一类非线性椭圆问题解的正则性

doi:10.3969/j.issn.1000-1565.2023.03.001

河北大学学报(自然科学版) ›› 2023, Vol. 43 ›› Issue (3): 225-229.DOI: 10.3969/j.issn.1000-1565.2023.03.001

• • 下一篇

一类非线性椭圆问题解的正则性

黄苗苗^1,2,赵雪漪¹,冯贺平³

收稿日期:2022-05-22 出版日期:2023-05-25 发布日期:2023-06-15
通讯作者: 冯贺平(1978—)
作者简介:黄苗苗(1993—),女,湖北十堰人,汉江师范学院助教,主要从事偏微分方程研究.
E-mail:huangmiaomiao@hjnu.edu.cn
基金资助:
应用数学湖北省重点实验室(湖北大学)开放基金资助项目(HBAM202003);湖北省高等学校优秀中青年科技创新团队项目(T2022035)

Regularity for some nonlinear elliptic problems

HUANG Miaomiao^1,2, ZHAO Xueyi¹, FENG Heping³

1.College of Mathematics and Computer Science, Hanjiang Normal University, Shiyan 442000, China; 2.Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan 430062, China; 3.Intelligent Engineering Department, Hebei Software Institute, Baoding 071002, China

Received:2022-05-22 Online:2023-05-25 Published:2023-06-15

摘要/Abstract

摘要： 研究椭圆型偏微分方程的源函数是关于自变量、自变量的未知函数及未知函数的偏导数的函数时,非线性椭圆方程的边值问题.利用推广的Stampacchia引理以及Sobolev空间的分析技巧,得到椭圆型方程的边值问题在强制性和控制增长条件下解的正则性.

关键词: 正则性, 椭圆型方程, 边值问题, Stampacchia引理

Abstract: When the source function of elliptic partial differential equation is a function of independent variable, unknown function of independent variable and partial derivative of unknown function, the boundary value problem of nonlinear elliptic equation is studied. By using the generalized Stampacchia lemma and the analysis technique of Sobolev space, the regularity of the solution of the boundary value problem of elliptic equation under some coercivity and controlled growth assumptions is obtained.

Key words: regularity, elliptic equation, boundary vale problem, Stampacchia lemma

中图分类号:

O175.23

黄苗苗,赵雪漪,冯贺平. 一类非线性椭圆问题解的正则性[J]. 河北大学学报(自然科学版), 2023, 43(3): 225-229.

HUANG Miaomiao, ZHAO Xueyi, FENG Heping. Regularity for some nonlinear elliptic problems[J]. Journal of Hebei University(Natural Science Edition), 2023, 43(3): 225-229.

