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Global gradient estimates for a borderline case of double phase problems with measure data. (English) Zbl 1466.35061

Summary: We obtain a global Calderón-Zygmund estimate for a borderline case of double phase problems with measure data in terms of the 1-fractional maximal function of the measure.

MSC:

35B45 A priori estimates in context of PDEs
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35R60 PDEs with randomness, stochastic partial differential equations
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[1] Aboulaich, R.; Meskine, D.; Souissi, A., New diffusion models in image processing, Comput. Math. Appl., 56, 4, 874-882 (2008) · Zbl 1155.35389
[2] Acerbi, E.; Mingione, G., Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal., 156, 121-140 (2001) · Zbl 0984.49020
[3] Acerbi, E.; Mingione, G., Regularity results for electrorheological fluids: the stationary case, C. R. Math. Acad. Sci. Paris, 334, 817-822 (2002) · Zbl 1017.76098
[4] Acerbi, E.; Mingione, G., Gradient estimates for the \(p(x)\)-Laplacean system, J. Reine Angew. Math., 584, 117-148 (2005) · Zbl 1093.76003
[5] Baroni, P.; Colombo, M.; Mingione, G., Harnack inequalities for double phase functionals, Nonlinear Anal., 121, 206-222 (2015) · Zbl 1321.49059
[6] Baroni, P.; Colombo, M.; Mingione, G., Non-autonomous functionals, borderline cases and related function classes, St. Petersburg Math. J., 27, 347-379 (2016) · Zbl 1335.49057
[7] Baroni, P.; Habermann, J., Elliptic interpolation estimates for non-standard growth operators, Ann. Acad. Sci. Fenn., Math., 39, 1, 119-162 (2014) · Zbl 1296.35205
[8] Beck, L.; Mingione, G., Lipschitz bounds and nonuniform ellipticity, Commun. Pure Appl. Math. (2020) · Zbl 1445.35140
[9] Boccardo, L.; Gallouët, T., Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87, 1, 149-169 (1989) · Zbl 0707.35060
[10] Boccardo, L.; Gallouët, T., Nonlinear elliptic equations with right-hand side measures, Commun. Partial Differ. Equ., 17, 3-4, 641-655 (1992) · Zbl 0812.35043
[11] Bögelein, V.; Habermann, J., Gradient estimates via non standard potentials and continuity, Ann. Acad. Sci. Fenn., Math., 35, 2, 641-678 (2010) · Zbl 1217.35200
[12] Byun, S.; Ok, J., Nonlinear parabolic equations with variable exponent growth in nonsmooth domains, SIAM J. Math. Anal., 48, 5, 3148-3190 (2016) · Zbl 1355.35112
[13] Byun, S.; Oh, J., Global gradient estimates for non-uniformly elliptic equations, Calc. Var. Partial Differ. Equ., 56, 2, Article 46 pp. (2017), 36 pp · Zbl 1378.35139
[14] Byun, S.; Oh, J., Global gradient estimates for the borderline case of double phase problems with BMO coefficients in nonsmooth domains, J. Differ. Equ., 263, 2, 1643-1693 (2017) · Zbl 1372.35115
[15] Byun, S.; Ok, J.; Ryu, S., Global gradient estimates for elliptic equations of \(p(x)\)-Laplacian type with BMO nonlinearity, J. Reine Angew. Math., 715, 1-38 (2016) · Zbl 1347.35095
[16] Byun, S.; Ok, J.; Youn, Y., Global gradient estimates for spherical quasi-minimizers of integral functionals with \(p(x)\)-growth, Nonlinear Anal., 177, 186-208 (2018) · Zbl 1402.35095
[17] Byun, S.; Wang, L., Elliptic equations with BMO coefficients in Reifenberg domains, Commun. Pure Appl. Math., 57, 10, 1283-1310 (2004) · Zbl 1112.35053
[18] Byun, S.; Youn, Y., Optimal gradient estimates via Riesz potentials for \(p(\cdot)\)-Laplacian type equations, Q. J. Math., 68, 4, 1071-1115 (2017) · Zbl 1412.35138
[19] Byun, S.; Youn, Y., Riesz potential estimates for a class of double phase problems, J. Differ. Equ., 264, 2, 1263-1316 (2018) · Zbl 1387.35276
[20] Chen, Y.; Levine, S.; Rao, M., Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66, 4, 1383-1406 (2006) · Zbl 1102.49010
[21] Cho, Y., Global gradient estimates for divergence-type elliptic problems involving general nonlinear operators, J. Differ. Equ., 264, 10, 6152-6190 (2018) · Zbl 1386.35085
[22] Cianchi, A.; Maz’ya, V., Global Lipschitz regularity for a class of quasilinear elliptic equations, Commun. Partial Differ. Equ., 36, 1, 100-133 (2011) · Zbl 1220.35065
