Work

Existence of maximal and minimal weak solutions and finite difference approximations for elliptic systems with nonlinear boundary conditions

Year: 2025

Type: article

Abstract: 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. For more information see https://ejde.math.txstate.edu/Volumes/2025/43/abstr.html

Source: Electronic Journal of Differential Equations

DOI: 10.58997/ejde.2025.43

Authors: Shalmali Bandyopadhyay, Thomas Lewis, Nsoki Mavinga

Institutions: University of Tennessee at Martin, University of North Carolina at Greensboro, Swarthmore College

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: National Science Foundation, Division of Mathematical Sciences

Awards: G202495085, Mathematical Sciences Research Institute (MSRI), Narrow-Stencil Numerical Methods for Approximating Nonlinear Elliptic Partial Differential Equations