RENORMALIZED SOLUTIONS OF ELLIPTIC EQUATIONS WITH ROBIN BOUNDARY CONDITIONS

doi:10.1016/S0252-9602(17)30046-2

数学物理学报(英文版) ›› 2017, Vol. 37 ›› Issue (4): 889-910.doi: 10.1016/S0252-9602(17)30046-2

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RENORMALIZED SOLUTIONS OF ELLIPTIC EQUATIONS WITH ROBIN BOUNDARY CONDITIONS

Olivier GUIBÉ¹, Alip OROPEZA^1,2

1. Laboratoire de Mathématiques Raphaël Salem, CNRS, UMR 6085, Université de Rouen, Avenue de l'Université, BP 12, 76801 Saint-Étienne du Rouvray cedex, France;
2. University of the Philippines Diliman, Institute of Mathematics, 1101 Diliman, Quezon City, Philippines

收稿日期:2015-12-28 修回日期:2016-10-26 出版日期:2017-08-25 发布日期:2017-08-25
通讯作者: Olivier GUIBÉ,E-mail:olivier.guibe@univ-rouen.fr E-mail:olivier.guibe@univ-rouen.fr
作者简介:Alip OROPEZA,Alip.Oropeza@etu.univ-rouen.fr,aaoropeza@math.upd.edu.ph

RENORMALIZED SOLUTIONS OF ELLIPTIC EQUATIONS WITH ROBIN BOUNDARY CONDITIONS

Olivier GUIBÉ¹, Alip OROPEZA^1,2

1. Laboratoire de Mathématiques Raphaël Salem, CNRS, UMR 6085, Université de Rouen, Avenue de l'Université, BP 12, 76801 Saint-Étienne du Rouvray cedex, France;
2. University of the Philippines Diliman, Institute of Mathematics, 1101 Diliman, Quezon City, Philippines

Received:2015-12-28 Revised:2016-10-26 Online:2017-08-25 Published:2017-08-25
Contact: Olivier GUIBÉ,E-mail:olivier.guibe@univ-rouen.fr E-mail:olivier.guibe@univ-rouen.fr
About author:Alip OROPEZA,Alip.Oropeza@etu.univ-rouen.fr,aaoropeza@math.upd.edu.ph

摘要/Abstract

摘要：

In the present paper, we consider elliptic equations with nonlinear and nonhomogeneous Robin boundary conditions of the type

where f and g are the element of L¹(Ω) and L¹(Γ₁), respectively. We define a notion of renormalized solution and we prove the existence of a solution. Under additional assumptions on the matrix field B we show that the renormalized solution is unique.

关键词: elliptic equations, renormalized solution, uniqueness, Robin boundary conditions, L¹ data

Abstract:

In the present paper, we consider elliptic equations with nonlinear and nonhomogeneous Robin boundary conditions of the type

where f and g are the element of L¹(Ω) and L¹(Γ₁), respectively. We define a notion of renormalized solution and we prove the existence of a solution. Under additional assumptions on the matrix field B we show that the renormalized solution is unique.

Key words: elliptic equations, renormalized solution, uniqueness, Robin boundary conditions, L¹ data

Olivier GUIBÉ, Alip OROPEZA. RENORMALIZED SOLUTIONS OF ELLIPTIC EQUATIONS WITH ROBIN BOUNDARY CONDITIONS[J]. 数学物理学报(英文版), 2017, 37(4): 889-910.

Olivier GUIBÉ, Alip OROPEZA. RENORMALIZED SOLUTIONS OF ELLIPTIC EQUATIONS WITH ROBIN BOUNDARY CONDITIONS[J]. Acta mathematica scientia,Series B, 2017, 37(4): 889-910.

