$p(x)$-Stability of the Dirichlet problem for Poisson's equation with variable exponents

TL;DR

This paper proves that solutions to the Dirichlet problem for variable exponent p(x)-Laplacians are stable under uniform limits of the exponent sequence, ensuring convergence to the solution with the limiting exponent.

Contribution

It establishes the stability of solutions to the p(x)-Laplacian Dirichlet problem under uniform convergence of the variable exponent sequence.

Findings

Solutions converge when p_j(x) increases uniformly to p(x).

Solutions also converge when p_j(x) decreases to p(x).

Results extend stability understanding for variable exponent problems.

Abstract

It is shown that if the sequence $(p_j(x))$ increases uniformly to $p(x)$ in a bounded, smooth domain $\Omega$, then the sequence $(u_i)$ of solutions to the Dirichlet problem for the $p_i(x)$-Laplacian with fixed boundary datum $\varphi$ converges (in a sense to be made precise) to the solution $u_p$ of the Dirichlet problem for the $p(x)$-Laplacian with boundary datum $\varphi$. A similar result is proved for a decreasing sequence $p_j\searrow p$