Lipschitz Bounds and Nonuniform Ellipticity
We consider nonuniformly elliptic variational problems and give optimal conditions guaranteeing the local Lipschitz regularity of solutions in terms of the regularity of the given data. The analysis catches the main model cases in the literature. Integrals with fast, exponential‐type growth conditio...
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| Published in | Communications on pure and applied mathematics Vol. 73; no. 5; pp. 944 - 1034 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Melbourne
John Wiley & Sons Australia, Ltd
01.05.2020
John Wiley and Sons, Limited |
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| Abstract | We consider nonuniformly elliptic variational problems and give optimal conditions guaranteeing the local Lipschitz regularity of solutions in terms of the regularity of the given data. The analysis catches the main model cases in the literature. Integrals with fast, exponential‐type growth conditions as well as integrals with unbalanced polynomial growth conditions are covered. Our criteria involve natural limiting function spaces and reproduce, in this very general context, the classical and optimal ones known in the linear case for the Poisson equation. Moreover, we provide new and natural growth a priori estimates whose validity was an open problem. Finally, we find new results also in the classical uniformly elliptic case. Beyond the specific results, the paper proposes a new approach to nonuniform ellipticity that, in a sense, allows us to reduce nonuniform elliptic problems to uniformly elliptic ones via potential theoretic arguments that are for the first time applied in this setting. © 2019 Wiley Periodicals, Inc. |
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| AbstractList | We consider nonuniformly elliptic variational problems and give optimal conditions guaranteeing the local Lipschitz regularity of solutions in terms of the regularity of the given data. The analysis catches the main model cases in the literature. Integrals with fast, exponential‐type growth conditions as well as integrals with unbalanced polynomial growth conditions are covered. Our criteria involve natural limiting function spaces and reproduce, in this very general context, the classical and optimal ones known in the linear case for the Poisson equation. Moreover, we provide new and natural growth a priori estimates whose validity was an open problem. Finally, we find new results also in the classical uniformly elliptic case. Beyond the specific results, the paper proposes a new approach to nonuniform ellipticity that, in a sense, allows us to reduce nonuniform elliptic problems to uniformly elliptic ones via potential theoretic arguments that are for the first time applied in this setting. © 2019 the Authors. Communications on Pure and Applied Mathematics is published by the Courant Institute of Mathematical Sciences and Wiley Periodicals, Inc. We consider nonuniformly elliptic variational problems and give optimal conditions guaranteeing the local Lipschitz regularity of solutions in terms of the regularity of the given data. The analysis catches the main model cases in the literature. Integrals with fast, exponential‐type growth conditions as well as integrals with unbalanced polynomial growth conditions are covered. Our criteria involve natural limiting function spaces and reproduce, in this very general context, the classical and optimal ones known in the linear case for the Poisson equation. Moreover, we provide new and natural growth a priori estimates whose validity