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Existence and Uniqueness of Entropy Solution for an Elliptic Problem with Nonlinear Boundary Conditions and Measure-data

Received: 15 July 2019     Accepted: 29 August 2019     Published: 16 September 2019
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Abstract

In this paper, we study a class of elliptic equations on a bounded domain with nonlinear boundary conditions of type graph and measure - data. First of all, give some spaces and basic assumptions. Next, we apply the classical variational approach. So, we need an essentially bounded estimate on the solution, which is not evident to obtain directly in our problem. The obstacles which we encounter is that we cannot get rid of the non-linear term evaluated as a zero gradient and it appear at the boundary, for the part of the measure-data, a term which cannot vanish, when one uses the integration by parts formula. To overcome this difficulties, we first redefine and extend the function which appears in the third Leray-Lions-type conditions and we add a penalization term on the boundary. Secondly, we consider a smooth domain in order to work with the Sobolev spaces that are the closure of indefinitely differentiable and null functions on the bounary, and to going back later to the classical Sobolev space. Then, we assume that the domain is extensible. Next, we obtain a priori estimates and convergence results in the approach problem, which allow us to delete the penalization term. To finish, we introduce a notion of entropy solution for our main problem and prove that it is the limit of the solution obtained in the variational case.

Published in American Journal of Applied Mathematics (Volume 7, Issue 4)
DOI 10.11648/j.ajam.20190704.13
Page(s) 114-126
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Elliptic Problem, Entropy Solution, Nonlinear Boundary Conditions, Graph, Measure-data

References
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[2] K. Ammar, Solutions entropiques et renormalisées de quelques E. D. P non linaires dans L1, Thesis, Strasbourg, 2003.
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[4] B. Andreianov, F. Bouhssis, Uniqueness for an elliptic-parabolic problem with Neumann boundary condition, J. Evol. Equ. 4 (2004), no. 2, 273-295.
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[7] Ph. B´enilan, L. Boccardo, T. Gallou¨et, R. Gariepy, M. Pierre, J. L. Vasquez, An L1 −theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa 22 (1995), 241-273.
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[9] Ph. B´enilan, J. Carrillo, P. Wittbold, Renormalized and entropy solutions of scalar conservation laws, Ann. Scuola Norm. Sup. Pisa 29 (2000), 313-327.
[10] Ph. B´enilan, B. G. Crandall, A. Pazy, Evolutions Equations Governed by accretive Operators, Forth- coming book.
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[12] L. Boccardo, T. Gallou¨et, L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. Henri Poincar´e, 13, 5 (1196), 539-551.
[13] G. Bouchitt´e, Calcul des variations en cadre non r´eflexif. Repr´esentation et relaxation de fonction-nelles int´egrales sur un espace de mesures. Applications en plasticit´e et homog´enisation. Th`ese de Doctorat d’Etat. Perpignan, 1987.
[14] G. Bouchitt´e, Conjugu´e et sous-diff´erentiel d’une fonctionnelle int´egrale sur un espace de Sobolev, C. R. Acad. Sci. Paris S´er. I Math. 307 (1988), no., 79-82.
[15] H. Br´ezis; Analyse Fonctionnelle: Th´eorie et Applications, Paris, Masson (1983).
[16] J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal, 147 (1999), 269-361.
[17] R. Diperna, P. L. Lions, On the Cauchy problem for the Boltzman equation: global existence and stability, Ann. of Math. 130 (1989), 321-366.
[18] N. Dunfort, L. schwartz, Linear Operators, part I, Pure and Applied Mathematics, Vol VII.
[19] C. B. Jr. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, 1966.
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    Arouna Ouedraogo. (2019). Existence and Uniqueness of Entropy Solution for an Elliptic Problem with Nonlinear Boundary Conditions and Measure-data. American Journal of Applied Mathematics, 7(4), 114-126. https://doi.org/10.11648/j.ajam.20190704.13

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    Arouna Ouedraogo. Existence and Uniqueness of Entropy Solution for an Elliptic Problem with Nonlinear Boundary Conditions and Measure-data. Am. J. Appl. Math. 2019, 7(4), 114-126. doi: 10.11648/j.ajam.20190704.13

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    AMA Style

    Arouna Ouedraogo. Existence and Uniqueness of Entropy Solution for an Elliptic Problem with Nonlinear Boundary Conditions and Measure-data. Am J Appl Math. 2019;7(4):114-126. doi: 10.11648/j.ajam.20190704.13

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  • @article{10.11648/j.ajam.20190704.13,
      author = {Arouna Ouedraogo},
      title = {Existence and Uniqueness of Entropy Solution for an Elliptic Problem with Nonlinear Boundary Conditions and Measure-data},
      journal = {American Journal of Applied Mathematics},
      volume = {7},
      number = {4},
      pages = {114-126},
      doi = {10.11648/j.ajam.20190704.13},
      url = {https://doi.org/10.11648/j.ajam.20190704.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20190704.13},
      abstract = {In this paper, we study a class of elliptic equations on a bounded domain with nonlinear boundary conditions of type graph and measure - data. First of all, give some spaces and basic assumptions. Next, we apply the classical variational approach. So, we need an essentially bounded estimate on the solution, which is not evident to obtain directly in our problem. The obstacles which we encounter is that we cannot get rid of the non-linear term evaluated as a zero gradient and it appear at the boundary, for the part of the measure-data, a term which cannot vanish, when one uses the integration by parts formula. To overcome this difficulties, we first redefine and extend the function which appears in the third Leray-Lions-type conditions and we add a penalization term on the boundary. Secondly, we consider a smooth domain in order to work with the Sobolev spaces that are the closure of indefinitely differentiable and null functions on the bounary, and to going back later to the classical Sobolev space. Then, we assume that the domain is extensible. Next, we obtain a priori estimates and convergence results in the approach problem, which allow us to delete the penalization term. To finish, we introduce a notion of entropy solution for our main problem and prove that it is the limit of the solution obtained in the variational case.},
     year = {2019}
    }
    

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  • TY  - JOUR
    T1  - Existence and Uniqueness of Entropy Solution for an Elliptic Problem with Nonlinear Boundary Conditions and Measure-data
    AU  - Arouna Ouedraogo
    Y1  - 2019/09/16
    PY  - 2019
    N1  - https://doi.org/10.11648/j.ajam.20190704.13
    DO  - 10.11648/j.ajam.20190704.13
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
    SP  - 114
    EP  - 126
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20190704.13
    AB  - In this paper, we study a class of elliptic equations on a bounded domain with nonlinear boundary conditions of type graph and measure - data. First of all, give some spaces and basic assumptions. Next, we apply the classical variational approach. So, we need an essentially bounded estimate on the solution, which is not evident to obtain directly in our problem. The obstacles which we encounter is that we cannot get rid of the non-linear term evaluated as a zero gradient and it appear at the boundary, for the part of the measure-data, a term which cannot vanish, when one uses the integration by parts formula. To overcome this difficulties, we first redefine and extend the function which appears in the third Leray-Lions-type conditions and we add a penalization term on the boundary. Secondly, we consider a smooth domain in order to work with the Sobolev spaces that are the closure of indefinitely differentiable and null functions on the bounary, and to going back later to the classical Sobolev space. Then, we assume that the domain is extensible. Next, we obtain a priori estimates and convergence results in the approach problem, which allow us to delete the penalization term. To finish, we introduce a notion of entropy solution for our main problem and prove that it is the limit of the solution obtained in the variational case.
    VL  - 7
    IS  - 4
    ER  - 

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Author Information
  • Department of Mathematics, Norbert Zongo University, Koudougou, Burkina Faso

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