbl 0922.58015
[179] P. Harjulehto, P. Hästö, and M. Koskenoja: The Dirichlet energy integral on intervals in variable exponent Sobolev spaces. Z. Anal. Anwend. 22 (2003), 911–923. · Zbl 1046.46027
[180] P. Hästö: Counter-examples of regularity in variable exponent Sobolev spaces. In: The p-harmonic Equation and Recent Advances in Analysis. Contemp. Math. 370 (P. Poggi-Corradini, ed.). Amer. Math. Soc., Providence, 2005, pp. 133–143.
[181] J. Heinonen, T. Kilpeläinen, and O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs. Clarendon Press, Oxford, 1993. · Zbl 0780.31001
[182] S. Hildebrandt, H. Kaul, and K.-O. Widman: An existence theorem for harmonic mappings of Riemanninan manifolds. Acta Math. 138 (1977), 1–16. · Zbl 0356.53015
[183] S. Hildebrandt, K.-O. Widman: Some regularity results for quasilinear elliptic systems of second order. Math. Z. 142 (1975), 67–86. · Zbl 0317.35040
[184] M.-C. Hong: Existence and partial regularity in the calculus of variations. Ann. Mat. Pura Appl., IV. Ser. 149 (1987), 311–328. · Zbl 0648.49008
[185] M.-C. Hong: Some remarks on the minimizers of variational integrals with nonstandard growth conditions. Boll. Unione Mat. Ital., VII. Ser., A 6 (1992), 91–101. · Zbl 0768.49022
[186] M.-C. Hong: On the minimality of the p-harmonic map x/B n n. Calc. Var. Partial Differ. Equ. 13 (2001), 459–468. · Zbl 0999.58009
[187] P.-A. Ivert: Regularitätsuntersuchungen von Lösungen elliptischer Systeme von quasilinearen Differentialgleichungen zweiter Ordnung. Manuscr. Math. 30 (1979), 53–88. (In German.) · Zbl 0429.35033
[188] P.-A. Ivert: Partial regularity of vector valued functions minimizing variational integrals. Preprint Univ. Bonn. 1982.
[189] T. Iwaniec: Projections onto gradient-fields and L p -estimates for degenerated elliptic operators. Stud. Math. 75 (1983), 293–312. · Zbl 0552.35034
[190] T. Iwaniec: p-harmonic tensors and quasiregular mappings. Ann. Math. 136 (1992), 589–624. · Zbl 0785.30009
[191] T. Iwaniec: The Gehring lemma. In: Quasiconformal Mappings and Analysis, Ann Arbor, 1995 (P. Duren, ed.). Springer-Verlag, New York, 1998, pp. 181–204. · Zbl 0888.30017
[192] T. Iwaniec, G. Martin: Geometric Function Theory and Non-linear Analysis. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2001.
[193] T. Iwaniec, C. Sbordone: Quasiharmonic-fields. Ann. Inst. Henri Poincaré, Anal. Non Linè aire 18 (2001), 519–572. · Zbl 1068.30011
[194] O. John, J. Malý, and J. Stará: Nowhere continuous solutions to elliptic systems. Commentat. Math. Univ. Carol. 30 (1989), 33–43. · Zbl 0691.35024
[195] J. Jost, M. Meier: Boundary regularity for minima of certain quadratic functionals. Math. Ann. 262 (1983), 549–561. · Zbl 0501.49009
[196] V. Kokilashvili, S. Samko: Maximal and fractional operators in weighted L p(x) spaces. Rev. Mat. Iberoam. 20 (2004), 493–515. · Zbl 1099.42021
[197] J. Kinnunen: Sharp Results on Reverse Hölder Inequalities. Ann. Acad. Sci. Fenn. Ser. A I. Math. Dissertationes No 95. Suomalainen Tiedeakatemia, Helsinki, 1994. · Zbl 0816.26008