was an open problem. Finally, we find new results also in the classical uniformly elliptic case. Beyond the specific results, the paper proposes a new approach to nonuniform ellipticity that, in a sense, allows us to reduce nonuniform elliptic problems to uniformly elliptic ones via potential theoretic arguments that are for the first time applied in this setting. © 2019 the Authors. Communications on Pure and Applied Mathematics is published by the Courant Institute of Mathematical Sciences and Wiley Periodicals, Inc. We consider nonuniformly elliptic variational problems and give optimal conditions guaranteeing the local Lipschitz regularity of solutions in terms of the regularity of the given data. The analysis catches the main model cases in the literature. Integrals with fast, exponential‐type growth conditions as well as integrals with unbalanced polynomial growth conditions are covered. Our criteria involve natural limiting function spaces and reproduce, in this very general context, the classical and optimal ones known in the linear case for the Poisson equation. Moreover, we provide new and natural growth a priori estimates whose validity was an open problem. Finally, we find new results also in the classical uniformly elliptic case. Beyond the specific results, the paper proposes a new approach to nonuniform ellipticity that, in a sense, allows us to reduce nonuniform elliptic problems to uniformly elliptic ones via potential theoretic arguments that are for the first time applied in this setting. © 2019 Wiley Periodicals, Inc. |
| Author | Beck, Lisa Mingione, Giuseppe |
| Author_xml | – sequence: 1 givenname: Lisa surname: Beck fullname: Beck, Lisa email: lisa.beck@math.uni-augsburg.de organization: Universität Augsburg – sequence: 2 givenname: Giuseppe surname: Mingione fullname: Mingione, Giuseppe email: giuseppe.mingione@unipr.it organization: Università di Parma |
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| Cites_doi | 10.1007/s00205-012-0562-z 10.1007/s00205-015-0859-9 10.1007/BF02392316 10.1007/s00526-017-1137-5 10.1016/j.anihpc.2010.07.002 10.1201/b18601 10.1007/BF02392793 10.1007/s10778-006-0110-3 10.3934/cpaa.2015.14.285 10.1016/j.na.2017.12.007 10.1007/s00205-008-0162-0 10.1007/s00205-006-0036-2 10.1016/j.jde.2017.09.038 10.1016/j.jde.2004.11.011 10.4171/JEMS/440 10.1007/s002291020227 10.1002/cpa.3160230409 10.1016/0022-0396(91)90158-6 10.1007/s00526-017-1148-2 10.1007/BF00251503 10.1142/S0129167X91000223 10.1007/BF01158049 10.1007/s00526-018-1323-0 10.4171/rmi/684 10.1007/s00205-013-0705-x 10.1006/jdeq.1998.3614 10.1080/03605302.2013.866959 10.1073/pnas.222494699 10.1007/s13373-013-0048-9 10.1007/BF01759317 10.1512/iumj.1976.25.25066 10.1007/s00526-018-1332-z 10.4171/jems/258 10.1080/03605301003657843 10.1016/0022-0396(88)90070-8 10.1016/j.na.2015.02.011 10.1016/j.anihpc.2011.02.005 10.1515/form.2002.011 10.1007/s00526-014-0768-z 10.1007/s00208-016-1362-9 10.1007/BF02921575 10.1142/5002 10.1016/S0022-247X(03)00584-5 10.1007/BF02192251 10.1007/s00526-003-0209-x 10.4171/JEMS/780 10.1016/j.jfa.2015.06.022 10.1080/03605309108820761 10.1051/cocv/2015034 |
