# Incompressible limit of a mechanical model for tissue growth with non-overlapping constraint. *

## TL;DR

This paper proves that a tissue growth model with a non-overlapping constraint converges to a Hele-Shaw free boundary problem in the incompressible limit, despite using a singular pressure law.

## Contribution

It introduces a non-overlapping constraint into a tissue growth model and shows convergence to the Hele-Shaw problem even with a singular pressure law.

## Key findings

- The model converges to the Hele-Shaw free boundary problem.
- The non-overlapping constraint is preserved in the limit.
- Singular pressure laws can enforce non-overlapping constraints.

## Abstract

A mathematical model for tissue growth is considered. This model describes the dynamics of the density of cells due to pressure forces and proliferation. It is known that such cell population model converges at the incompressible limit towards a Hele-Shaw type free boundary problem. The novelty of this work is to impose a non-overlapping constraint. This constraint is important to be satisfied in many applications. One way to guarantee this non-overlapping constraint is to choose a singular pressure law. The aim of this paper is to prove that, although the pressure law has a singularity, the incompressible limit leads to the same Hele-Shaw free boundary problem.