| Title:
|
$C^{1,\alpha }$ regularity for elliptic equations with the general nonstandard growth conditions (English) |
| Author:
|
Kim, Sungchol |
| Author:
|
Ri, Dukman |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
149 |
| Issue:
|
3 |
| Year:
|
2024 |
| Pages:
|
365-396 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study elliptic equations with the general nonstandard growth conditions involving Lebesgue measurable functions on $\Omega $. We prove the global $C^{1, \alpha }$ regularity of bounded weak solutions of these equations with the Dirichlet boundary condition. Our results generalize the $C^{1, \alpha }$ regularity results for the elliptic equations in divergence form not only in the variable exponent case but also in the constant exponent case. (English) |
| Keyword:
|
nonstandard growth |
| Keyword:
|
$C^{1, \alpha }$ regularity |
| Keyword:
|
Hölder continuity |
| Keyword:
|
bounded weak solution |
| Keyword:
|
partial differential equations |
| MSC:
|
35B65 |
| MSC:
|
35D30 |
| MSC:
|
35J25 |
| DOI:
|
10.21136/MB.2023.0055-23 |
| . |
| Date available:
|
2024-09-11T13:47:06Z |
| Last updated:
|
2024-09-11 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152539 |
| . |
| Reference:
|
[1] Acerbi, E., Mingione, G.: Regularity results for a class of functionals with non-standard growth.Arch. Ration. Mech. Anal. 156 (2001), 121-140. Zbl 0984.49020, MR 1814973, 10.1007/s002050100117 |
| Reference:
|
[2] Adamowicz, T., Toivanen, O.: Hölder continuity of quasiminimizers with nonstandard growth.Nonlinear Anal., Theory Methods Appl., Ser. A 125 (2015), 433-456. Zbl 1322.49059, MR 3373594, 10.1016/j.na.2015.05.023 |
| Reference:
|
[3] Adams, R. A., Fournier, J. J. F.: Sobolev Spaces.Pure and Mathematics 140. Elsevier, Amsterdam (2003). Zbl 1098.46001, MR 2424078 |
| Reference:
|
[4] Antontsev, S., Shmarev, S.: Evolution PDEs with Nonstandard Growth Conditions: Existence, Uniqueness, Localization, Blow-up.Atlantis Studies in Differential Equations 4. Atlantis Press, Amsterdam (2015). Zbl 1410.35001, MR 3328376 |
| Reference:
|
[5] Baroni, P., Colombo, M., Mingione, G.: Non-autonomous functionals, borderline cases and related function classes.St. Petersbg. Math. J. 27 (2016), 347-379. Zbl 1335.49057, MR 3570955, 10.1090/spmj/1392 |
| Reference:
|
[6] Beck, L.: Elliptic Regularity Theory: A First Course.Lecture Notes of the Unione Matematica Italiana 19. Springer, Cham (2016). Zbl 1346.35001, MR 3468875, 10.1007/978-3-319-27485-0 |
| Reference:
|
[7] Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration.SIAM J. Appl. Math. 66 (2006), 1383-1406. Zbl 1102.49010, MR 2246061, 10.1137/050624522 |
| Reference:
|
[8] Piat, V. Chiadò, Coscia, A.: Hölder continuity of minimizers of functionals with variable growth exponent.Manuscr. Math. 93 (1997), 283-299. Zbl 0878.49010, MR 1457729, 10.1007/BF02677472 |
| Reference:
|
[9] Colombo, M., Mingione, G.: Bounded minimizers of double phase variational integrals.Arch. Ration. Mech. Anal. 218 (2015), 219-273. Zbl 1325.49042, MR 3360738, 10.1007/s00205-015-0859-9 |
| Reference:
|
[10] Colombo, M., Mingione, G.: Regularity for double phase variational problems.Arch. Ration. Mech. Anal. 215 (2015), 443-496. Zbl 1322.49065, MR 3294408, 10.1007/s00205-014-0785-2 |