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| References | 1968; 7 2002; 14 1991; 16 2007; 184 2013; 207 1994; 172 2002; 99 1988; 76 2014a; 212 2011; 13 1996a; 23 2003; 18 1993; 164 2010; 27 1989; 105 2018; 170 2014; 4 1991; 146 2014b; 16 1981; 113 1976b; 282 1992; 431 1970; 23 1986 1991; 90 2013; 674 2014; 13 1983 2015; 218 2012; 28 1977; 138 2011; 28 2016; 270 2001; 13 1992; 2 2003; 287 1991; 2 2018; 264 2015; 14 1986; 50 2006; 51 1976a; 25 2015; 53 2015; 120 2016; 366 2003 2011; 36 1992; 33 2018; 20 1957; 3 1987; 59 1972; 27 2000; 102 2009; 193 2017; 56 1999; 157 2015 2014; 39 1996b; 90 2006; 221 2018; 57 2016; 22 Beck L. (e_1_2_1_4_1) 2013; 674 e_1_2_1_20_1 Bildhauer M. (e_1_2_1_6_1) 2001; 13 Lieberman G. M. (e_1_2_1_41_1) 1992; 33 e_1_2_1_24_1 e_1_2_1_45_1 e_1_2_1_62_1 e_1_2_1_22_1 e_1_2_1_43_1 e_1_2_1_28_1 e_1_2_1_49_1 Zhikov V. V (e_1_2_1_65_1) 1986; 50 e_1_2_1_26_1 Simon J (e_1_2_1_58_1) 1976; 282 De Giorgi E. (e_1_2_1_23_1) 1957; 3 Stein E. M. (e_1_2_1_60_1) 1981; 113 e_1_2_1_54_1 e_1_2_1_8_1 Kuusi T. (e_1_2_1_34_1) 2012; 28 e_1_2_1_12_1 e_1_2_1_35_1 Rao M. M. (e_1_2_1_56_1) 1991 e_1_2_1_10_1 e_1_2_1_33_1 e_1_2_1_52_1 e_1_2_1_2_1 e_1_2_1_16_1 e_1_2_1_39_1 e_1_2_1_14_1 e_1_2_1_37_1 e_1_2_1_18_1 Marcellini P. (e_1_2_1_47_1) 1996; 23 ceva N. N. (e_1_2_1_63_1) 1968; 7 Hamburger C (e_1_2_1_31_1) 1992; 431 e_1_2_1_42_1 e_1_2_1_40_1 e_1_2_1_46_1 e_1_2_1_61_1 e_1_2_1_21_1 e_1_2_1_44_1 e_1_2_1_27_1 Mingione G. (e_1_2_1_53_1) 2011; 13 e_1_2_1_25_1 e_1_2_1_48_1 Havin V. P. A (e_1_2_1_50_1) 1972; 27 e_1_2_1_29_1 e_1_2_1_7_1 Urdaletova A. B. (e_1_2_1_64_1) 1983 e_1_2_1_30_1 e_1_2_1_55_1 e_1_2_1_5_1 e_1_2_1_57_1 e_1_2_1_3_1 e_1_2_1_13_1 e_1_2_1_51_1 e_1_2_1_32_1 e_1_2_1_17_1 e_1_2_1_38_1 e_1_2_1_15_1 e_1_2_1_36_1 Carozza M. (e_1_2_1_11_1) 2014; 13 e_1_2_1_59_1 e_1_2_1_9_1 e_1_2_1_19_1 |
| References_xml | – volume: 212 start-page: 129 issue: 1 year: 2014a end-page: 177 article-title: Global boundedness of the gradient for a class of nonlinear elliptic systems publication-title: Arch. Ration. Mech. Anal. – volume: 14 start-page: 245 issue: 2 year: 2002 end-page: 272 article-title: Regularity results for minimizers of irregular integrals with growth publication-title: Forum Math. – volume: 366 start-page: 1403 issue: 3‐4 year: 2016 end-page: 1450 article-title: Global Lipschitz continuity for minima of degenerate problems publication-title: Math. Ann. – volume: 287 start-page: 593 issue: 2 year: 2003 end-page: 608 article-title: Everywhere regularity for a class of vectorial functionals under subquadratic general growth conditions publication-title: J. Math. Anal. Appl. – volume: 27 start-page: 67 year: 1972 end-page: 138 article-title: nonlinear potential theory. publication-title: Math. Surveys – volume: 57 year: 2018 article-title: On the higher differentiability of solutions to a class of variational problems of fast growth publication-title: Calc. Var. Partial Differential Equations – volume: 221 start-page: 412 issue: 2 year: 2006 end-page: 443 article-title: Nonlinear elliptic systems with general growth publication-title: J. Differential Equations – volume: 674 start-page: 113 year: 2013 end-page: 194 article-title: On the Dirichlet problem for variational integrals in BV publication-title: J. Reine Angew. Math. – volume: 16 start-page: 311 year: 1991 end-page: 361 article-title: The natural generalization of the natural conditions of Ladyzhenskaya and Ural tseva for elliptic equations publication-title: Comm. Partial Differential Equations – volume: 105 start-page: 267 issue: 3 year: 1989 end-page: 284 article-title: Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions publication-title: Arch. Rational Mech. Anal. – volume: 193 start-page: 311 issue: 2 year: 2009 end-page: 337 article-title: Regularity of relaxed minimizers of quasiconvex variational integrals with ‐growth publication-title: Arch. Ration. Mech. Anal. – volume: 76 start-page: 203 issue: 2 year: 1988 end-page: 212 article-title: Regularity for minima of functionals with ‐growth publication-title: J. Differential Equations – volume: 