Mathematica Bohemica, first online, pp. 1-32


$C^{1,\alpha}$ regularity for elliptic equations with the general nonstandard growth conditions

Sungchol Kim, Dukman Ri

Received March 29, 2023.   Published online August 21, 2023.

Abstract:  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.
Keywords:  nonstandard growth; $C^{1, \alpha}$ regularity; Hölder continuity; bounded weak solution; partial differential equations
Classification MSC:  35B65, 35D30, 35J25

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References:
[1] E. Acerbi, G. Mingione: Regularity results for a class of functionals with non-standard growth. Arch. Ration. Mech. Anal. 156 (2001), 121-140. DOI 10.1007/s002050100117 | MR 1814973 | Zbl 0984.49020
[2] T. Adamowicz, O. Toivanen: Hölder continuity of quasiminimizers with nonstandard growth. Nonlinear Anal., Theory Methods Appl., Ser. A 125 (2015), 433-456. DOI 10.1016/j.na.2015.05.023 | MR 3373594 | Zbl 1322.49059
[3] R. A. Adams, J. J. F. Fournier: Sobolev Spaces. Pure and Mathematics 140. Elsevier, Amsterdam (2003). MR 2424078 | Zbl 1098.46001
[4] S. Antontsev, S. Shmarev: Evolution PDEs with Nonstandard Growth Conditions: Existence, Uniqueness, Localization, Blow-up. Atlantis Studies in Differential Equations 4. Atlantis Press, Amsterdam (2015). MR 3328376 | Zbl 1410.35001
[5] P. Baroni, M. Colombo, G. Mingione: Non-autonomous functionals, borderline cases and related function classes. St. Petersbg. Math. J. 27 (2016), 347-379. DOI 10.1090/spmj/1392 | Zbl 1335.49057
[6] L. Beck: Elliptic Regularity Theory: A First Course. Lecture Notes of the Unione Matematica Italiana 19. Springer, Cham (2016). DOI 10.1007/978-3-319-27485-0 | MR 3468875 | Zbl 1346.35001
[7] Y. Chen, S. Levine, M. Rao: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66 (2006), 1383-1406. DOI 10.1137/050624522 | MR 2246061 | Zbl 1102.49010
[8] V. Chiadò Piat, A. Coscia: Hölder continuity of minimizers of functionals with variable growth exponent. Manuscr. Math. 93 (1997), 283-299. DOI 10.1007/BF02677472 | MR 1457729 | Zbl 0878.49010
[9] M. Colombo, G. Mingione: Bounded minimizers of double phase variational integrals. Arch. Ration. Mech. Anal. 218 (2015), 219-273. DOI 10.1007/s00205-015-0859-9 | MR 3360738 | Zbl 1325.49042
[10] M. Colombo, G. Mingione: Regularity for double phase variational problems. Arch. Ration. Mech. Anal. 215 (2015), 443-496. DOI 10.1007/s00205-014-0785-2 | MR 3294408 | Zbl 1322.49065
[11] A. Coscia, G. Mingione: Hölder continuity of the gradient of $p(x)$-harmonic mappings. C. R. Acad. Sci., Paris, Sér. I, Math. 328 (1999), 363-368. DOI 10.1016/S0764-4442(99)80226-2 | MR 1675954 | Zbl 0920.49020
[12] D. V. Cruz-Uribe, A. Fiorenza: Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Heidelberg (2013). DOI 10.1007/978-3-0348-0548-3 | MR 3026953 | Zbl 1268.46002
[13] L. Diening, P. Harjulehto, P. Hästö, M. Růžička: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics 2017. Springer, Berlin (2011). DOI 10.1007/978-3-642-18363-8 | MR 2790542 | Zbl 1222.46002
[14] L. Diening, S. Schwarzacher: Global gradient estimates for the $p(\cdot)$-Laplacian. Nonlinear Anal., Theory Methods Appl., Ser. A 106 (2014), 70-85. DOI 10.1016/j.na.2014.04.006 | MR 3209686 | Zbl 1291.35070
[15] M. Eleuteri: 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. MR 2044264 | Zbl 1178.49045
[16] X. Fan: Global $C^{1,\alpha}$ regularity for variable exponent elliptic equations in divergence form. J. Differ. Equations 235 (2007), 397-417. DOI 10.1016/j.jde.2007.01.008 | MR 2317489 | Zbl 1143.35040
[17] X. Fan, D. Zhao: A class of De Giorgi type and Hölder continuity. Nonlinear Anal., Theory Methods Appl. 36 (1999), 295-318. DOI 10.1016/S0362-546X(97)00628-7 | MR 1688232 | Zbl 0927.46022
[18] X. Fan, D. Zhao: The quasi-minimizer of integral functionals with $m(x)$ growth conditions. Nonlinear Anal., Theory Methods Appl., Ser. A 39 (2000), 807-816. DOI 10.1016/S0362-546X(98)00239-9 | MR 1736389 | Zbl 0943.49029
[19] N. Fusco, C. Sbordone: Some remarks on the regularity of minima of anisotropic integrals. Commun. Partial Differ. Equations 18 (1993), 153-167. DOI 10.1080/03605309308820924 | MR 1211728 | Zbl 0795.49025