[198] J. Kinnunen, J. L. Lewis: Higher integrability for parabolic systems of p-Laplacian type. Duke Math. J. 102 (2000), 253–271. · Zbl 0994.35036
[199] J. Kinnunen, S. Zhou: A local estimate for nonlinear equations with discontinuous coefficients. Commun. Partial Differ. Equations 24 (1999), 2043–2068. · Zbl 0941.35026
[200] B. Kirchheim, S. Müller, and V. Šverák: Studying nonlinear pde by geometry in matrix space. In: Geometric Analysis and Nonlinear Partial Differential Equations (S. Hildebrandt, ed.). Springer-Verlag, Berlin, 2003, pp. 347–395. · Zbl 1290.35097
[201] A. Koshelev: Regularity Problem for Quasilinear Elliptic and Parabolic Systems. Lecture Notes in Mathematics 1614. Springer-Verlag, Berlin, 1995. · Zbl 0847.35023
[202] J. Kristensen: On the non-locality of quasiconvexity. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 16 (1999), 1–13. · Zbl 0932.49015
[203] J. Kristensen: Lower semicontinuity in spaces of weakly differentiable functions. Math. Ann. 313 (1999), 653–710. · Zbl 0924.49012
[204] J. Kristensen, G. Mingione: The singular set of {\(\omega\)}-minima. Arch. Ration. Mech. Anal. 177 (2005), 93–114. · Zbl 1082.49036
[205] J. Kristensen, G. Mingione: Non-differentiable functionals and singular sets of minima. C. R. Acad. Sci. Paris 340 (2005), 93–98. · Zbl 1058.49012
[206] J. Kristensen, G. Mingione: The singular set of minima of integral functionals. Arch. Ration. Mech. Anal. 180 (2006), 331–398. · Zbl 1116.49010
[207] J. Kristensen, G. Mingione: The singular set of Lipschitzian minima of multiple integrals. Arch. Ration. Mech. Anal. To appear. · Zbl 1114.49038
[208] J. Kristensen, G. Mingione: The singular set of solutions to elliptic problems with rough coefficients. In preparation. · Zbl 1082.49036
[209] J. Kristensen, A. Taheri: Partial regularity of strong local minimizers in the multidimensional calculus of variations. Arch. Ration. Mech. Anal. 170 (2003), 63–89. · Zbl 1030.49040
[210] M. Kronz: Quasimonotone systems of higher order. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 6 (2003), 459–480. · Zbl 1150.35385
[211] M. Kronz: Boundary regularity for almost minimizers of quasiconvex variational problems. Nonlinear Differ. Equations Appl. 12 (2005), 351–382. · Zbl 1116.49019
[212] M. Kronz: Habilitation Thesis. University of Erlangen-Nürneberg, Erlangen, 2006
[213] O. A. Ladyzhenskaya, N. N. Ural’tseva: Linear and Quasilinear Elliptic Equations. Mathematics in Science and Engineering 46. Academic Press, New York-London, 1968.
[214] O. A. Ladyzhenskaya, N. N. Ural’tseva: Local estimates for gradients of solutions of non-uniformly elliptic and parabolic equations. Commun. Pure Appl. Math. Ser. IX 23 (1970), 677–703. · Zbl 0193.07202
[215] O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva: Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, Vol. 23. Amer. Math. Soc., Providence, 1968.
