Regularized Subjective-Surface Flow with Monotone Reaction: Viscosity Well-Posedness and Convergence

Abstract

We study the vanishing-regularization limit of the regularized subjective-surface model introduced for touching and dividing cell nuclei and previously analyzed for fixed positive regularization parameters. On a smooth bounded domain $\Omega \subset {\mathbb{R}}^d$, the model is \[ \begin{aligned} \partial_tu^{\eps,ν} ={}&νΔu^{\eps,ν} +A_\eps(\nabla u^{\eps,ν}) \operatorname{div}\!\left( G(x)\frac{\nabla u^{\eps,ν}} {A_\eps(\nabla u^{\eps,ν})} \right) -μR_η(x,u^{\eps,ν}),\\ &A_\eps(p)=\sqrt{\eps^2+|p|^2}, \qquad R_η(x,r)=Λ(x)H_η(r-q), \end{aligned} \] subject to homogeneous Dirichlet data and a prescribed initial profile. We formulate the limiting weighted level-set mean-curvature equation as a viscosity initial-boundary value problem and establish global existence, uniqueness, preservation of $[0,1]$, and nonexpansive dependence on the initial data. We determine the direct upper and lower limiting values of the regularized principal operators at zero gradient and identify the additional argument needed to recover the geometric directional values. For every $T>0$, the regularized solutions converge uniformly to the unique viscosity solution on $[0,T]\times \overline{\Omega }$ as $\eps,ν\to0$, independently of their relative rates of decay. This determines the geometric limit of the regularized model used in 3D and 3D+time microscopy segmentation.