参考文献

[1] BOCCADO L, CROCE G. Elliptic partial differential equations[M]. Berlin:De Gruyter, 2014.
[2] BOCCADO L, AGLIO A D, ORSINA L. Existence and regularity results for some elliptic equations with degenerate coercivity[J]. Atti Sem Mat Fis Univ Modena, 1998,46: 51-81.
[3] BOCCADO L, GIACHETTI D. Existence results via regularity for some nonlinear elliptic problems[J]. Comm Partial Differential Equations, 1989,14: 663-680.
[4] BOCCADO L, GALLOUET T. Nonlinear elliptic and parabolic equations involving measure data[J]. J Funct Anal, 1989, 87(1):149-169. DOI:10.1016/0022-1236(89)90005-0.
[5] BOCCADO L, MURAT F, PUEI J P. L^∞estimate for nonlinear elliptic partial differential equations and application to an existence result[J]. SIAM J Math Anal, 1992, 23(2): 326-333. DOI:10.1137/0523016.
[6] EVANS L C. Partial differential equations[M]. Rhode Island: American Mathematical Society, 1997.
[7] GIAQUINTA M. Multiple integrals in the calculus of variations and nonlinear elliptic systems[M]. Princeton: Princeton University Press, 1983.
[8] GIACHRTTI D, PORZIO M M. Local regularity result for minima of functionals of the calculus of variation[J]. Nonlinear Analysis, 2000,39: 463-482. DOI:10.1016/s0362-546x(98)00215-6.
[9] GILBARG D, TRUDINGER N S. Elliptic partial differential equations of second order.2nd[M]. Berlin: Springer, 1998.
[10] HEINONEN J, KILPELAINEN T, MARTIO O. Nonlinear potential theory of degenerate elliptic equations[M]. Oxford: Dover Publications, 2006.
[11] LADYZHENSKAYA O A, URALTSEVA G. Linear and quasilinear equations of elliptic type[M]. New York: Academic press, 1968.
[12] PASSARELLI DI NAPOLI A. A regularity result for a class of polyconvex functionals[J]. Ricerche Mat, 1999, 48:379-393.
[13] BYUN S S, PALAGACHEY D K. Boundness of the weak solution to quasilinear elliptic equations with morrey date[J]. Indiana University Mathematics Journal, 2013, 62:1565-1585. DOI:10.1512/iumj.2013.62.5115.
[14] GAO H, LEONETTI F, WANG L. Remarks on Stampacchia Lemma[J]. J Math Anal Appl, 2018,458: 112-122. DOI:10.1016/j.jmaa.2017.08.056.
[15] STAMPACCHIA G. Formes bilineaires coercitives sur les ensembles convexes[J]. C R Acad Sci Paris, 1964,258: 4413-4416.
[16] STAMPACCHIA G. Equations elliptiques du second ordrea coefficients discontinus[J]. Seminaire Jean Leray, 1963-1964,3: 1-77.
[17] KOVALEVSKII A A, VOITOVICH M V. On the impeovement of summability of generalized solutions of the dirichlet problem for nonlinear equations of the fourth order with strengthened ellipticity[J]. Ukr Math J, 2006,58: 1511-1524. DOI:10.1007/s11253-006-0164-8.
[18] GAO H Y, HUANG M M, REN W. Regularity for entropy solutions to degenerate elliptic equations[J]. J Math Anal Appl, 2020, 491(1):124251. DOI:10.1016/j.jmaa.2020.124251. (

一类非线性椭圆问题解的正则性

Regularity for some nonlinear elliptic problems

PDF (PC)

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 14

编辑推荐

Metrics

本文评价

[1]	高红亚, 高斯宇. Stampacchia引理、推广及应用[J]. 河北大学学报(自然科学版), 2021, 41(5): 481-487.
[2]	黄苗苗,王凤莉,高红亚. Holder不等式的推广及应用[J]. 河北大学学报(自然科学版), 2019, 39(6): 568-572.
[3]	张琪,王莲虹,高红亚. 熵解的一个正则性结果[J]. 河北大学学报(自然科学版), 2019, 39(2): 113-117.
[4]	刘健,封汉颍. 二阶非线性常微分方程组周期边值问题的正解[J]. 河北大学学报(自然科学版), 2014, 34(5): 455-459.
[5]	高红亚,乔金静. 微分形式障碍问题解的正则性[J]. 河北大学学报(自然科学版), 2011, 31(5): 453-455.
[6]	尚仲平,金燕,鲁淑霞. 时标上一阶脉冲方程的周期边值问题[J]. 河北大学学报(自然科学版), 2011, 31(4): 341-347.
[7]	刘洋,赵军生,于涛. 一类推广的IMBq方程解的爆破[J]. 河北大学学报(自然科学版), 2011, 31(3): 240-243.
[8]	张红晔. 巴拿赫空间中二阶微分方程的多点边值问题[J]. 河北大学学报(自然科学版), 2009, 29(6): 574-576.
[9]	姚庆六. 一类非线性弹性梁方程的正周期解[J]. 河北大学学报(自然科学版), 2009, 29(3): 225-228.
[10]	李云红,吕文静,郭彦平. 测度链上二阶边值问题多个正解的存在性[J]. 河北大学学报(自然科学版), 2008, 28(5): 462-465.
[11]	王培光,卢艳霞. 时间尺度上三点边值问题的拟线性方法[J]. 河北大学学报(自然科学版), 2008, 28(1): 1-3,21.
[12]	安敏,高红亚,刘红. 非齐次A-调和方程障碍问题解的局部正则性[J]. 河北大学学报(自然科学版), 2005, 25(2): 140-143.
[13]	苏晓红,郑连存. 一类非线性边界值问题正解的存在性[J]. 河北大学学报(自然科学版), 2005, 25(1): 5-8.
[14]	高红亚,高春霞. 空间拟正则映射理论与相关问题[J]. 河北大学学报(自然科学版), 2003, 23(4): 440-444.