A Model of Controlled Growth

TL;DR

This paper models biological tissue growth using a PDE system with a free boundary, where tissue shape results from energy minimization constrained by morphogen-driven volume growth, proving local existence and uniqueness of solutions.

Contribution

It introduces a novel free boundary PDE model for tissue growth incorporating elastic energy minimization and morphogen dynamics, with rigorous proof of local solution existence and uniqueness.

Findings

Proved local existence and uniqueness of solutions.

Established the model's well-posedness for smooth initial domains.

Demonstrated the role of energy minimization in shape determination.

Abstract

We consider a free boundary problem for a system of PDEs, modeling the growth of a biological tissue. A morphogen, controlling volume growth, is produced by specific cells and then diffused and absorbed throughout the domain. The geometric shape of the growing tissue is determined by the instantaneous minimization of an elastic deformation energy, subject to a constraint on the volumetric growth. For an initial domain with $C^{2,\alpha}$ boundary, our main result establishes the local existence and uniqueness of a classical solution, up to a rigid motion.