On a class of coupled obstacle systems

TL;DR

This paper introduces and analyzes a new class of coupled obstacle systems, establishing existence, regularity, and free boundary properties, and explores their limiting behavior as parameters vary.

Contribution

It develops the theory for coupled obstacle systems, including existence, regularity, and free boundary analysis, which are novel extensions of classical obstacle problem results.

Findings

Existence of solutions via variational methods.

Optimal regularity of solutions established.

Free boundary regularity and singularity analysis conducted.

Abstract

In this paper, we explore cooperative and competitive coupled obstacle systems, which, up to now, are new type obstacle systems and formed by coupling two equations belonging to classical obstacle problem. On one hand, applying the constrained minimizer in variational methods we establish the existence of solutions for the systems. Moreover, the optimal regularity of solutions is obtained, which is the cornerstone for further research on so-called free boundary. Furthermore, as coefficient $\lambda\to0$, there exists a sequence of solutions converging to solutions of the single classical obstacle equation. On the other hand, motivated by the heartstirring ideas of single classical obstacle problem, based on the corresponding blowup methods, Weiss type monotonicity formula and Monneau type monotonicity formula of systems to be studied, we investigate the regularity of free boundary, and on the regular and singular points in particular, as it should be, which is more challenging but exceedingly meaningful in solving free boundary problems.