Geometric \(C^{1 + \alpha}\) regularity estimates for nonlinear evolution models. (English) Zbl 1418.35056
Summary: In this survey we establish geometric \(C^{1 + \alpha}\) regularity estimates for bounded solutions of a number of nonlinear evolution models in divergence and non-divergence form. The main insights to obtain such estimates are based on geometric tangential methods, and make use of systematic oscillation mechanisms combined with intrinsic scaling techniques.
MSC:
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35K55 | Nonlinear parabolic equations |
| 35K65 | Degenerate parabolic equations |
| 35J60 | Nonlinear elliptic equations |
| 35J70 | Degenerate elliptic equations |
Keywords:
divergence and non-divergence form; geometric tangential methods; intrinsic scaling techniquesReferences:
| [1] | Acerbi, E.; Mingione, G., Gradient estimates for a class of parabolic systems, Duke Math. J., 136, 2, 285-320 (2007) · Zbl 1113.35105 |
| [2] | Acerbi, E.; Mingione, G.; Seregin, G. A., Regularity results for parabolic systems relates to a class of non-Newtonian fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21, 1, 25-60 (2004) · Zbl 1052.76004 |
| [3] | Amaral, M.; da Silva, J. V.; Ricarte, G. C.; Teymurazyan, R., Sharp regularity estimates for quasilinear evolution equations, Israel J. Math. (2019), (in press) http://arxiv.org/abs/180411140 · Zbl 1433.35192 |
| [4] | Attouchi, A.; Parviainen, M.; Ruosteenoja, E., \(C^{1, \alpha}\) Regularity for the normalized \(p\)-Poisson problem, J. Math. Pures Appl. (9), 108, 4, 553-591 (2017) · Zbl 1375.35170 |
| [5] | Benedek, A.; Panzone, R., The space \(L^p\), with mixed norm, Duke Math. J., 28, 301-324 (1961) · Zbl 0107.08902 |
| [6] | Besov, O. V.; Il’in, V. P.; Nikolskii, S. M., (Mitchel, H.; Taibleson, V. H., Integral Representations of Functions and Embedding Theorems. Integral Representations of Functions and Embedding Theorems, Scripta Series in Mathematics, vol. 1 (1978), Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons]: Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons] New York-Toronto, Ont.-London), viii+345, Translated from the Russian · Zbl 0392.46022 |
| [7] | Bögelein, V.; Duzaar, F.; Mingione, G., The regularity of general parabolic systems with degenerate diffusions, Mem. Amer. Math. Soc., 221, 1041, vi+143 (2013), ISBN: 978-0-8218-8975-6 · Zbl 1297.35066 |
| [8] | Caffarelli, L., Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2), 130, 1, 189-213 (1989) · Zbl 0692.35017 |
| [9] | Caffarelli, L.; Stefanelli, U., A counterexample to \(C^{2, 1}\) regularity for parabolic fully nonlinear equations, Comm. Partial Differential Equations, 33, 7-9, 1216-1234 (2008) · Zbl 1162.35042 |
| [10] | Crandall, M. G.; (Ken )Fok, P.-K.; Kocan, M.; Świȩch, A., Remarks on nonlinear uniformly parabolic equations, Indiana Univ. Math. J., 47, 4, 1293-1326 (1998) · Zbl 0933.35091 |
| [11] | Crandall, M. G.; Kocan, M.; Świȩch, A., \(L^p\)-Theory for fully nonlinear uniformly parabolic equations, Comm. Partial Differential Equations, 25, 11-12, 1997-2053 (2000) · Zbl 0973.35097 |
| [12] | DiBenedetto, E., Degenerate Parabolic Equations, xvi+387 (1993), Universitext. Springer-Verlag: Universitext. Springer-Verlag New York · Zbl 0794.35090 |
| [13] | DiBenedetto, E.; Friedman, A., Hölder Estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math., 357, 1-22 (1985) · Zbl 0549.35061 |
| [14] | DiBenedetto, E.; Urbano, J. M.; Vespri, V., Current issues on singular and degenerate evolution equations, (Evolutionary equations. Evolutionary equations, Handb. Differ. Equ., vol. I (2004), North-Holland: North-Holland Amsterdam), 169-286 · Zbl 1082.35002 |
