A free boundary problem in accretive growth

TL;DR

This paper establishes existence, regularity, and variational properties of solutions for a free boundary problem modeling accretive growth, coupling Hamilton-Jacobi and elliptic equations.

Contribution

It introduces a novel free boundary model for accretive growth using a level-set approach and proves existence and regularity of solutions for the coupled system.

Findings

Existence of solutions for the free boundary problem.

Regularity and variational representation of solutions.

Successful iterative method for the coupled system.

Abstract

We prove an existence result for a free boundary problem inspired by the modelization of accretive growth. The growth process is formulated through a level-set approach, leading to a boundary-value problem for a Hamilton-Jacobi equation within a prescribed constraining set. Existence, variational representability, and regularity of solutions to the growth subproblem are investigated. The full system arises from coupling the growth dynamics with an elliptic equation for the activation field. Existence of solutions to the fully coupled free boundary problems is obtained via an iterative procedure.