| Reference:
|
[11] Coscia, A., Mingione, G.: 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, MR 1675954, 10.1016/S0764-4442(99)80226-2 |
| Reference:
|
[12] Cruz-Uribe, D. V., Fiorenza, A.: Variable Lebesgue Spaces: Foundations and Harmonic Analysis.Applied and Numerical Harmonic Analysis. Birkhäuser, Heidelberg (2013). Zbl 1268.46002, MR 3026953, 10.1007/978-3-0348-0548-3 |
| Reference:
|
[13] Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents.Lecture Notes in Mathematics 2017. Springer, Berlin (2011). Zbl 1222.46002, MR 2790542, 10.1007/978-3-642-18363-8 |
| Reference:
|
[14] Diening, L., Schwarzacher, S.: Global gradient estimates for the $p(\cdot)$-Laplacian.Nonlinear Anal., Theory Methods Appl., Ser. A 106 (2014), 70-85. Zbl 1291.35070, MR 3209686, 10.1016/j.na.2014.04.006 |
| Reference:
|
[15] Eleuteri, M.: Hölder continuity results for a class of functionals with non-standard growth.Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 7 (2004), 129-157. Zbl 1178.49045, MR 2044264 |
| Reference:
|
[16] Fan, X.: Global $C^{1,\alpha}$ regularity for variable exponent elliptic equations in divergence form.J. Differ. Equations 235 (2007), 397-417. Zbl 1143.35040, MR 2317489, 10.1016/j.jde.2007.01.008 |
| Reference:
|
[17] Fan, X., Zhao, D.: A class of De Giorgi type and Hölder continuity.Nonlinear Anal., Theory Methods Appl. 36 (1999), 295-318. Zbl 0927.46022, MR 1688232, 10.1016/S0362-546X(97)00628-7 |
| Reference:
|
[18] Fan, X., Zhao, D.: The quasi-minimizer of integral functionals with $m(x)$ growth conditions.Nonlinear Anal., Theory Methods Appl., Ser. A 39 (2000), 807-816. Zbl 0943.49029, MR 1736389, 10.1016/S0362-546X(98)00239-9 |
| Reference:
|
[19] Fusco, N., Sbordone, C.: Some remarks on the regularity of minima of anisotropic integrals.Commun. Partial Differ. Equations 18 (1993), 153-167. Zbl 0795.49025, MR 1211728, 10.1080/03605309308820924 |
| Reference:
|
[20] Giannetti, F., Napoli, A. Passarelli di: Regularity results for a new class of functionals with non-standard growth conditions.J. Differ. Equations 254 (2013), 1280-1305. Zbl 1255.49064, MR 2997371, 10.1016/j.jde.2012.10.011 |
| Reference:
|
[21] Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order.Classics in Mathematics. Springer, Berlin (2001). Zbl 1042.35002, MR 1814364, 10.1007/978-3-642-61798-0 |
| Reference:
|
[22] Giusti, E.: Direct Methods in the Calculus of Variations.World Scientific, Singapore (2003). Zbl 1028.49001, MR 1962933, 10.1142/5002 |
| Reference:
|
[23] Gordadze, E., Meskhi, A., Ragusa, M. A.: On some extrapolation in generalized grand Morrey spaces and applications to partial differential equations.Trans. A. Razmadze Math. Inst. 176 (2022), 435-441. Zbl 1522.42052, MR 4524235 |
| Reference:
|
[24] Harjulehto, P., Hästö, P., Lê, Ú. V., Nuortio, M.: Overview of differential equations with non-standard growth.Nonlinear Anal., Theory Methods Appl., Ser. A 72 (2010), 4551-4574. Zbl 1188.35072, MR 2639204, 10.1016/j.na.2010.02.033 |
| Reference:
|
[25] Harjulehto, P., Kuusi, T., Lukkari, T., Marola, N., Parviainen, M.: Harnack's inequality for quasiminimizers with nonstandard growth conditions.J. Math. Anal. Appl. 344 (2008), 504-520. Zbl 1145.49023, MR 2416324, 10.1016/j.jmaa.2008.03.018 |
