Regularity theory for non-autonomous problems with a priori assumptions. (English) Zbl 1529.35109
Summary: We study weak solutions and minimizers \(u\) of the non-autonomous problems \(\operatorname{div}A(x, Du) = 0\) and \(\min_v\int_\Omega F(x, Dv)\,dx\) with quasi-isotropic \((p, q)\)-growth. We consider the case that \(u\) is bounded, Hölder continuous or lies in a Lebesgue space and establish a sharp connection between assumptions on \(A\) or \(F\) and the corresponding norm of \(u\). We prove a Sobolev-Poincaré inequality, higher integrability and the Hölder continuity of \(u\) and \(Du\). Our proofs are optimized and streamlined versions of earlier research that can more readily be further extended to other settings. Connections between assumptions on \(A\) or \(F\) and assumptions on \(u\) are known for the double phase energy \(F(x, \xi) = |\xi|^p + a(x)|\xi|^q\). We obtain slightly better results even in this special case. Furthermore, we also cover perturbed variable exponent, Orlicz variable exponent, degenerate double phase, Orlicz double phase, triple phase, double variable exponent as well as variable exponent double phase energies and the results are new in most of these special cases.
MSC:
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35A15 | Variational methods applied to PDEs |
| 35J62 | Quasilinear elliptic equations |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 49N60 | Regularity of solutions in optimal control |
Keywords:
perturbed variable exponent; Orlicz variable exponent; degenerate double phase; Orlicz double phase; triple phaseReferences:
| [1] | Acerbi, E.; Fusco, N., An approximation lemma for \(W^{1, p}\) functions, Material instabilities in continuum mechanics (Edinburgh, 1985-1986), 1-5 (1988), New York: Oxford Univ. Press, New York |
| [2] | Baasandorj, S., Byun, S.S.: Regularity for Orlicz phase problems, Memoirs of American Mathematical Society, to appear. arXiv:2106.15131 · Zbl 1480.35249 |
| [3] | Baasandorj, S.; Byun, SS; Lee, H-S, Gradient estimates for Orlicz double phase problems with variable exponents, Nonlinear Anal., 221, 112891 (2022) · Zbl 1491.35206 · doi:10.1016/j.na.2022.112891 |
| [4] | Baasandorj, S.; Byun, SS; Oh, J., Gradient estimates for multi-phase problems, Calc Var. Partial Differential Equ., 60, 104 (2021) · Zbl 1511.35180 · doi:10.1007/s00526-021-01940-8 |
| [5] | Balci, AKh; Diening, L.; Surnachev, M., New examples on Lavrentiev gap using fractals, Calc Var. Partial Differential Equ., 59, 180 (2020) · Zbl 1453.35082 · doi:10.1007/s00526-020-01818-1 |
| [6] | Baroni, P.; Colombo, M.; Mingione, G., Harnack inequalities for double phase functionals, Nonlinear Anal., 121, 206-222 (2015) · Zbl 1321.49059 · doi:10.1016/j.na.2014.11.001 |
| [7] | Baroni, P.; Colombo, M.; Mingione, G., Nonautonomous functionals, borderline cases and related function classes, St. Petersburg Math. J., 27, 3, 347-379 (2016) · Zbl 1335.49057 · doi:10.1090/spmj/1392 |
| [8] | Baroni, P.; Colombo, M.; Mingione, G., Regularity for general functionals with double phase, Calc. Var. Partial Differential Equ., 57, 2 (2018) · Zbl 1394.49034 · doi:10.1007/s00526-018-1332-z |
| [9] | Beck, L.; Mingione, G., Lipschitz bounds and non-uniform ellipticity, Comm. Pure Appl. Math., 73, 944-1034 (2020) · Zbl 1445.35140 · doi:10.1002/cpa.21880 |
| [10] | Bella, P.; Schäffner, M., On the regularity of minimizers for scalar integral functionals with \((p, q)\)-growth, Anal. PDE, 13, 7, 2241-2257 (2020) · Zbl 1460.49027 · doi:10.2140/apde.2020.13.2241 |
| [11] | Benyaiche, A.; Harjulehto, P.; Hästö, P.; Karppinen, A., The weak Harnack inequality for unbounded supersolutions of equations with generalized Orlicz growth, J. Differential Equ., 275, 790-814 (2021) · Zbl 1455.35088 · doi:10.1016/j.jde.2020.11.007 |
| [12] | Benyaiche, A.; Khlifi, I., Harnack inequality for quasilinear elliptic equations in generalized Orlicz-Sobolev spaces, Pot. Anal., 53, 631-643 (2020) · Zbl 1445.35099 · doi:10.1007/s11118-019-09781-z |