[216] F. Leonetti: Maximum principle for vector-valued minimizers of some integral functionals. Boll. Unione Mat. Ital., VII. Ser., A 5 (1991), 51–56. · Zbl 0729.49015
[217] F. Leonetti: Higher differentiability for weak solutions of elliptic systems with nonstandard growth conditions. Ric. Mat. 42 (1993), 101–122. · Zbl 0855.35022
[218] F. Leonetti: Higher integrability for minimizers of integral functionals with nonstandard growth. J. Differ. Equations 112 (1994), 308–324. · Zbl 0813.49030
[219] F. Leonetti: Regularity results for minimizers of integral functionals with nonstandard growth. Atti Sem. Mat. Fis. Univ. Modena 43 (1995), 425–429. · Zbl 0851.49025
[220] F. Leonetti: Pointwise estimates for a model problem in nonlinear elasticity. Forum Mathematicum 18 (2006), 529–535. · Zbl 1125.49029
[221] F. Leonetti, V. Nesi: Quasiconformal solutions to certain-first order systems and the proof of a conjecture of G.W. Milton. J. Math. Pures Appl. 76 (1997), 109–124. · Zbl 0869.35019
[222] G. M. Lieberman: Gradient estimates for a class of elliptic systems. Ann. Mat. Pura Appl., IV. Ser. 164 (1993), 103–120. · Zbl 0819.35019
[223] G. M. Lieberman: The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations. Commun. Partial Differ. Equations 16 (1991), 311–361. · Zbl 0742.35028
[224] G. M. Lieberman: On the regularity of the minimizer of a functional with exponential growth. Commentat. Math. Univ. Carol. 33 (1992), 45–49. · Zbl 0776.49026
[225] G. M. Lieberman: Gradient estimates for a new class of degenerate elliptic and parabolic equations. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 21 (1994), 497–522. · Zbl 0839.35018
[226] G. M. Lieberman: Gradient estimates for anisotropic elliptic equations. Adv. Differ. Equ. 10 (2005), 767–182.
[227] P. Lindqvist: On the definition and properties of p-superharmonic functions. J. Reine Angew. Math. 365 (1986), 67–79. · Zbl 0572.31004
[228] J.-L. Lions: Quelques méthodes de résolution des problèmes aux limites non liné aires. Dunod, Gauthier-Villars, Paris, 1969. (In French.)
[229] G. Lucas & co.: ”Star Wars Episode 5: The empire strikes back” – Act, Darth Vader: ”Come with me to the Dark Side...”.
[230] J. Malý, W. P. Ziemer: Fine Regularity of Solutions of Elliptic Partial Differential Equations. Mathematical Surveys and Monographs, 51. Amer. Math. Soc., Providence, 1997. · Zbl 0882.35001
[231] J. J. Manfredi: Regularity for minima of functionals with p-growth. J. Differ. Equations 76 (1988), 203–212. · Zbl 0674.35008
[232] J. J. Manfredi: Regularity of the gradient for a class of nonlinear possibly degenerate elliptic equations. PhD. Thesis. University of Washington, St. Louis.
[233] P. Marcellini: On the definition and the lower semicontinuity of certain quasiconvex integrals. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3 (1986), 391–409. · Zbl 0609.49009
[234] P. Marcellini: Un esemple de solution discontinue d’un problème variationnel dans le case scalaire. Ist. Mat. ”U. Dini” No. 11, Firenze, 1987. (In Italian.)
[235] P. Marcellini: Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions. Arch. Ration. Mech. Anal. 105 (1989), 267–284. · Zbl 0667.49032
[236] P. Marcellini: Regularity and existence of solutions of elliptic equations with p, q-growth conditions. J. Differ. Equations 90 (1991), 1–30. · Zbl 0724.35043
[237] P. Marcellini: Regularity for elliptic equations with general growth conditions. J. Differ. Equations 105 (1993), 296–333. · Zbl 0812.35042
[238] P. Marcellini: Everywhere regularity for a class of elliptic systems without growth conditions. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 23 (1996), 1–25. · Zbl 0922.35031
[239] P. Marcellini: Regularity for some scalar variational problems under general growth conditions. J. Optimization Theory Appl. 90 (1996), 161–181. · Zbl 0901.49030
[240] P. Marcellini: Alcuni recenti sviluppi nei problemi 19-esimo e 20-esimo di Hilbert. Boll. Unione Mat. Ital., VII. Ser., A 11 (1997), 323–352. (In Italian.)
