A rigorous sharp interface limit for diffuse interface models related to tumor growth

In this paper, we study the rigorous sharp interface limit of a diffuse interface model related to the dynamics of tumor growth, when a parameter ε, representing the interface thickness between the tumorous and non-tumorous cells, tends to zero. More in particular, we analyze here a gradient-flow-type model arising from a modification of the recently introduced model for tumor growth dynamics in Hawkins-Daruud et al. (Int J Numer Math Biomed Eng 28:3–24, 2011) (cf. also Hilhorst et al. Math Models Methods Appl Sci 25:1011–1043, 2015). Exploiting the techniques related to both gradient flows and gamma convergence, we recover a condition on the interface Γ relating the chemical and double-well potentials, the mean curvature, and the normal velocity.