| [15] | Kim, D., Elliptic and Parabolic equations with measurable coefficients in \(L_p\)-spaces with mixed norms, Methods Appl. Anal., 15, 4, 437-467 (2008) · Zbl 1223.35127 |
| [16] | Kim, D., Parabolic equations with partially BMO coefficients and boundary value problems in Sobolev spaces with mixed norms, Potential Anal., 33, 1, 17-46 (2010) · Zbl 1195.35163 |
| [17] | Krylov, N., Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat., 47, 1, 75-108 (1983), English transl. in Math. USSR Izv. 22 (1984) (1) 67-98 · Zbl 0578.35024 |
| [18] | Krylov, N., Parabolic equations in \(L_p -\) spaces with mixed norms, Algebra i Analiz, 14, 4, 91-106 (2002), translation in St. Petersburg Math. J 14 (2003) (4) 603-614 · Zbl 1032.35046 |
| [19] | Krylov, N., Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms, J. Funct. Anal., 250, 2, 521-558 (2007) · Zbl 1133.35052 |
| [20] | Krylov, N., (Lectures on Elliptic and Parabolic Equations in Sobolev Spaces. Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Graduate Studies in Mathematics, vol. 96 (2008), American Mathematical Society: American Mathematical Society Providence, RI), xviii+357 · Zbl 1147.35001 |
| [21] | Krylov, N., On \(C^{1 + \alpha}\) regularity of solutions of Isaacs parabolic equations with VMO coefficients, NoDEA Nonlinear Differential Equations Appl., 21, 1, 63-85 (2014) · Zbl 1304.35173 |
| [22] | Krylov, N., \(C^{1 + \alpha -}\) Regularity of viscosity solutions of general nonlinear parabolic equations, J. Math. Sci. (N.Y.), 232, 4, 403-427 (2018), Problems in mathematical analysis. (93) (Russian) · Zbl 1402.35061 |
| [23] | Krylov, N.; Safonov, M., A certain property of solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat., 44, 1, 161-175 (1980), Math. USSR-Izv. 16:1 (1981) 151-164 · Zbl 0464.35035 |
| [24] | Kuusi, T.; Mingione, G., Gradient regularity for nonlinear parabolic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 12, 4, 755-822 (2013) · Zbl 1288.35145 |
| [25] | Kuusi, T.; Mingione, G., The wolff gradient bound for degenerate parabolic equations, J. Eur. Math. Soc., 16, 835-892 (2014) · Zbl 1303.35120 |
| [26] | Ladyzhenskaya, O. A.; Solonnikov, V. A.; Ural’tseva, N. N., (Linear and Quasilinear Equations of Parabolic Type. Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23 (1968), American Mathematical Society: American Mathematical Society Providence, R.I.), xi+648, (Russian) Translated from the Russian by S Smith · Zbl 0174.15403 |
| [27] | Lieberman, G. M., Second Order Parabolic Differential Equations, xii+439 (1996), World Scientific Publishing Co. Inc.: World Scientific Publishing Co. Inc. River Edge, NJ · Zbl 0884.35001 |
| [28] | Lindqvist, P., On the time derivative in a quasilinear equation, Skr. K. Nor. Vidensk. Selsk., 2, 1-7 (2008) · Zbl 1165.35402 |
| [29] | Nadirashvili, N.; Vlăduţ, S., Nonclassical solutions of fully nonlinear elliptic equations, Geom. Funct. Anal., 17, 4, 1283-1296 (2007) · Zbl 1132.35036 |
| [30] | Nadirashvili, N.; Vlăduţ, S., Singular viscosity solutions to fully nonlinear elliptic equations, J. Math. Pures Appl. (9), 89, 2, 107-113 (2008) · Zbl 1133.35049 |
| [31] | Nadirashvili, N.; Vlăduţ, S., Octonions and singular solutions of Hessian elliptic equations, Geom. Funct. Anal., 21, 2, 483-498 (2011) · Zbl 1216.35041 |
| [32] | Nadirashvili, N.; Vlăduţ, S., Singular solutions of Hessian elliptic equations in five dimensions, J. Math. Pures Appl. (9), 100, 6, 769-784 (2013) · Zbl 1283.35040 |