[241] P. Marcellini, G. Papi: Nonlinear elliptic systems with general growth. J. Differ. Equations 221 (2006), 412–443. · Zbl 1330.35131
[242] P. Marcellini, C. Sbordone: On the existence of minima of multiple integrals of the calculus of variations. J. Math. Pures Appl., IX. Sér. 62 (1983), 1–9. · Zbl 0516.49011
[243] E. Mascolo, G. Migliorini: Everywhere regularity for vectorial functionals with general growth. ESAIM, Control Optim. Calc. Var. 9 (2003), 399–418. · Zbl 1066.49023
[244] E. Mascolo, G. Papi: Local boundedness of minimizers of integrals of the calculus of variations. Ann. Mat. Pura Appl., IV. Ser. 167 (1994), 323–339. · Zbl 0819.49023
[245] E. Mascolo, G. Papi: Harnack inequality for minimizers of integral functionals with general growth conditions. NoDEA, Nonlinear Differ. Equ. Appl. 3 (1996), 231–244. · Zbl 0855.49027
[246] P. Mattila: Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability. Cambridge Studies in Adv. Math., 44.Cambridge University Press, Cambridge, 1995. · Zbl 0819.28004
[247] V. Maz’ya: Examples of nonregular solutions of quasilinear elliptic equations with analytic coefficients. Funct. Anal. Appl. 2 (1968), 230–234. · Zbl 0179.43601
[248] G. Mingione: The singular set of solutions to non-differentiable elliptic systems. Arch. Ration. Mech. Anal. 166 (2003), 287–301. · Zbl 1142.35391
[249] G. Mingione: Bounds for the singular set of solutions to non linear elliptic systems. Calc. Var. Partial Differ. Equ. 18 (2003), 373–400. · Zbl 1045.35024
[250] G. Mingione, D. Mucci: Integral functionals and the gap problem: sharp bounds for relaxation and energy concentration. SIAM J. Math. Anal. 36 (2005), 1540–1579. · Zbl 1080.49014
[251] G. Mingione, F. Siepe: Full C 1,{\(\alpha\)}-regularity for minimizers of integral functionals with L log L-growth. Z. Anal. Anwend. 18 (1999), 1083–1100. · Zbl 0954.49025
[252] C. B. Morrey: Review to [77]. Mathematical Reviews MR0093649 (20 #172).
[253] C. B. Morrey: Second order elliptic equations in several variables and Hölder continuity. Math. Z. 72 (1959), 146–164. · Zbl 0094.07802
[254] C. B. Morrey: Quasi-convexity and the lower semicontinuity of multiple integrals. Pac. J. Math. 2 (1952), 25–53. · Zbl 0046.10803
[255] C. B. Morrey: Multiple Integrals in the Calculus of Variations. Die Grundlehren der mathematischen Wissenschaften, 130. Springer-Verlag, Berlin-Heidelberg-New York, 1966.
[256] C. B. Morrey: Partial regularity results for non-linear elliptic systems. J. Math. Mech. 17 (1968), 649–670. · Zbl 0175.11901
[257] G. Moscariello, L. Nania: Hölder continuity of minimizers of functionals with nonstandard growth conditions. Ric. Mat. 40 (1991), 259–273. · Zbl 0773.49019
[258] J. Moser: A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations. Commun. Pure Appl. Math. 13 (1960), 457–468. · Zbl 0111.09301
[259] J. Moser: On Harnack’s theorem for elliptic differential equations. Commun. Pure Appl. Math. 14 (1961), 577–591. · Zbl 0111.09302
[260] J. Moser: A Harnack inequality for parabolic differential equations. Commun. Pure Appl. Math. 17 (1964), 101–134. · Zbl 0149.06902
[261] S. Müller: Variational models for microstructure and phase transitions. In: Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996). Lecture Notes in Math., 1713 (S. Hildebrandt, ed.). Springer-Verlag, Berlin, 1999, pp. 85–210.
[262] S. Müller, V. Šverák: Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. Math. 157 (2003), 715–742. · Zbl 1083.35032
[263] J. Musielak: Orlicz spaces and Modular spaces. Lecture Notes in Mathematics, 1034. Springer-Verlag, Berlin, 1983. · Zbl 0557.46020
[264] J. Nash: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80 (1958), 931–954. · Zbl 0096.06902
[265] J. Nečas: Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity. In: Theor. Nonlin. Oper., Constr. Aspects. Proc. 4th Int. Summer School. Akademie-Verlag, Berlin, 1975, pp. 197–206.