| Reference:
|
[26] Kim, S., Ri, D.: Global boundedness and Hölder continuity of quasiminimizers with the general nonstandard growth conditions.Nonlinear Anal., Theory Methods Appl., Ser. A 185 (2019), 170-192. Zbl 1419.49045, MR 3926581, 10.1016/j.na.2019.02.016 |
| Reference:
|
[27] Lieberman, G. M.: Boundary regularity for solutions of degenerate elliptic equations.Nonlinear Anal., Theory Methods Appl. 12 (1988), 1203-1219. Zbl 0675.35042, MR 0969499, 10.1016/0362-546X(88)90053-3 |
| Reference:
|
[28] Lieberman, G. M.: 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, MR 1104103, 10.1080/03605309108820761 |
| Reference:
|
[29] Marcellini, P.: 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, MR 0969900, 10.1007/BF00251503 |
| Reference:
|
[30] Mingione, G.: Regularity of minima: An invitation to the dark side of the calculus of variations.Appl. Math., Praha 51 (2006), 355-426. Zbl 1164.49324, MR 2291779, 10.1007/s10778-006-0110-3 |
| Reference:
|
[31] Rădulescu, V. D., Repovš, D. D.: Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis.Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2015). Zbl 1343.35003, MR 3379920, 10.1201/b18601 |
| Reference:
|
[32] Rajagopal, K. R., Růžička, M.: Mathematical modeling of electrorheological materials.Contin. Mech. Thermodyn. 13 (2001), 59-78. Zbl 0971.76100, 10.1007/s001610100034 |
| Reference:
|
[33] Růžička, M.: Electrorheological Fluids: Modeling and Mathematical Theory.Lecture Notes in Mathematics 1748. Springer, Berlin (2000). Zbl 0962.76001, MR 1810360, 10.1007/BFb0104029 |
| Reference:
|
[34] Toivanen, O.: Local boundedness of general minimizers with nonstandard growth.Nonlinear Anal., Theory Methods Appl., Ser. A 81 (2013), 62-69. Zbl 1275.49068, MR 3016440, 10.1016/j.na.2012.10.024 |
| Reference:
|
[35] Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations.J. Differ. Equations 51 (1984), 126-150. Zbl 0488.35017, MR 0727034, 10.1016/0022-0396(84)90105-0 |
| Reference:
|
[36] Yao, F.: Local Hölder regularity of the gradients for the elliptic $p(x)$-Laplacian equation.Nonlinear Anal., Theory Methods Appl., Ser. A 78 (2013), 79-85. Zbl 1320.35169, MR 2992987, 10.1016/j.na.2012.09.017 |
| Reference:
|
[37] Yu, C., Ri, D.: Global $L^\infty$-estimates and Hölder continuity of weak solutions to elliptic equations with the general nonstandard growth conditions.Nonlinear Anal., Theory Methods Appl., Ser. A 156 (2017), 144-166. Zbl 1375.35127, MR 3634773, 10.1016/j.na.2017.02.019 |
| Reference:
|
[38] Zhang, C., Zhou, S.: Hölder regularity for the gradients of solutions of the strong $p(x)$-Laplacian.J. Math. Anal. Appl. 389 (2012), 1066-1077. Zbl 1234.35122, MR 2879280, 10.1016/j.jmaa.2011.12.047 |
| Reference:
|
[39] Zhang, H.: A global regularity result for the 2D generalized magneto-micropolar equations.J. Funct. Spaces 2022 (2022), 1501851, 6 pages. Zbl 1485.35097, MR 4389530, 10.1155/2022/1501851 |
| Reference:
|
[40] Zhikov, V. V.: Averaging of functionals of the calculus of variations and elasticity theory.Math. USSR, Izv. 29 (1987), 33-66. Zbl 0599.49031, MR 0864171, 10.1070/IM1987v029n01ABEH000958 |
| . |
