Numerical verification of positiveness for solutions to semilinear elliptic problems

Abstract

In this paper, we propose a numerical method for verifying the positiveness of solutions to semilinear elliptic boundary value problems. We provide a sufficient condition for a solution to an elliptic problem to be positive in the domain of the problem, which can be checked numerically without requiring a complicated computation. Although we focus on the homogeneous Dirichlet case in this paper (in fact, it is often possible that solutions are not positive near the boundary in this case), our method can be applied naturally to other boundary conditions. We present some numerical examples.