[266] J. Nečas, O. John, and J. Stará: Counterexample to the regularity of weak solution of elliptic systems. Commentat. Math. Univ. Carol. 21 (1980), 145–154. · Zbl 0442.35034
[267] L. Nirenberg: Remarks on strongly elliptic partial differential equations. Commun. Pure Appl. Math. 8 (1955), 649–675. · Zbl 0067.07602
[268] D. Phillips: On one-homogeneous solutions to elliptic systems in two dimensions. C. R., Math., Acad. Sci. Paris 335 (2002), 39–42. · Zbl 1006.35031
[269] L. Piccinini, S. Spagnolo: On the Hölder continuity of solutions of second order elliptic equations in two variables. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 26 (1972), 391–402. · Zbl 0237.35028
[270] K. R. Rajagopal, T. Ružička: Mathematical modeling of electrorheological materials. Contin. Mech. and Thermodyn. 13 (2001), 59–78. · Zbl 0971.76100
[271] M. M. Rao, Z. D. Ren: Theory of Orlicz spaces. Marcel Dekker, New York, 1991.
[272] J.-P. Raymond: Lipschitz regularity of solutions of some asymptotically convex problems. Proc. R. Soc. Edinb., Sect. A 117 (1991), 59–73. · Zbl 0725.49012
[273] T. Rivière: Everywhere discontinuous harmonic maps into spheres. Acta Math. 175 (1995), 197–226. · Zbl 0898.58011
[274] M. Ružička: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, 1748. Springer-Verlag, Berlin, 2000. · Zbl 0962.76001
[275] A. Salli: On the Minkowski dimension of strongly porous fractal sets in \(\mathbb{R}\)n. Proc. Lond. Math. Soc., III. Ser. 62 (1991), 353–372. · Zbl 0716.28006
[276] S. Samko: On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators. Integral Transforms Spec. Funct. 16 (2005), 461–482. · Zbl 1069.47056
[277] C. Sbordone: Rearrangement of functions and reverse Hölder inequalities. Ennio De Giorgi colloq., H. Poincaré Inst., Paris, 1983. Res. Notes Math. 125 (1985), 139–148.
[278] R. Schoen, K. Uhlenbeck: A regularity theory for harmonic maps. J. Differ. Geom. 17 (1982), 307–335. · Zbl 0521.58021
[279] J. Serrin: Local behavior of solutions of quasi-linear equations. Acta Math. 111 (1964), 247–302. · Zbl 0128.09101
[280] L. Simon: Global estimates of Hölder continuity for a class of divergence-form elliptic equations. Arch. Ration. Mech. Anal. 56 (1974), 253–272. · Zbl 0295.35027
[281] L. Simon: Interior gradient bounds for non-uniformly elliptic equations. Indiana Univ. Math. J. 25 (1976), 821–855. · Zbl 0346.35016
[282] L. Simon: Rectifiability of the singular set of energy minimizing maps. Calc. Var. Partial Differ. Equ. 3 (1995), 1–65. · Zbl 0818.49023
[283] L. Simon: Lectures on Regularity and Singularities of Harmonic Maps. Birkhäuser-Verlag, Basel-Boston-Berlin, 1996.
[284] J. Souček: Singular solutions to linear elliptic systems. Commentat. Math. Univ. Carol. 25 (1984), 273–281. · Zbl 0564.35008
[285] G. Stampacchia: Problemi al contorno ellitici, con dati discontinui, dotati di soluzionie hö lderiane. Ann. Mat. Pura Appl., IV. Ser. 51 (1960), 1–37. (In Italian.) · Zbl 0204.42001
[286] G. Stampacchia: Hilbert’s twenty-third problem: extensions of the calculus of variations. In: Mathematical developments arising from Hilbert problems. Proc. Symp. Pure Math. 28, De Kalb 1974. 1976, pp. 611–628.