[23] Cianchi, A.; Maz’ya, V., Global boundedness of the gradient for a class of nonlinear elliptic systems, Arch. Ration. Mech. Anal., 212, 1, 129-177 (2014) · Zbl 1298.35070
[24] Colombo, M.; Mingione, G., Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 218, 1, 219-273 (2015) · Zbl 1325.49042
[25] Colombo, M.; Mingione, G., Regularity for double phase variational problems, Arch. Ration. Mech. Anal., 215, 2, 443-496 (2015) · Zbl 1322.49065
[26] Colombo, M.; Mingione, G., Calderón-Zygmund estimates and non-uniformly elliptic operators, J. Funct. Anal., 270, 4, 1416-1478 (2016) · Zbl 1479.35158
[27] Coscia, A.; Mingione, G., Hölder continuity of the gradient of \(p(x)\)-harmonic mappings, C. R. Acad. Sci. Paris Sér. I Math., 328, 4, 363-368 (1999) · Zbl 0920.49020
[28] De Filippis, C.; Mingione, G., A borderline case of Calderón-Zygmund estimates for nonuniformly elliptic problems, St. Petersburg Math. J. (2020), in press · Zbl 1435.35127
[29] De Filippis, C.; Oh, J., Regularity for multi-phase variational problems, J. Differ. Equ., 267, 3, 1631-1670 (2019) · Zbl 1422.49037
[30] Diening, L., Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math., 129, 8, 657-700 (2005) · Zbl 1096.46013
[31] Diening, L.; Ettwein, F., Fractional estimates for non-differentiable elliptic systems with general growth, Forum Math., 20, 3, 523-556 (2008) · Zbl 1188.35069
[32] Eleuteri, M.; Marcellini, P.; Mascolo, E., Lipschitz continuity for energy integrals with variable exponents, Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl., 27, 1, 61-87 (2016) · Zbl 1338.35169
[33] Fan, X.; Zhao, D., A class of De Giorgi type and Hölder continuity, Nonlinear Anal., 36, 3, 295-318 (1999) · Zbl 0927.46022
[34] Harjulehto, P.; Hästö, P., Orlicz Spaces and Generalized Orlicz Spaces, Lecture Notes in Mathematics (2019), Springer: Springer Cham · Zbl 1436.46002
[35] Harjulehto, P.; Hästö, P.; Karppinen, A., Local higher integrability of the gradient of a quasiminimizer under generalized Orlicz growth conditions, Nonlinear Anal., 177, 543-552 (2018) · Zbl 1403.49034
[36] Harjulehto, P.; Hästö, P.; Toivanen, O., Hölder regularity of quasiminimizers under generalized growth conditions, Calc. Var. Partial Differ. Equ., 56, 2, Article 22 pp. (2017), 26 pp · Zbl 1366.35036
[37] Hästö, P.; Ok, J., Calderón-Zygmund estimates in generalized Orlicz spaces, J. Differ. Equ., 267, 5, 2792-2823 (2019) · Zbl 1420.35087
[38] Iwaniec, T., p-harmonic tensors and quasiregular mappings, Ann. Math. (2), 136, 3, 589-624 (1992) · Zbl 0785.30009
[39] Iwaniec, T.; Verde, A., On the operator \(\mathcal{L}(f) = f \log | f |\), J. Funct. Anal., 169, 391-420 (1999) · Zbl 0961.47020
[40] Lemenant, A.; Milakis, E.; Spinolo, L., On the extension property of Reifenberg-flat domains, Ann. Acad. Sci. Fenn., Math., 39, 1, 51-71 (2014) · Zbl 1292.49042
[41] Lieberman, G. M., The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations, Commun. Partial Differ. Equ., 16, 23, 311-361 (1991) · Zbl 0742.35028
[42] Lukkari, T.; Maeda, F. Y.; Marola, N., Wolff potential estimates for elliptic equations with nonstandard growth and applications, Forum Math., 22, 6, 1061-1087 (2010) · Zbl 1203.35099
[43] Necas, J.; Málek, J.; Rokyta, M.; Ruzicka, M., Weak and Measure-Valued Solutions to Evolutionary PDEs, vol. 13 (1996), CRC Press · Zbl 0851.35002
[44] Růžička, M., Flow of shear dependent electrorheological fluids, C. R. Acad. Sci. Paris Sér. I Math., 329, 5, 393-398 (1999) · Zbl 0954.76097
[45] Růžička, M., Electrorheological Fluids: Modeling and Mathematical Theory (2000), Springer-Verlag: Springer-Verlag Berlin, xvi+176 pp · Zbl 0968.76531
[46] Rao, M. M.; Ren, Z. D., Theory of Orlicz Spaces (1991), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York, xii+449 pp · Zbl 0724.46032
[47] Zhikov, V. V., Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR, Ser. Mat., 50, 4, 675-710 (1986)
[48] Zhikov, V. V., Lavrentiev phenomenon and homogenization for some variational problems, C. R. Acad. Sci. Paris Sér. I Math., 316, 5, 435-439 (1993) · Zbl 0783.35005
[49] Zhikov, V. V., On Lavrentiev’s phenomenon, Russ. J. Math. Phys., 3, 2, 249-269 (1995) · Zbl 0910.49020
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