Free boundary regularity for a tumor growth model with obstacle

TL;DR

This paper establishes existence and regularity results for a tumor growth model with a free boundary constrained by an obstacle, using viscosity solutions and flatness improvement techniques.

Contribution

It introduces a new regularity theory for a geometric free boundary problem with obstacle constraints in tumor growth models.

Findings

Existence of viscosity solutions via Perron's method.

Interior regularity of the free boundary near regular points.

Boundary regularity showing the free boundary meets the obstacle as a C^{1,α} graph.

Abstract

We develop an existence and regularity theory for solutions to a geometric free boundary problem motivated by models of tumor growth. In this setting, the tumor invades an accessible region $D$, its motion is directed along a constant vector $V$, and it cannot penetrate another region $K$ acting as an obstacle to the spread of the tumor. Due to the non variational structure of the problem, we show existence of viscosity solutions via Perron's method. Subsequently, we prove interior regularity for the free boundary near regular points by means of an improvement of flatness argument. We further analyze the boundary regularity and we prove that the free boundary meets the obstacle as a $C^{1,\alpha}$ graph. A key step in the analysis of the boundary regularity involves the study of a thin obstacle problem with oblique boundary conditions, for which we establish $C^{1,\alpha}$ estimates.