参考文献

[1] Andreu F, Mazón J M, Segura de León S, Toledo J. Quasi-linear elliptic and parabolic equations in L¹ with nonlinear boundary conditions. Adv Math Sci Appl, 1997, 7(1):183-213
[2] Andreu F, Mazón J M, Segura de León S, Toledo J. Existence and uniqueness for a degenerate parabolic equation with L¹-data. Trans Amer Math Soc, 1999, 351(1):285-306
[3] Artola M. Sur une classe de problèmes paraboliques quasi-linéaires. Boll Un Mat Ital B (6), 1986, 5(1):51-70
[4] Bénilan P, Boccardo L, Gallouët T, Gariepy R, Pierre M, Vázquez J L. An L¹-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann Scuola Norm Sup Pisa Cl Sci (4), 1995, 22(2):241-273
[5] Blanchard D, Désir F, Guibé O. Quasi-linear degenerate elliptic problems with L¹ data. Nonlinear Anal, 2005, 60(3):557-587
[6] Boccardo L, Gallouët T. On some nonlinear elliptic and parabolic equations involving measure data. J Funct Anal, 1989, 87:149-169
[7] Boccardo L, Gallouët T, Murat F. Unicité de la solution de certaines équations elliptiques non linéaires. C R Acad Sci Paris Sér I Math, 1992, 315(11):1159-1164
[8] Bonzi B K, Ouaro S, Zongo F D Y. Entropy solutions to nonlinear elliptic anisotropic problem with Robin boundary condition. Matematiche (Catania), 2013, 68(2):53-76
[9] Cabarrubias B, Donato P. Existence and uniqueness for a quasilinear elliptic problem with nonlinear Robin conditions. Carpathian J Math, 2011, 27(2):173-184
[10] Cabarrubias B, Donato P. Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions. Appl Anal, 2012, 91(6):1111-1127
[11] Carrillo J, Chipot M. On some nonlinear elliptic equations involving derivatives of the nonlinearity. Proc Roy Soc Edinburgh Sect A, 1985, 100(3/4):281-294
[12] Chipot M, Michaille G. Uniqueness results and monotonicity properties for strongly nonlinear elliptic variational inequalities. Ann Scuola Norm Sup Pisa Cl Sci (4), 1989, 16(1):137-166
[13] Dal Maso G, Murat F, Orsina L, Prignet A. Renormalized solutions of elliptic equations with general measure data. Ann Scuola Norm Sup Pisa Cl Sci (4), 1999, 28(4):741-808
[14] Dall'Aglio A. Approximated solutions of equations with L¹ data. Application to the H-convergence of quasi-linear parabolic equations. Ann Mat Pura Appl (4), 1996, 170:207-240
[15] Di Nardo R, Feo F, Guibé O. Uniqueness result for nonlinear anisotropic elliptic equations. Adv Differ Equ, 2013, 18(5/6):433-458
[16] DiPerna R -J, Lions P -L. On the Cauchy problem for Boltzmann equations:global existence and weak stability. Ann Math, 1989, 130(1):321-366
[17] Droniou J. Solving convection-diffusion equations with mixed, Neumann and Fourier boundary conditions and measures as data, by a duality method. Adv Differ Equ, 2000, 5(10/12):1341-1396
[18] Guibé O. Uniqueness of the renormalized solution to a class of nonlinear elliptic equations//On the Notions of Solution to Nonlinear Elliptic Problems:Results and Developments. volume 23 of Quad Mat, Dept Math, Seconda Univ Napoli, Caserta, 2008:255-282
[19] Lions P -L, Murat F. Sur les solutions renormalisées d'équations elliptiques. (manuscript)
[20] Murat F. Soluciones renormalizadas de EDP elipticas non lineales. Technical Report R93023, Laboratoire d'Analyse Numérique, Paris VI, 1993. Cours à l'Université de Séville
[21] Ouaro S, Ouedraogo A. Entropy solution to an elliptic problem with nonlinear boundary conditions. An Univ Craiova Ser Mat Inform, 2012, 39(2):148-181
[22] Porretta A. Uniqueness of solutions for some nonlinear Dirichlet problems. Nonl Differ Equ Appl, 2004, 11(4):407-430
[23] Sbihi K, Wittbold P. Entropy solution of a quasilinear elliptic problem with nonlinear boundary condition. Commun Appl Anal, 2007, 11(2):299-325
[24] Serrin J. Pathological solution of elliptic differential equations. Ann Scuola Norm Sup Pisa Cl Sci, 1964, 18:385-387

RENORMALIZED SOLUTIONS OF ELLIPTIC EQUATIONS WITH ROBIN BOUNDARY CONDITIONS

RENORMALIZED SOLUTIONS OF ELLIPTIC EQUATIONS WITH ROBIN BOUNDARY CONDITIONS

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