Uniqueness of the strong positive solution for a general quasilinear elliptic problem with variable exponents and homogeneous Neumann boundary conditions using a generalization of the $p(x)$-Díaz-Saa inequality
Bogdan Maxim
TL;DR
The paper develops a generalized Díaz-Saa inequality for nonlinear elliptic problems with variable exponents in $W^{1,p(x)}(\Omega)$ under homogeneous Neumann boundaries. It provides an elementary proof of the inequality in this broader setting, analyzes equality cases, and uses these insights to prove a uniqueness result for strong positive weak solutions of a general quasilinear elliptic problem. Two applications are explored: a multi-phase elliptic problem and an image-processing model, demonstrating the practical impact of the theory even when certain standard hypotheses fail. Overall, the work advances the theory of variable-exponent elliptic equations and strengthens uniqueness guarantees for Neumann problems in applied contexts.
Abstract
In this paper, we study a generalization of the Díaz-Saa inequality and its applications to nonlinear elliptic problems. We first present the necessary hypotheses and preliminary results before introducing an improved version of the inequality, which holds in a broader functional setting and allows applications to problems with homogeneous Neumann boundary conditions. The significance of cases where the inequality becomes an equality is also analyzed, leading to uniqueness results for certain classes of partial differential equations. Furthermore, we provide a detailed proof of a uniqueness theorem for strong positive solutions and illustrate our findings with two concrete applications: a multiple-phase problem and an elliptic quasilinear equation relevant to image processing. The paper concludes with possible directions for future research.