157 start-page: 414 issue: 2 year: 1999 end-page: 438 article-title: Higher integrability for minimizers of integral functionals with growth publication-title: J. Differential Equations – volume: 13 start-page: 459 issue: 2 year: 2011 end-page: 486 article-title: Gradient potential estimates publication-title: J. Eur. Math. Soc. (JEMS) – volume: 13 start-page: 537 issue: 4 year: 2001 end-page: 560 article-title: Partial regularity for variational integrals with ‐growth. publication-title: Partial Differential Equations – volume: 16 start-page: 571 issue: 3 year: 2014b end-page: 595 article-title: Gradient regularity via rearrangements for ‐Laplacian type elliptic boundary value problems publication-title: J. Eur. Math. Soc. (JEMS) – volume: 172 start-page: 137 issue: 1 year: 1994 end-page: 161 article-title: The Wiener test and potential estimates for quasilinear elliptic equations publication-title: Acta Math. – volume: 18 start-page: 373 issue: 4 year: 2003 end-page: 400 article-title: Bounds for the singular set of solutions to non linear elliptic systems publication-title: Calc. Var. Partial Differential Equations – volume: 51 start-page: 355 issue: 4 year: 2006 end-page: 426 article-title: Regularity of minima: an invitation to the dark side of the calculus of variations publication-title: Appl. Math. – year: 1986 – volume: 36 start-page: 100 issue: 1 year: 2011 end-page: 133 article-title: Global Lipschitz regularity for a class of quasilinear elliptic equations publication-title: Comm. Partial Differential Equations – volume: 23 start-page: 677 year: 1970 end-page: 703 article-title: Local estimates for gradients of solutions of non‐uniformly elliptic and parabolic equations publication-title: Comm. Pure Appl. Math – volume: 14 start-page: 285 issue: 1 year: 2015 end-page: 311 article-title: Global gradient estimates in elliptic problems under minimal data and domain regularity publication-title: Commun. Pure Appl. Anal. – volume: 25 start-page: 821 issue: 9 year: 1976a end-page: 855 article-title: Interior gradient bounds for non‐uniformly elliptic equations publication-title: Indiana Univ. Math. J. – volume: 22 start-page: 862 issue: 3 year: 2016 end-page: 871 article-title: Strict convexity and the regularity of solutions to variational problems publication-title: ESAIM Control Optim. Calc. Var. – volume: 170 start-page: 1 year: 2018 end-page: 20 article-title: Higher integrability for constrained minimizers of integral functionals with ‐growth in low dimension publication-title: Nonlinear Anal. – year: 2015 – volume: 3 start-page: 25 year: 1957 end-page: 43 article-title: Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari publication-title: Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) – volume: 2 start-page: 395 issue: 4 year: 1991 end-page: 408 article-title: Regularity of exponentially harmonic functions publication-title: Internat. J. Math. – volume: 56 issue: 2 year: 2017 article-title: Regularity of ‐minimizers for a class of functionals with non‐standard growth publication-title: Calc. Var. Partial Differential Equations – start-page: 50 year: 1983 end-page: 56 article-title: The boundedness of the gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations publication-title: Vestnik Leningrad. Univ. Mat. Mekh. Astronom – volume: 2 start-page: 499 year: 1992 end-page: 515 article-title: Maximizing the norm of the gradient of solutions to the Poisson equation publication-title: J. Geom. Anal. – volume: 90 start-page: 161 issue: 1 year: 1996b end-page: 181 article-title: Regularity for some scalar variational problems