Existence of a unique solution to an elliptic partial differential equation when the average value is known
The purpose of this paper is to prove the existence of a unique classical solution $u(\mathbf{x})$ to the quasilinear elliptic partial differential equation $\nabla \cdot(a(u) \nabla u)=f$ for $\mathbf{x} \in \Omega$, which satisfies the condition that the average value $\frac{1}{|\Omega|}\int_{\Ome...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2021-11-01
|
| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/10.3934/math.2021031/fulltext.html |
| _version_ | 1818452736627179520 |
|---|---|
| author | Diane Denny |
| author_facet | Diane Denny |
| author_sort | Diane Denny |
| collection | DOAJ |
| description | The purpose of this paper is to prove the existence of a unique classical solution $u(\mathbf{x})$ to the quasilinear elliptic partial differential equation $\nabla \cdot(a(u) \nabla u)=f$ for $\mathbf{x} \in \Omega$, which satisfies the condition that the average value $\frac{1}{|\Omega|}\int_{\Omega} u d\mathbf{x}=u_0$, where $u_0$ is a given constant and $\frac{1}{|\Omega|}\int_{\Omega} f d\mathbf{x}=0$. Periodic boundary conditions will be used. That is, we choose for our spatial domain the N-dimensional torus $\mathbb{T}^N$, where $N=2$ or $N=3$. The key to the proof lies in obtaining a priori estimates for $u$. |
| first_indexed | 2024-12-14T21:27:50Z |
| format | Article |
| id | doaj.art-f0aed10669864b0ab94c4b55de487fc2 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-14T21:27:50Z |
| publishDate | 2021-11-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-f0aed10669864b0ab94c4b55de487fc22022-12-21T22:46:46ZengAIMS PressAIMS Mathematics2473-69882021-11-016151853110.3934/math.2021031Existence of a unique solution to an elliptic partial differential equation when the average value is knownDiane Denny0Department of Mathematics and Statistics, Texas A&M University-Corpus Christi, 6300 Ocean Drive, Corpus Christi, TX 78412, USAThe purpose of this paper is to prove the existence of a unique classical solution $u(\mathbf{x})$ to the quasilinear elliptic partial differential equation $\nabla \cdot(a(u) \nabla u)=f$ for $\mathbf{x} \in \Omega$, which satisfies the condition that the average value $\frac{1}{|\Omega|}\int_{\Omega} u d\mathbf{x}=u_0$, where $u_0$ is a given constant and $\frac{1}{|\Omega|}\int_{\Omega} f d\mathbf{x}=0$. Periodic boundary conditions will be used. That is, we choose for our spatial domain the N-dimensional torus $\mathbb{T}^N$, where $N=2$ or $N=3$. The key to the proof lies in obtaining a priori estimates for $u$.https://www.aimspress.com/article/10.3934/math.2021031/fulltext.htmlellipticexistenceuniqueness |
| spellingShingle | Diane Denny Existence of a unique solution to an elliptic partial differential equation when the average value is known AIMS Mathematics elliptic existence uniqueness |
| title | Existence of a unique solution to an elliptic partial differential equation when the average value is known |
| title_full | Existence of a unique solution to an elliptic partial differential equation when the average value is known |
| title_fullStr | Existence of a unique solution to an elliptic partial differential equation when the average value is known |
| title_full_unstemmed | Existence of a unique solution to an elliptic partial differential equation when the average value is known |
| title_short | Existence of a unique solution to an elliptic partial differential equation when the average value is known |
| title_sort | existence of a unique solution to an elliptic partial differential equation when the average value is known |
| topic | elliptic existence uniqueness |
| url | https://www.aimspress.com/article/10.3934/math.2021031/fulltext.html |
| work_keys_str_mv | AT dianedenny existenceofauniquesolutiontoanellipticpartialdifferentialequationwhentheaveragevalueisknown |