[287] P. Sternberg, G. Williams, and W. P. Ziemer: Existence, uniqueness, and regularity for functions of least gradient. J. Reine Angew. Math. 430 (1992), 35–60. · Zbl 0756.49021
[288] E. W. Stredulinsky: Higher integrability from reverse Hölder inequalities. Indiana Univ. Math. J. 29 (1980), 407–413. · Zbl 0442.35064
[289] B. Stroffolini: Global boundedness of solutions of anisotropic variational problems. Boll. Unione Mat. Ital., VII. Ser., A 5 (1991), 345–352. · Zbl 0754.49026
[290] V. Šverák, X. Yan: A singular minimizer of a smooth strongly convex functional in three dimensions. Calc. Var. Partial Differ. Equ. 10 (2000), 213–221. · Zbl 1013.49027
[291] V. Šverák, X. Yan: Non-Lipschitz minimizers of smooth uniformly convex variational integrals. Proc. Natl. Acad. Sci. USA 99 (2002), 15269–15276. · Zbl 1106.49046
[292] L. Székelyhidi, Jr.: The regularity of critical points of polyconvex functionals. Arch. Ration. Mech. Anal. 172 (2004), 133–152. · Zbl 1049.49017
[293] L. Székelyhidi, Jr.: Rank-one convex hulls in \(\mathbb{R}\)2{\(\times\)}2. Calc. Var. Partial Differ. Equ. 22 (2005), 253–281. · Zbl 1104.49013
[294] G. Talenti: Boundedness of minimizers. Hokkaido Math. J. 19 (1990), 259–279. · Zbl 0723.58015
[295] Qi Tang: Regularity of minimizers of nonisotropic integrals of the calculus of variations. Ann. Mat. Pura Appl., IV. Ser. 164 (1993), 77–87. · Zbl 0796.49037
[296] N. S. Trudinger: On Harnack type inequalities and their application to quasilinear elliptic equations. Commun. Pure Appl. Math. 20 (1967), 721–747. · Zbl 0153.42703
[297] N. S. Trudinger: Pointwise estimates and quasilinear parabolic equations. Commun. Pure Appl. Math. 21 (1968), 205–226. · Zbl 0159.39303
[298] N. S. Trudinger: On the regularity of generalized solutions of linear, non-uniformly elliptic equations. Arch. Ration. Mech. Anal. 42 (1971), 50–62. · Zbl 0218.35035
[299] N. S. Trudinger: Linear elliptic operators with measurable coefficients. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 27 (1973), 265–308.
[300] 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. · Zbl 1067.35023
[301] K. Uhlenbeck: Regularity for a class of non-linear elliptic systems. Acta Math. 138 (1977), 219–240. · Zbl 0372.35030
[302] N. N. Ural’tseva: Degenerate quasilinear elliptic systems. Semin. in Mathematics, V. A. Steklov Math. Inst., Leningrad 7 (1968), 83–99.
[303] N. N. Ural’tseva, A. B. Urdaletova: The boundedness of the gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations. Vestn. Leningr. Univ., Math. 16 (1984), 263–270.
[304] K.-O. Widman: Hölder continuity of solutions of elliptic systems. Manuscr. Math. 5 (1971), 299–308. · Zbl 0223.35044
[305] J. Wolf: Partial regularity of weak solutions to nonlinear elliptic systems satisfying a Dini condition. Z. Anal. Anwend. 20 (2001), 315–330. · Zbl 1163.35329
[306] K.-W. Zhang: On the Dirichlet problem for a class of quasilinear elliptic systems of partial differential equations in divergence form. In: Partial Differential Equations, Proc. ymp. Tianjin/China, 1986. Lect. Notes Math. 1306. Springer-Verlag, Berlin, 1988, pp. 262–277.
[307] V. V. Zhikov: Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), 675–710.
[308] V. V. Zhikov: On Lavrentiev’s phenomenon. Russ. J. Math. Phys. 3 (1995), 249–269. · Zbl 0910.49020
[309] V. V. Zhikov: On some variational problems. Russ. J. Math. Phys. 5 (1997), 105–116.
[310] V. V. Zhikov: Meyer-type estimates for solving the nonlinear Stokes system. Differ. Equations 33 (1997), 108–115.
[311] V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik: Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin, 1994.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.