under general growth conditions publication-title: J. Optim. Theory Appl. – volume: 39 start-page: 574 issue: 3 year: 2014 end-page: 590 article-title: Borderline estimates for fully nonlinear elliptic equations publication-title: Comm. Partial Differential Equations – volume: 282 start-page: A1351 year: 1976b end-page: A1354 article-title: Régularité de solutions de problèmes nonlinéaires publication-title: C. R. Acad. Sci. Paris Sér. A‐B – year: 2003 – volume: 28 start-page: 535 issue: 2 year: 2012 end-page: 576 article-title: Potential estimates and gradient boundedness for nonlinear parabolic systems publication-title: Rev. Mat. Iberoam. – volume: 23 start-page: 1 issue: 1 year: 1996a end-page: 25 article-title: Everywhere regularity for a class of elliptic systems without growth conditions publication-title: Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) – volume: 184 start-page: 341 issue: 2 year: 2007 end-page: 369 article-title: The singular set of Lipschitzian minima of multiple integrals publication-title: Arch. Ration. Mech. Anal. – volume: 7 start-page: 184 year: 1968 end-page: 222 article-title: Degenerate quasilinear elliptic systems publication-title: Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) – volume: 207 start-page: 215 issue: 1 year: 2013 end-page: 246 article-title: Linear potentials in nonlinear potential theory publication-title: Arch. Ration. Mech. Anal. – volume: 50 start-page: 675 year: 1986 end-page: 710 article-title: Averaging of functionals of the calculus of variations and elasticity theory publication-title: Izv. Akad. Nauk SSSR Ser. Mat – volume: 13 start-page: 1065 issue: 4 year: 2014 end-page: 1089 article-title: Regularity of minimizers of autonomous convex variational integrals publication-title: Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) – volume: 113 start-page: 383 year: 1981 end-page: 385 publication-title: Ann. of Math. (2) – volume: 59 start-page: 245 issue: 2 year: 1987 end-page: 248 article-title: Growth conditions and regularity, a counterexample publication-title: Manuscripta Math. – volume: 164 start-page: 103 year: 1993 end-page: 120 article-title: Gradient estimates for a class of elliptic systems publication-title: Ann. Mat. Pura Appl. (4) – volume: 27 start-page: 1361 issue: 6 year: 2010 end-page: 1396 article-title: Local Lipschitz regularity for degenerate elliptic systems publication-title: Ann. Inst. H. Poincaré Anal. Non Linéaire – volume: 264 start-page: 1263 issue: 2 year: 2018 end-page: 1316 article-title: Riesz potential estimates for a class of double phase problems publication-title: J. Differential Equations – volume: 102 start-page: 227 year: 2000 end-page: 250 publication-title: Manuscripta Math – volume: 56 year: 2017 article-title: Global gradient estimates for non‐uniformly elliptic equations publication-title: Calc. Var. Partial Differential Equations – volume: 431 start-page: 7 year: 1992 end-page: 64 article-title: Regularity of differential forms minimizing degenerate elliptic functionals publication-title: J. Reine Angew. Math. – volume: 20 start-page: 929 issue: 4 year: 2018 end-page: 1004 article-title: Vectorial nonlinear potential theory publication-title: J. Eur. Math. Soc. (JEMS) – volume: 90 year: 1991 article-title: Regularity and existence of solutions of elliptic equations with ‐growth conditions publication-title: J. Differential Equations – volume: 99 start-page: 15269 issue: 24 year: 2002 end-page: 15276 article-title: Non‐Lipschitz minimizers of smooth uniformly convex variational integrals publication-title: Proc. Natl. Acad. Sci. USA – volume: 120 start-page: 86 year: 2015 end-page: 106 article-title: Interior gradient regularity for BV minimizers of singular variational problems publication-title: Nonlinear Anal. – volume: 33 start-page: 45 issue: 1 year: 1992 end-page: 49 article-title: On the regularity of the minimizer of a functional with exponential growth publication-title: Comment. Math. Univ. Carolin. – volume: 53 start-page: 803 issue: 3‐4 year: 2015 end-page: 846 article-title: Riesz potential estimates for a general class of quasilinear equations publication-title: Calc. Var. Partial Differential Equations – volume: 57 year: 2018 article-title: Regularity for general functionals with double phase publication-title: Calc. Var. Partial Differential Equations – volume: 218 start-page: 219 issue: 1 year: 2015 end-page: 273 article-title: Bounded minimisers of double phase variational integrals publication-title: Arch. Ration. Mech. Anal. – volume: 270 start-page: 1416 issue: 4 year: 2016 end-page: 1478 article-title: Calderón‐Zygmund estimates and non‐uniformly elliptic operators publication-title: J. Funct. Anal. – volume: 146 year: 1991 – volume: 4 start-page: 1 issue: 1 year: 2014 end-page: 82 article-title: Guide to nonlinear potential estimates publication-title: Bull. Math. Sci. – volume: 28 start-page: 395 issue: 3 year: 2011 end-page: 411 article-title: Higher differentiability of minimizers of convex variational integrals publication-title: Ann. Inst. H. Poincaré Anal. Non Linéaire – volume: 138 start-page: 219 issue: 3‐4 year: 1977 end-page: 240 article-title: Regularity for a class of non‐linear elliptic systems publication-title: Acta Math. – ident: e_1_2_1_36_1 doi: 10.1007/s00205-012-0562-z – ident: e_1_2_1_19_1 doi: 10.1007/s00205-015-0859-9 – volume: 3 start-page: 25 year: 1957 ident: e_1_2_1_23_1 article-title: Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari publication-title: Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) – ident: e_1_2_1_62_1 doi: 10.1007/BF02392316 – ident: e_1_2_1_54_1 doi: 10.1007/s00526-017-1137-5 – ident: e_1_2_1_43_1 – ident: e_1_2_1_25_1 doi: 10.1016/j.anihpc.2010.07.002 – ident: e_1_2_1_55_1 doi: 10.1201/b18601 – ident: e_1_2_1_32_1 doi: 10.1007/BF02392793 – ident: e_1_2_1_52_1 doi: 10.1007/s10778-006-0110-3 – volume: 13 start-page: 537 issue: 4 year: 2001 ident: e_1_2_1_6_1 article-title: Partial regularity for variational integrals with (s, μ, q)‐growth. Calc. Var publication-title: Partial Differential Equations – ident: e_1_2_1_18_1 doi: 10.3934/cpaa.2015.14.285 – ident: e_1_2_1_22_1 doi: 10.1016/j.na.2017.12.007 – ident: e_1_2_1_59_1 doi: 10.1007/s00205-008-0162-0 – ident: e_1_2_1_33_1 doi: 10.1007/s00205-006-0036-2 – ident: e_1_2_1_9_1 doi: 10.1016/j.jde.2017.09.038 – volume-title: Theory of Orlicz spaces. Monographs and Textbooks in Pure and Applied Mathematics year: 1991 ident: e_1_2_1_56_1 – ident: e_1_2_1_49_1 doi: 10.1016/j.jde.2004.11.011 – ident: e_1_2_1_17_1 doi: 10.4171/JEMS/440 – volume: 674 start-page: 113 year: 2013 ident: e_1_2_1_4_1 article-title: On the Dirichlet problem for variational integrals in BV publication-title: J. Reine Angew. Math. – ident: e_1_2_1_28_1 doi: 10.1007/s002291020227 – ident: e_1_2_1_38_1 doi: 10.1002/cpa.3160230409 – ident: e_1_2_1_46_1 doi: 10.1016/0022-0396(91)90158-6 – ident: e_1_2_1_8_1 doi: 10.1007/s00526-017-1148-2 – ident: e_1_2_1_45_1 doi: 10.1007/BF00251503 – volume: 27 start-page: 67 year: 1972 ident: e_1_2_1_50_1 article-title: nonlinear potential theory. Russ publication-title: Math. Surveys – ident: e_1_2_1_24_1 doi: 10.1142/S0129167X91000223 – ident: