Regularity and Nondegeneracy for Tumor Growth with Nutrients

In this paper, we study a tumor growth model where the growth is driven by a diffusing nutrient and the tumor expands according to Darcy’s law with a mechanical pressure resulting from the incompressibility of the cells. Our focus is on the free boundary regularity of the tumor patch that holds beyond topological changes. A crucial element in our analysis is establishing the regularity of the hitting time T(x), namely the first time the tumor patch reaches a given point. We achieve this by introducing a novel Hamilton-Jacobi-Bellman (HJB) interpretation of the pressure, which is of independent interest. The HJB structure is obtained by viewing the model as a limit of the Porous Media Equation (PME) and building upon a new variant of the AB estimate. Using the HJB structure, we establish a new Hopf-Lax type formula for the pressure variable. Combined with barrier arguments, the formula allows us to show that T is \(C^{\alpha }\) with \(\alpha =\alpha (d)\), which translates into a mild nondegeneracy of the tumor patch evolution. Building on this and obstacle problem theory, we show that the tumor patch boundary is regular in \({ \mathbb {R}}^d\times (0,\infty )\) except on a set of Hausdorff dimension at most \(d-\alpha \). On the set of regular points, we further show that the tumor patch is locally \(C^{1,\alpha }\) in space-time. This conclusively establishes that instabilities in the boundary evolution do not amplify arbitrarily high frequencies.