[20] F. Giannetti, A. Passarelli di Napoli: Regularity results for a new class of functionals with non-standard growth conditions. J. Differ. Equations 254 (2013), 1280-1305. DOI 10.1016/j.jde.2012.10.011 | MR 2997371 | Zbl 1255.49064
[21] D. Gilbarg, N. S. Trudinger: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001). DOI 10.1007/978-3-642-61798-0 | MR 1814364 | Zbl 1042.35002
[22] E. Giusti: Direct Methods in the Calculus of Variations. World Scientific, Singapore (2003). DOI 10.1142/5002 | MR 1962933 | Zbl 1028.49001
[23] E. Gordadze, A. Meskhi, M. A. Ragusa: On some extrapolation in generalized grand Morrey spaces and applications to partial differential equations. Trans. A. Razmadze Math. Inst. 176 (2022), 435-441. MR 4524235
[24] P. Harjulehto, P. Hästö, Ú. V. Lê, M. Nuortio: Overview of differential equations with non-standard growth. Nonlinear Anal., Theory Methods Appl., Ser. A 72 (2010), 4551-4574. DOI 10.1016/j.na.2010.02.033 | MR 2639204 | Zbl 1188.35072
[25] P. Harjulehto, T. Kuusi, T. Lukkari, N. Marola, M. Parviainen: Harnack's inequality for quasiminimizers with nonstandard growth conditions. J. Math. Anal. Appl. 344 (2008), 504-520. DOI 10.1016/j.jmaa.2008.03.018 | MR 2416324 | Zbl 1145.49023
[26] S. Kim, D. Ri: Global boundedness and Hölder continuity of quasiminimizers with the general nonstandard growth conditions. Nonlinear Anal., Theory Methods Appl., Ser. A 185 (2019), 170-192. DOI 10.1016/j.na.2019.02.016 | MR 3926581 | Zbl 1419.49045
[27] G. M. Lieberman: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal., Theory Methods Appl. 12 (1988), 1203-1219. DOI 10.1016/0362-546X(88)90053-3 | MR 0969499 | Zbl 0675.35042
[28] G. M. Lieberman: The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations. Commun. Partial Differ. Equations 16 (1991), 311-361. DOI 10.1080/03605309108820761 | MR 1104103 | Zbl 0742.35028
[29] P. Marcellini: Regularity of minimizers of integrals of the calculus of variations with non-standard growth conditions. Arch. Ration. Mech. Anal. 105 (1989), 267-284. DOI 10.1007/BF00251503 | MR 0969900 | Zbl 0667.49032
[30] G. Mingione: Regularity of minima: An invitation to the dark side of the calculus of variations. Appl. Math., Praha 51 (2006), 355-426. DOI 10.1007/s10778-006-0110-3 | MR 2291779 | Zbl 1164.49324
[31] V. D. Rădulescu, D. D. Repovš: Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2015). DOI 10.1201/b18601 | MR 3379920 | Zbl 1343.35003
[32] K. R. Rajagopal, M. Růžička: Mathematical modeling of electrorheological materials. Contin. Mech. Thermodyn. 13 (2001), 59-78. DOI 10.1007/s001610100034 | Zbl 0971.76100
[33] M. Růžička: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics 1748. Springer, Berlin (2000). DOI 10.1007/BFb0104029 | MR 1810360 | Zbl 0962.76001
[34] O. Toivanen: Local boundedness of general minimizers with nonstandard growth. Nonlinear Anal., Theory Methods Appl., Ser. A 81 (2013), 62-69. DOI 10.1016/j.na.2012.10.024 | MR 3016440 | Zbl 1275.49068
[35] P. Tolksdorf: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equations 51 (1984), 126-150. DOI 10.1016/0022-0396(84)90105-0 | MR 0727034 | Zbl 0488.35017
[36] F. Yao: Local Hölder regularity of the gradients for the elliptic $p(x)$-Laplacian equation. Nonlinear Anal., Theory Methods Appl., Ser. A 78 (2013), 79-85. DOI 10.1016/j.na.2012.09.017 | MR 2992987 | Zbl 1320.35169
[37] C. Yu, D. Ri: 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. DOI 10.1016/j.na.2017.02.019 | MR 3634773 | Zbl 1375.35127
[38] C. Zhang, S. Zhou: Hölder regularity for the gradients of solutions of the strong $p(x)$-Laplacian. J. Math. Anal. Appl. 389 (2012), 1066-1077. DOI 10.1016/j.jmaa.2011.12.047 | MR 2879280 | Zbl 1234.35122
[39] H. Zhang: A global regularity result for the 2D generalized magneto-micropolar equations. J. Funct. Spaces 2022 (2022), 1501851, 6 pages. DOI 10.1155/2022/1501851 | MR 4389530 | Zbl 1485.35097
[40] V. V. Zhikov: Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR, Izv. 29 (1987), 33-66. DOI 10.1070/IM1987v029n01ABEH000958 | MR 0864171 | Zbl 0599.49031

Affiliations:   Sungchol Kim, Dukman Ri (corresponding author), Department of Mathematics, University of Science, Kwahak-dong, Unjong District, Pyongyang, Democratic People's Republic of Korea, e-mail: ksc@star-co.net.kp, ridukman@star-co.net.kp


 
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