| [13] | Borowski, M.; Chlebicka, I., Modular density of smooth functions in inhomogeneous and fully anisotropic Musielak-Orlicz-Sobolev spaces, J. Funct. Anal., 283, 12 (2022) · Zbl 1509.46020 · doi:10.1016/j.jfa.2022.109716 |
| [14] | Chlebicka, I.; Gwiazda, P.; Świerczewska-Gwiazda, A.; Wróblewska-Kamińska, A., Partial Differential Equations in Anisotropic Musielak-Orlicz Spaces (2021), Springer, Cham: Springer Monographs in Mathematics, Springer, Cham · Zbl 1490.35001 · doi:10.1007/978-3-030-88856-5 |
| [15] | Colombo, M.; Mingione, G., Regularity for double phase variational problems, Arch. Ration. Mech. Anal., 215, 2, 443-496 (2015) · Zbl 1322.49065 · doi:10.1007/s00205-014-0785-2 |
| [16] | Colombo, M.; Mingione, G., Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 218, 1, 219-273 (2015) · Zbl 1325.49042 · doi:10.1007/s00205-015-0859-9 |
| [17] | Colombo, M.; Mingione, G., Calderón-Zygmund estimates and non-uniformly elliptic operators, J. Funct. Anal., 270, 4, 1416-1478 (2016) · Zbl 1479.35158 · doi:10.1016/j.jfa.2015.06.022 |
| [18] | Crespo-Blanco, Á.; Gasiński, L.; Harjulehto, P.; Winkert, P., A new class of double phase variable exponent problems: existence and uniqueness, J. Differential Equ., 323, 182-228 (2022) · Zbl 1489.35041 · doi:10.1016/j.jde.2022.03.029 |
| [19] | Cruz-Uribe, D.; Diening, L., Sharp \({\cal{A} } \)-harmonic approximation, Appl. Anal., 98, 1-2, 374-380 (2019) · Zbl 1421.35104 · doi:10.1080/00036811.2017.1422729 |
| [20] | De Filippis, C., Optimal gradient estimates for multi-phase integrals, Math. Eng., 4, 5, 1-36 (2022) · Zbl 1529.49004 · doi:10.3934/mine.2022043 |
| [21] | De Filippis, C.; Mingione, G., Lipschitz bounds and nonautonomous integrals, Arch. Ration. Mech. Anal., 242, 973-1057 (2021) · Zbl 1483.49050 · doi:10.1007/s00205-021-01698-5 |
| [22] | De Filippis, C.,Mingione, G.: Nonuniformly elliptic schauder theory, Invent. Math. (to appear) · Zbl 1435.35127 |
| [23] | DiBenedetto, E., \(C^{1+\alpha }\) local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7, 8, 827-850 (1983) · Zbl 0539.35027 · doi:10.1016/0362-546X(83)90061-5 |
| [24] | Diening, L.; Harjulehto, P.; Hästö, P.; Růžička, M., Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics (2011), Heidelberg: Springer, Heidelberg · Zbl 1222.46002 · doi:10.1007/978-3-642-18363-8 |
| [25] | Diening, L.; Málek, J.; Steinhauer, M., On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications, ESAIM Control Optim. Calc. Var., 14, 2, 211-232 (2008) · Zbl 1143.35037 · doi:10.1051/cocv:2007049 |
| [26] | Diening, L.; Kaplický, P.; Schwarzacher, S., BMO estimates for the \(p\)-Laplacian, Nonlinear Anal., 75, 2, 637-650 (2012) · Zbl 1233.35056 · doi:10.1016/j.na.2011.08.065 |
| [27] | Diening, L.; Stroffolini, B.; Verde, A., The \(\varphi \)-harmonic approximation and the regularity of \(\varphi \)-harmonic maps, J. Differential Equ., 253, 7, 1943-1958 (2012) · Zbl 1260.35048 · doi:10.1016/j.jde.2012.06.010 |
| [28] | Esposito, L.; Leonetti, F.; Mingione, G., Sharp regularity for functionals with \((p, q)\) growth, J. Differential Equ., 204, 1, 5-55 (2004) · Zbl 1072.49024 · doi:10.1016/j.jde.2003.11.007 |
| [29] | Evans, LC, A new proof of local \(C^{1,\alpha }\) regularity for solutions of certain degenerate elliptic p.d.e., J. Differential Equ., 45, 3, 356-373 (1982) · Zbl 0508.35036 · doi:10.1016/0022-0396(82)90033-X |
| [30] | Fang, Y.; Rădulescu, V.; Zhang, C.; Zhang, X., Gradient estimates for multi-phase problems in Campanato spaces, Indiana Univ. Math. J., 71, 3, 1079-1099 (2022) · Zbl 1500.35172 · doi:10.1512/iumj.2022.71.8947 |
| [31] | Giusti, E., Direct Methods in the Calculus of Variations (2003), River Edge: World Scientific, River Edge · Zbl 1028.49001 · doi:10.1142/5002 |
| [32] | Hadzhy, O.V., Skrypnik, I.I., Voitovych, M.V.:Interior continuity, continuity up to the boundary and Harnack’s inequality for double-phase elliptic equations with non-logarithmic conditions, Math. Nachr., to appear. doi:10.1002/mana.202000574 · Zbl 1533.35051 |
