Existence of maximal and minimal weak solutions and finite difference approximations for elliptic systems with nonlinear boundary conditions
We establish the existence of maximal and minimal weak solutions between ordered pairs of weak sub- and super-solutions for a coupled system of elliptic equations with quasimonotone nonlinearities on the boundary. We also formulate a finite difference method to approximate the solutions and est...
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| Format: | Article |
| Language: | English |
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Texas State University
2025-04-01
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| Series: | Electronic Journal of Differential Equations |
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| Online Access: | http://ejde.math.txstate.edu/Volumes/2025/43/abstr.html |
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| author | Shalmali Bandyopadhyay Thomas Lewis Nsoki Mavinga |
| author_facet | Shalmali Bandyopadhyay Thomas Lewis Nsoki Mavinga |
| author_sort | Shalmali Bandyopadhyay |
| collection | DOAJ |
| description | We establish the existence of maximal and minimal weak solutions
between ordered pairs of weak sub- and super-solutions for a coupled
system of elliptic equations with quasimonotone nonlinearities on the
boundary. We also formulate a finite difference method to approximate the
solutions and establish the existence of maximal and minimal approximations
between ordered pairs of discrete sub- and super-solutions.
Monotone iterations are formulated for constructing the maximal and minimal
solutions when the nonlinearity is monotone.
Numerical simulations are used to explore existence, nonexistence,
uniqueness and non-uniqueness properties of positive solutions.
When the nonlinearities do not satisfy the monotonicity condition,
we prove the existence of weak maximal and minimal solutions using Zorn’s
lemma and a version of Kato’s inequality up to the boundary. |
| format | Article |
| id | doaj-art-b5b58f6f8e7f4a619e3d7ba1a3b18453 |
| institution | Kabale University |
| issn | 1072-6691 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj-art-b5b58f6f8e7f4a619e3d7ba1a3b184532025-08-20T03:43:40ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912025-04-01202543,121Existence of maximal and minimal weak solutions and finite difference approximations for elliptic systems with nonlinear boundary conditionsShalmali Bandyopadhyay0Thomas Lewis1Nsoki Mavinga2 Univ. of Tennessee, Martin, TN, USA The Univ. of North Carolina, Greensboro, NC, USA Swarthmore College, Swarthmore, PA, USA We establish the existence of maximal and minimal weak solutions between ordered pairs of weak sub- and super-solutions for a coupled system of elliptic equations with quasimonotone nonlinearities on the boundary. We also formulate a finite difference method to approximate the solutions and establish the existence of maximal and minimal approximations between ordered pairs of discrete sub- and super-solutions. Monotone iterations are formulated for constructing the maximal and minimal solutions when the nonlinearity is monotone. Numerical simulations are used to explore existence, nonexistence, uniqueness and non-uniqueness properties of positive solutions. When the nonlinearities do not satisfy the monotonicity condition, we prove the existence of weak maximal and minimal solutions using Zorn’s lemma and a version of Kato’s inequality up to the boundary.http://ejde.math.txstate.edu/Volumes/2025/43/abstr.htmlweak solutionsquasimonotonesubsolutionsupersolutionzorn's lemmafinite difference methodkato's inequality |
| spellingShingle | Shalmali Bandyopadhyay Thomas Lewis Nsoki Mavinga Existence of maximal and minimal weak solutions and finite difference approximations for elliptic systems with nonlinear boundary conditions Electronic Journal of Differential Equations weak solutions quasimonotone subsolution supersolution zorn's lemma finite difference method kato's inequality |
| title | Existence of maximal and minimal weak solutions and finite difference approximations for elliptic systems with nonlinear boundary conditions |
| title_full | Existence of maximal and minimal weak solutions and finite difference approximations for elliptic systems with nonlinear boundary conditions |
| title_fullStr | Existence of maximal and minimal weak solutions and finite difference approximations for elliptic systems with nonlinear boundary conditions |
| title_full_unstemmed | Existence of maximal and minimal weak solutions and finite difference approximations for elliptic systems with nonlinear boundary conditions |
| title_short | Existence of maximal and minimal weak solutions and finite difference approximations for elliptic systems with nonlinear boundary conditions |
| title_sort | existence of maximal and minimal weak solutions and finite difference approximations for elliptic systems with nonlinear boundary conditions |
| topic | weak solutions quasimonotone subsolution supersolution zorn's lemma finite difference method kato's inequality |
| url | http://ejde.math.txstate.edu/Volumes/2025/43/abstr.html |
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