e_1_2_1_29_1 doi: 10.1007/BF01158049 – volume: 13 start-page: 1065 issue: 4 year: 2014 ident: e_1_2_1_11_1 article-title: Regularity of minimizers of autonomous convex variational integrals publication-title: Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) – ident: e_1_2_1_13_1 doi: 10.1007/s00526-018-1323-0 – volume: 28 start-page: 535 issue: 2 year: 2012 ident: e_1_2_1_34_1 article-title: Potential estimates and gradient boundedness for nonlinear parabolic systems publication-title: Rev. Mat. Iberoam. doi: 10.4171/rmi/684 – ident: e_1_2_1_16_1 doi: 10.1007/s00205-013-0705-x – ident: e_1_2_1_26_1 doi: 10.1006/jdeq.1998.3614 – volume: 7 start-page: 184 year: 1968 ident: e_1_2_1_63_1 article-title: Degenerate quasilinear elliptic systems publication-title: Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) – ident: e_1_2_1_21_1 doi: 10.1080/03605302.2013.866959 – ident: e_1_2_1_61_1 doi: 10.1073/pnas.222494699 – ident: e_1_2_1_35_1 doi: 10.1007/s13373-013-0048-9 – ident: e_1_2_1_42_1 doi: 10.1007/BF01759317 – ident: e_1_2_1_57_1 doi: 10.1512/iumj.1976.25.25066 – volume: 113 start-page: 383 year: 1981 ident: e_1_2_1_60_1 publication-title: Ann. of Math. (2) – volume: 431 start-page: 7 year: 1992 ident: e_1_2_1_31_1 article-title: Regularity of differential forms minimizing degenerate elliptic functionals publication-title: J. Reine Angew. Math. – volume: 50 start-page: 675 year: 1986 ident: e_1_2_1_65_1 article-title: Averaging of functionals of the calculus of variations and elasticity theory publication-title: Izv. Akad. Nauk SSSR Ser. Mat – volume: 23 start-page: 1 issue: 1 year: 1996 ident: e_1_2_1_47_1 article-title: Everywhere regularity for a class of elliptic systems without growth conditions publication-title: Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) – volume: 282 start-page: A1351 year: 1976 ident: e_1_2_1_58_1 article-title: Régularité de solutions de problèmes nonlinéaires publication-title: C. R. Acad. Sci. Paris Sér. A‐B – ident: e_1_2_1_3_1 doi: 10.1007/s00526-018-1332-z – volume: 13 start-page: 459 issue: 2 year: 2011 ident: e_1_2_1_53_1 article-title: Gradient potential estimates publication-title: J. Eur. Math. Soc. (JEMS) doi: 10.4171/jems/258 – ident: e_1_2_1_15_1 doi: 10.1080/03605301003657843 – ident: e_1_2_1_44_1 doi: 10.1016/0022-0396(88)90070-8 – ident: e_1_2_1_5_1 doi: 10.1016/j.na.2015.02.011 – ident: e_1_2_1_10_1 doi: 10.1016/j.anihpc.2011.02.005 – ident: e_1_2_1_27_1 doi: 10.1515/form.2002.011 – volume: 33 start-page: 45 issue: 1 year: 1992 ident: e_1_2_1_41_1 article-title: On the regularity of the minimizer of a functional with exponential growth publication-title: Comment. Math. Univ. Carolin. – start-page: 50 year: 1983 ident: e_1_2_1_64_1 article-title: The boundedness of the gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations publication-title: Vestnik Leningrad. Univ. Mat. Mekh. Astronom – ident: e_1_2_1_2_1 doi: 10.1007/s00526-014-0768-z – ident: e_1_2_1_7_1 doi: 10.1007/s00208-016-1362-9 – ident: e_1_2_1_14_1 doi: 10.1007/BF02921575 – ident: e_1_2_1_30_1 doi: 10.1142/5002 – ident: e_1_2_1_39_1 doi: 10.1016/S0022-247X(03)00584-5 – ident: e_1_2_1_48_1 doi: 10.1007/BF02192251 – ident: e_1_2_1_51_1 doi: 10.1007/s00526-003-0209-x – ident: e_1_2_1_37_1 doi: 10.4171/JEMS/780 – ident: e_1_2_1_20_1 doi: 10.1016/j.jfa.2015.06.022 – ident: e_1_2_1_40_1 doi: 10.1080/03605309108820761 – ident: e_1_2_1_12_1 doi: 10.1051/cocv/2015034 |
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| Title | Lipschitz Bounds and Nonuniform Ellipticity |
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