| [33] | Harjulehto, P.; Hästö, P., Orlicz Spaces and Generalized Orlicz Spaces. Lecture Notes in Mathematics (2019), Cham: Springer, Cham · Zbl 1436.46002 · doi:10.1007/978-3-030-15100-3 |
| [34] | Harjulehto, P.; Hästö, P., Double phase image restoration, J. Math. Anal. Appl., 501, 1 (2021) · Zbl 1472.49026 · doi:10.1016/j.jmaa.2019.123832 |
| [35] | Harjulehto, P.; Hästö, P.; Juusti, J., Bloch estimates in non-doubling generalized Orlicz spaces, Math. Eng., 5, 3, 1-21 (2023) · Zbl 1536.49016 · doi:10.3934/mine.2023052 |
| [36] | Harjulehto, P., Hästö, P., Lee, M.: Hölder continuity of \(\omega \)-minimizers of functionals with generalized Orlicz growth, Ann. Sc. Norm. Super. Pisa Cl. Sci. XXII (2), 549-582 (2021) · Zbl 1482.46034 |
| [37] | Harjulehto, P.; Hästö, P.; Toivanen, O., Hölder regularity of quasiminimizers under generalized growth conditions, Calc. Var. Partial Differential Equ., 56, 2, 22 (2017) · Zbl 1366.35036 · doi:10.1007/s00526-017-1114-z |
| [38] | Harjulehto, P.; Hästö, P.; Karppinen, A., Local higher integrability of the gradient of a quasiminimizer under generalized Orlicz growth conditions, Nonlinear Anal., 177, part B, 543-552 (2018) · Zbl 1403.49034 · doi:10.1016/j.na.2017.09.010 |
| [39] | Hästö, P., The maximal operator on generalized Orlicz spaces, J. Funct. Anal., 269, 12, 4038-4048 (2015) · Zbl 1338.47032 · doi:10.1016/j.jfa.2015.10.002 |
| [40] | Hästö, P., A fundamental condition for harmonic analysis in anisotropic generalized Orlicz spaces, J. Geom. Anal., 33, 7 (2023) · Zbl 1509.46018 · doi:10.1007/s12220-022-01052-5 |
| [41] | Hästö, P.; Ok, J., Maximal regularity for local minimizers of non-autonomous functionals, J. Eur. Math. Soc. (JEMS), 24, 4, 1285-1334 (2022) · Zbl 1485.49044 · doi:10.4171/JEMS/1118 |
| [42] | Hästö, P.; Ok, J., Regularity theory for non-autonomous partial differential equations without Uhlenbeck structure, Arch. Ration. Mech. Anal., 245, 3, 1401-1436 (2022) · Zbl 1511.35172 · doi:10.1007/s00205-022-01807-y |
| [43] | Lewis, J., Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J., 32, 6, 849-858 (1983) · Zbl 0554.35048 · doi:10.1512/iumj.1983.32.32058 |
| [44] | Maeda, FY; Mizuta, Y.; Ohno, T.; Shimomura, T., Boundedness of maximal operators and Sobolev’s inequality on Musielak-Orlicz-Morrey spaces, Bull. Sci. Math., 137, 76-96 (2013) · Zbl 1267.46045 · doi:10.1016/j.bulsci.2012.03.008 |
| [45] | Maeda, FY; Mizuta, Y.; Ohno, T.; Shimomura, T., Trudinger’s inequality for double phase functionals with variable exponents, Czechosolvak Math. J., 71, 2, 511-528 (2021) · Zbl 1513.46054 · doi:10.21136/CMJ.2021.0506-19 |
| [46] | Manfredi, J. J.: Regularity of the gradients for a class of nonlinear possibly degenerate elliptic equations, Thesis (Ph.D.) Washington University in St. Louis. 58 pp (1986) |
| [47] | Marcellini, P., Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions, Arch. Ration. Mech. Anal., 105, 267-284 (1989) · Zbl 0667.49032 · doi:10.1007/BF00251503 |
| [48] | Marcellini, P., Regularity and existence of solutions of elliptic equations with \((p, q)\)-growth conditions, J. Differential Equ., 90, 1-30 (1991) · Zbl 0724.35043 · doi:10.1016/0022-0396(91)90158-6 |
| [49] | Mizuta, Y.; Ohno, T.; Shimomura, T., Boundedness of fractional maximal operators for double phase functionals with variable exponents, J. Math. Anal. Appl., 501, 1 (2021) · Zbl 1478.46028 · doi:10.1016/j.jmaa.2020.124360 |
| [50] | Ok, J., Regularity for double phase problems under additional integrability assumptions, Nonlinear Anal., 194 (2020) · Zbl 1437.49052 · doi:10.1016/j.na.2018.12.019 |
| [51] | Rădulescu, V., Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Anal., 121, 336-369 (2015) · Zbl 1321.35030 · doi:10.1016/j.na.2014.11.007 |
| [52] | N.N. Uralćeva: Degenerate quasilinear elliptic systems (Russian), Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 184-222 |
| [53] | Zhikov, VV, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50, 675-710 (1986) |
| [54] | Zhikov, VV, On Lavrentiev’s phenomenon, Russian J. Math. Phys., 3, 2, 249-269 (1995) · Zbl 0910.49020 |
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