多相椭圆问题弱解的全局正则性

Global regularity of weak solutions to multi-phase elliptic problems

摘要

研究与多相泛函相对应的非一致椭圆方程弱解的全局正则性。利用 Young 不等式、Hölder 不等式、Sobolev-Poincaré 不等式、Gehring 引理等,提高了可积指数,获得了该方程弱解的全局正则性。

Abstract

Global regularity of weak solutions to the nonlinear elliptic equations corresponding to multi-phase functionals is considered. By using the Young inequality, the Hölder inequality, the Sobolev-Poincaré inequality and the Gehring lemma, the integrable exponent is improved. The global regularity of the weak solution of the equation is obtained.

关键词

正则性 / 椭圆问题 / 弱解 / 多相

Key words

regularity / elliptic problem / weak solution / multi-phase

参考文献

[1] Baroni P, Colombo M, Mingione G. Harnack inequalities for double phase functionals[J]. Nonlinear Analysis: Theory, Methods & Applications, 2015, 121: 206-222.

[2] Colombo M, Mingione G. Regularity for double phase variational problems[J]. Archive for Rational Mechanics and Analysis, 2015, 215(2): 443-496.

[3] Mingione G, Rădulescu V. Recent developments in problems with nonstandard growth and nonuniform ellipticity[J]. Journal of Mathematical Analysis and Applications, 2021, 501(1): 125197.

[4] Zhikov V V. Averaging of functionals of the calculus of variations and elasticity theory[J]. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 1986, 50: 675-710.

[5] Jikov B V V, Kozlov S M, Oleinik O A. Homogenization of differential operators and integral functionals[M]. Berlin: Springer—Verlag, 1994: 112.

[6] Perera K, Squassina M. Existence results for double—phase problems via Morse theory[J]. Communications in Contemporary Mathematics, 2018, 20(2): 1750023.

[7] Liu Wulong, Dai Guowei. Existence and multiplicity results for double phase problem[J]. Journal of Differential Equations, 2018, 265(9): 4311-4334.

[8] 葛斌, 陈志远. 一类具有奇异位势函数的双相问题[J]. 应用数学学报, 2022, 45(4): 624-636.

[9] Ge Bin, Chen Zhiyuan. On a double phase problem with singular weights[J]. Acta Mathematicae Applicatae Sinica, 2022, 45(4): 624-636.

[10] 郭艳敏, 赵崧, 佟玉霞. 一类非一致椭圆方程障碍问题的全局 BMO 估计[J]. 山东大学学报(理学版), 2023, 58(6): 46-56.

[11] Guo Yanmin, Zhao Song, Tong Yuxia. Global BMO estimations for obstacle problems of a class of non—uniformly elliptic equations[J]. Journal of Shandong University(Natural Science), 2023, 58(6): 46-56.

[12] Liang Shuang, Zheng Shenzhou. Calderón—Zygmund estimate for asymptotically regular non—uniformly elliptic equations[J]. Journal of Mathematical Analysis and Applications, 2020, 484(2): 123749.

[13] Byun S S, Oh J. Global gradient estimates for non—uniformly elliptic equations[J]. Calculus of Variations and Partial Differential Equations, 2017, 56(2): 46.

[14] De Filippis C, Oh J. Regularity for multi—phase variational problems[J]. Journal of Differential Equations, 2019, 267(3): 1631-1670.

[15] Baasandorj S, Byun S S, Oh J. Gradient estimates for multi—phase problems[J]. Calculus of Variations and Partial Differential Equations, 2021, 60(3): 104.

[16] Feng Jiangshan, Liang Shuang. Regularity for asymptotically regular elliptic double obstacle problems of multi—phase[J]. Results in Mathematics, 2023, 78(6): 232.

[17] De Filippis F, Piccinini M. Regularity for multi—phase problems at nearly linear growth[J]. Journal of Differential Equations, 2024, 410: 832-868.

[18] 沈毅, 马梦璐, 佟玉霞. 多相椭圆问题的局部高阶可积性[J]. 广西大学学报(自然科学版), 2024, 49(5): 1120-1125.

[19] Shen Yi, Ma Menglu, Tong Yuxia. Local higher integrability of multi—phase elliptic problems[J]. Journal of Guangxi University(Natural Science Edition), 2024, 49(5): 1120-1125.

[20] Giaquinta M. Multiple integrals in the calculus of variations and nonlinear elliptic systems[M]. New Jersey: Princeton University Press, 1984: 1-42.

[21] 周树清, 胡振华, 彭冬云. 一类 A—调和方程的障碍问题的很弱解的全局正则性[J]. 数学物理学报, 2014, 34(1): 27-38.

[22] Zhou Shuqing, Hu Zhenhua, Peng Dongyun. Global regularity for very weak solutions to obstacle promlems corresponding to a class of A—harmonic equations[J]. Acta Mathematica Scientia, 2014, 34(1): 27-38.

[23] Fan Xianling. An imbedding theorem for Musielak—Sobolev spaces[J]. Nonlinear Analysis: Theory, Methods & Applications, 2012, 75(4): 1959-1971.

[24] Vally M S E. Strongly nonlinear elliptic problems in Musielak—Orlicz—Sobolev spaces[J]. Advances in Dynamical Systems and Applications, 2013, 8(1): 115-124.

[25] Benkirane A, Sidi El Vally M . Variational inequalities in Musielak—Orlicz—Sobolev spaces[J]. Bulletin of the Belgian Mathematical Society—Simon Stevin, 2014, 21(5): 787-811.

[26] Byun S S, Lim M. Calderón—Zygmund estimates for non—uniformly elliptic equations with discontinuous nonlinearities on nonsmooth domains[J]. Journal of Differential Equations, 2022, 312: 374-406.

[27] Bögelein V, Zatorska—Goldstein A. Higher integrability of very weak solutions of systems of p(x)—Laplace ant type[J]. Journal of Mathematical Analysis and Applications, 2007, 336(1): 480-497.

[28] Colombo M, Mingione G. Calderón—Zygmund estimates and non—uniformly elliptic operators[J]. Journal of Functional Analysis, 2016, 270(4): 1416-1478.

[29] Baroni P, Colombo M, Mingione G. Regularity for general functionals with double phase[J]. Calculus of Variations and Partial Differential Equations, 2018, 57(2): 62.

[30] Ok J. Partial regularity for general systems of double phase type with continuous coefficients[J]. Nonlinear Analysis, 2018, 177: 673-698.

基金资助

河北省教育厅重点项目(ZD2022070)