| [33] | da Silva, J. V.; Ochoa, P., Fully nonlinear parabolic dead core problems, Pacific J. Math. (2019), (in press) · Zbl 1439.35237 |
| [34] | da Silva, J. V.; Ochoa, P.; Silva, A., Regularity for degenerate evolution equations with strong absorption, J. Differential Equations, 264, 12, 7270-7293 (2018) · Zbl 1393.35066 |
| [35] | da Silva, J. V.; dos Prazeres, D., Schauder type estimates for “flat” viscosity solutions to non-convex fully nonlinear parabolic equations and applications, Potential Anal., 50, 149-170 (2019) · Zbl 1407.35037 |
| [36] | da Silva, J. V.; Rossi, J. D.; Salort, A., Regularity properties for \(p -\) dead core problems and their asymptotic limit as \(p \to \infty \), J. Lond. Math. Soc., 99, 69-96 (2019) · Zbl 1412.35152 |
| [37] | da Silva, J. V.; Salort, A., Sharp regularity estimates for quasi-linear elliptic dead core problems and applications, Calc. Var. Partial Differential Equations, 57, 3, 83 (2018), 24 pp · Zbl 1398.35081 |
| [38] | da Silva, J. V.; Teixeira, E. V., Sharp regularity estimates for second order fully nonlinear parabolic equations, Math. Ann., 369, 3-4, 1623-1648 (2017) · Zbl 1377.35051 |
| [39] | Teixeira, E. V., Sharp regularity for general Poisson equations with borderline sources, J. Math. Pures Appl. (9), 99, 2, 150-164 (2013) · Zbl 1263.35071 |
| [40] | Teixeira, E. V., Regularity for quasilinear equations on degenerate singular sets, Math. Ann., 358, 1-2, 241-256 (2014) · Zbl 1286.35119 |
| [41] | Teixeira, E. V., Universal moduli of continuity for solutions to fully nonlinear elliptic equations, Arch. Ration. Mech. Anal., 211, 3, 911-927 (2014) · Zbl 1292.35132 |
| [42] | Teixeira, E. V., Hessian continuity at degenerate points in nonvariational elliptic problems, Int. Math. Res. Not. IMRN, 16, 6893-6906 (2015) · Zbl 1326.35100 |
| [43] | Teixeira, E. V., Geometric regularity estimates for elliptic equations, (Mathematical Congress of the Americas. Mathematical Congress of the Americas, Contemp. Math., vol. 656 (2016), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 185-201 · Zbl 1348.35046 |
| [44] | Teixeira, E. V.; Urbano, J. M., A geometric tangential approach to sharp regularity for degenerate evolution equations, Anal. PDE, 7, 3, 733-744 (2014) · Zbl 1295.35296 |
| [45] | Teixeira, E. V.; Urbano, J. M., An intrinsic Liouville theorem for degenerate parabolic equations, Arch. Math. (Basel), 102, 5, 483-487 (2014) · Zbl 1295.35146 |
| [46] | Teixeira, E. V.; Urbano, J. M., Geometric tangential analysis and sharp regularity for degenerate PDEs, (Proceedings of the INdAM Meeting “Harnack Inequalities and Nonlinear Operators” in honour of Prof. E. DiBenedetto. Proceedings of the INdAM Meeting “Harnack Inequalities and Nonlinear Operators” in honour of Prof. E. DiBenedetto, Springer INdAM Series (2019)), (in press) |
| [47] | Urbano, J. M., The method of intrinsic scaling, (A Systematic Approach to Regularity for Degenerate and Singular PDEs. A Systematic Approach to Regularity for Degenerate and Singular PDEs, Lecture Notes in Mathematics, vol. 1930 (2008), Springer-Verlag: Springer-Verlag Berlin), x+150 · Zbl 1158.35003 |
| [48] | Wang, L., On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl. Math., 45, 1, 27-76 (1992) · Zbl 0832.35025 |
| [49] | Wang, L., On the regularity theory of fully nonlinear parabolic equations. II, Comm. Pure Appl. Math., 45, 2, 141-178 (1992) · Zbl 0774.35042 |
| [50] | Wiegner, M., On \(C_{\alpha -}\) regularity of the gradient of solutions of degenerate parabolic systems, Ann. Mat. Pura Appl. (4), 145, 385-405 (1986) · Zbl 0642.35046 |
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