The classical Stampacchia lemma has been widely used in the regularity theory of elliptic partial differential equations and systems as well as variational problems. This paper summarizes the Stampacchia lemma and their generalizations, and give an application to the regularity theory of degenerate elliptic partial differential equations.
GAO Hongya
,
GAO Siyu
. Stampacchia lemma, generalizations and application[J]. Journal of Hebei University(Natural Science Edition), 2021
, 41(5)
: 481
-487
.
DOI: 10.3969/j.issn.1000-1565.2021.05.004
[1] STAMPACCHIA G. Regularisation des solutions de problemes aux limites elliptiques a données discontinues [J].Proceedings of the International Symposium on Linear Spaces, 1961, 2(1): 339-408.
[2] STAMPACCHIA G. Équations elliptiques du second ordreà coficients discontinus [J]. Séminaire Jean Leray, 1963, 3(1): 1-77.
[3] BOCCARDO L. The summability of solutions to variational problems since Guido Stampacchia [J].Rev R Acad Cien Serie A Mat, 2003, 97(1): 413-421.
[4] ALVINO A, BOCCARDO L, FERONE V, et al. Existence results for nonlinear elliptic equations with degenerate coercivity[J]. Ann Di Matematica Pura Ed Appl, 2003, 182(1): 53-79. DOI:10.1007/s10231-002-0056-y.
[5] BOCCARDO L, CROCE G. Elliptic partial differential equations [M]. Berlin: De Gruyter, 2014.
[6] BOCCARDO L, GIACHETTI D. Existence results via regularity for some nonlinear elliptic problems[J]. Commun Partial Differ Equ, 1989, 14(5): 663-680. DOI:10.1080/03605308908820624.
[7] BOCCARDO L, GALLOUET T, MARCELLINI P. Anisotropic equations in [J]. Differential Integral Equations, 1996, 9(1): 209-212.
[8] CIRMI G R, LEONARDI S. Regularity results for the gradient of solutions linear elliptic equations with L ^1, λ data[J]. Ann Di Matematica Pura Ed Appl, 2006, 185(4): 537-553. DOI:10.1007/s10231-005-0167-3.
[9] GAO H Y, DI Q H, MA D N. Integrability for solutions to some anisotropic obstacle problems[J]. Manuscripta Math, 2015, 146(3): 433-444. DOI:10.1007/s00229-014-0705-7.
[10] GAO H Y, LIU C, TIAN H.Remarks on a paper by Leonetti and Siepe [J]. J Math Anal Appl, 2013, 401(2): 881-887.
[11] GAO H Y, LEONETTI F, WAGN L H. Remarks on Stampacchia Lemma [J]. J Math Anal Appl, 2018, 458(1):112-122. DOI: 10.1016/j.jmaa.2017.08.056.
[12] INNAMORATI A, LEONETTI F. Global integrability for weak solutoins to some anisotropic elliptic equations [J]. Nonlinear Anal, 2015, 113(1): 430-434. DOI: 10.1016/j.na.2014.09.027.
[13] KOVALEVSKY A A. Integrability and boundedness of solutions to some anisotropic problems [J]. J Math Anal Appl,2015, 432(2):820-843. DOI: 10.1016/j.jmaa.2015.07.013.
[14] KOVALEVSKII A A, VOITOVICH M V. On the improvement of summability of generalized solutions of the Dirichlet problem for nonlinear equations of the fourth order with strengthened ellipticity[J]. Ukrainian Math J, 2006, 58(11): 1717-1733. DOI:10.1007/s11253-006-0164-8.
[15] LEONETTI F, SIEPE F. Global integrability for minimizers of anisotropic functionals[J]. Manuscripta Math, 2014, 144(1/2): 91-98. DOI:10.1007/s00229-013-0641-y.
[16] LEONETTI F, SIEPE F. Integrability for solutions to some anisotropic elliptic equations [J].Nonlinear Analysis, 2012, 75(5): 2867-2873. DOI: 10.1016/j.na.2011.11.028.
[17] GAO HY, 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.
[18] GAO H Y, DENG H, HUANG M M, et al. Generalizations of Stampacchia lemma and applications to quasilinear elliptic systems [J].Nonlinear Analysis, 2021, 208(1):112297. DOI: 10.1016/j.na.2021.112297.
[19] BOCCARDO L, DALLAGLIO A, ORSINA L. Existence and regularity results for some elliptic equations with degenerate coercivity [J]. Atti Sem Mat Fis Univ Modena, 1998, 46(1): 51-81. (
