Article
| Title: | Boundedness and Hölder continuity for a class of double phase variable exponent variational problems (English) |
| Author: | Ri, Dukman |
| Author: | Rim, Kuksung |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 76 |
| Issue: | 2 |
| Year: | 2026 |
| Pages: | 349-377 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We study the variable exponent double phase functionals with the critical growth. We show the sharp conditions for quasiminimizers of these functionals to be bounded or Hölder continuous. Our results generalize to variable exponent case results obtained by several authors in constant exponent case. (English) |
| Keyword: | double phase |
| Keyword: | boundedness |
| Keyword: | Hölder continuity |
| Keyword: | variable exponent |
| MSC: | 35J20 |
| MSC: | 49N60 |
| DOI: | 10.21136/CMJ.2026.0032-24 |
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| Date available: | 2026-05-22T11:17:15Z |
| Last updated: | 2026-05-25 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153638 |
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| Reference: | [1] Bahrouni, A., Rădulescu, V. D., Repovš, D. D.: Double phase transonic flow problems with variable growth: Nonlinear patterns and stationary waves.Nonlinearity 32 (2019), 2481-2495. Zbl 1419.35056, MR 3957220, 10.1088/1361-6544/ab0b03 |
| Reference: | [2] Baroni, P., Colombo, M., Mingione, G.: Harnack inequalities for double phase functionals.Nonlinear Anal., Theory Methods Appl., Ser. A 121 (2015), 206-222. Zbl 1321.49059, MR 3348922, 10.1016/j.na.2014.11.001 |
| Reference: | [3] 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: | [4] Baroni, P., Colombo, M., Mingione, G.: Regularity for general functionals with double phase.Calc. Var. Partial Differ. Equ. 57 (2018), Article ID 62, 48 pages. Zbl 1394.49034, MR 3775180, 10.1007/s00526-018-1332-z |
| Reference: | [5] Byun, S.-S., Oh, J.: Regularity results for generalized double phase functionals.Anal. PDE 13 (2020), 1269-1300. Zbl 1477.49057, MR 4149062, 10.2140/apde.2020.13.1269 |
| Reference: | [6] Colombo, M., Mingione, G.: Bounded minimisers 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: | [7] 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: | [8] Cupini, G., Marcellini, P., Mascolo, E.: Local boundedness of minimizers with limit growth conditions.J. Optim. Theory Appl. 166 (2015), 1-22. Zbl 1325.49043, MR 3366102, 10.1007/s10957-015-0722-z |
| Reference: | [9] Filippis, C. De, Oh, J.: Regularity for multi-phase variational problems.J. Differ. Equations 267 (2019), 1631-1670. Zbl 1422.49037, MR 3945612, 10.1016/j.jde.2019.02.015 |
| Reference: | [10] 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: | [11] Esposito, L., Leonetti, F., Mingione, G.: Sharp regularity for functionals with $(p,q)$ growth.J. Differ. Equations 204 (2004), 5-55. Zbl 1072.49024, MR 2076158, 10.1016/j.jde.2003.11.007 |
| Reference: | [12] Fan, X., Zhao, D.: On the spaces $L^{p(x)}$ and $W^{m,p(x)}(\Omega)$.J. Math. Anal. Appl. 263 (2001), 424-446. Zbl 1028.46041, MR 1866056, 10.1006/jmaa.2000.7617 |
| Reference: | [13] Fonseca, I., Malý, J., Mingione, G.: Scalar minimizers with fractal singular sets.Arch. Ration. Mech. Anal. 172 (2004), 295-307. Zbl 1049.49015, MR 2058167, 10.1007/s00205-003-0301-6 |
| Reference: | [14] 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: | [15] Giusti, E.: Direct Methods in the Calculus of Variations.World Scientific, Singapore (2003). Zbl 1028.49001, MR 1962933, 10.1142/5002 |
| Reference: | [16] Harjulehto, P., Hästö, P.: Double phase image restoration.J. Math. Anal. Appl. 501 (2021), Article ID 123832, 12 pages. Zbl 1472.49026, MR 4258782, 10.1016/j.jmaa.2019.123832 |
| Reference: | [17] Harjulehto, P., Hästö, P., Toivanen, O.: Hölder regularity of quasiminimizers under generalized growth conditions.Calc. Var. Partial Differ. Equ. 56 (2017), Article ID 22, 26 pages. Zbl 1366.35036, MR 3606780, 10.1007/s00526-017-1114-z |
| Reference: | [18] 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: | [19] Kováčik, O., Rákosník, J.: On spaces $L^{p(x)}$ and $W^{k,p(x)}$.Czech. Math. J. 41 (1991), 592-618. Zbl 0784.46029, MR 1134951, 10.21136/CMJ.1991.102493 |
| Reference: | [20] Ladyzhenskaya, O. A., Ural'tseva, N. N.: Linear and Quasilinear Elliptic Equations.Nauka, Moskva (1973), Russian. Zbl 0269.35029, MR 0509265 |
| Reference: | [21] Leonetti, F., Petricca, P. V.: Regularity for minimizers of integrals with nonstandard growth.Nonlinear Anal., Theory Methods Appl., Ser. A 129 (2015), 258-264. Zbl 1327.49064, MR 3414930, 10.1016/j.na.2015.09.009 |
| Reference: | [22] 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: | [23] 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: | [24] Marcellini, P.: Regularity and existence of solutions of elliptic equations with $p,q$-growth conditions.J. Differ. Equations 90 (1991), 1-30. Zbl 0724.35043, MR 1094446, 10.1016/0022-0396(91)90158-6 |
| Reference: | [25] Rădulescu, V. D.: Isotropic and anisotropic double-phase problems: Old and new.Opusc. Math. 39 (2019), 259-279. Zbl 1437.35315, MR 3897817, 10.7494/OpMath.2019.39.2.259 |
| Reference: | [26] Ragusa, M. A., Tachikawa, A.: Regularity for minimizers for functionals of double phase with variable exponents.Adv. Nonlinear Anal. 9 (2020), 710-728. Zbl 1420.35145, MR 3985000, 10.1515/anona-2020-0022 |
| Reference: | [27] 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: | [28] 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 |
| Reference: | [29] Zhikov, V. V.: On Lavrentiev's phenomenon.Russ. J. Math. Phys. 3 (1995), 249-269. Zbl 0910.49020, MR 1350506 |
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