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

Arouna Ouedraogo

doi:doi:10.11648/j.ajam.20190704.13

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

Arouna Ouedraogo

Published in American Journal of Applied Mathematics (Volume 7, Issue 4)

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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APA Style

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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ACS 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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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

Arouna Ouedraogo

Department of Mathematics, Norbert Zongo University, Koudougou, Burkina Faso

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APA Style

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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ACS 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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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}
}

Copy | Download

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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