The nonlinear N-membranes evolution problem

TL;DR

This paper investigates the evolution of the N-membranes problem involving the p-Laplacian, focusing on approximation, regularity, stability, and long-term behavior of solutions with ordering constraints.

Contribution

It extends the characterization of Lagrange multipliers to the evolutionary case and analyzes the asymptotic behavior of solutions and coincidence sets.

Findings

Lagrange multipliers characterized via characteristic functions

Continuous dependence of solutions established

Solutions and sets converge to stationary states as time approaches infinity

Abstract

The parabolic N-membranes problem for the p-Laplacian and the complete order constraint on the components of the solution is studied in what concerns the approximation, the regularity and the stability of the variational solutions. We extend to the evolutionary case the characterization of the Lagrange multipliers associated with the ordering constraint in terms of the characteristic functions of the coincidence sets. We give continuous dependence results, and study the asymptotic behavior as $t \to \infty$ of the solution and the coincidence sets, showing that they converge to their stationary counterparts.