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A free boundary problem in accretive growth

Ulisse Stefanelli

TL;DR

This work addresses a free boundary problem modeling accretive growth through a level-set formulation with front time-of-attachment $v$ and an activation field $u$ driven by a space-time regularized front input $Ku$. The coupled system consists of a stationary Hamilton-Jacobi equation $H(x,Ku(x,v(x)),\nabla v)=0$ in the evolving domain and a quasistatic elliptic equation $-\Delta u=1$ on the growing region, with mixed boundary conditions and a variational interpretation of the elliptic problem. Existence is established via an iterative scheme alternately solving the HJ subproblem (in the viscosity sense with a variational representation) and the elliptic subproblem (in the weak sense on John domains), together with regularity results showing John-domain properties and Lipschitz/Hölder controls on the front. The paper situates the problem in the broader viscosity-solution framework, extending known results to its specific constrained, front-interacting setting, and connects to related front-propagation and invasion models, including morpho3 and tumor-growth analyses. Overall, it provides a rigorous existence theory for a biologically motivated, front-driven growth model with a regularized front coupling and a robust functional-analytic backbone for the two-equation coupled system.

Abstract

We prove an existence result for a free boundary problem inspired by the modelization of accretive growth. The growth process is formulated through a level-set approach, leading to a boundary-value problem for a Hamilton-Jacobi equation within a prescribed constraining set. Existence, variational representability, and regularity of solutions to the growth subproblem are investigated. The full system arises from coupling the growth dynamics with an elliptic equation for the activation field. Existence of solutions to the fully coupled free boundary problems is obtained via an iterative procedure.

A free boundary problem in accretive growth

TL;DR

This work addresses a free boundary problem modeling accretive growth through a level-set formulation with front time-of-attachment and an activation field driven by a space-time regularized front input . The coupled system consists of a stationary Hamilton-Jacobi equation in the evolving domain and a quasistatic elliptic equation on the growing region, with mixed boundary conditions and a variational interpretation of the elliptic problem. Existence is established via an iterative scheme alternately solving the HJ subproblem (in the viscosity sense with a variational representation) and the elliptic subproblem (in the weak sense on John domains), together with regularity results showing John-domain properties and Lipschitz/Hölder controls on the front. The paper situates the problem in the broader viscosity-solution framework, extending known results to its specific constrained, front-interacting setting, and connects to related front-propagation and invasion models, including morpho3 and tumor-growth analyses. Overall, it provides a rigorous existence theory for a biologically motivated, front-driven growth model with a regularized front coupling and a robust functional-analytic backbone for the two-equation coupled system.

Abstract

We prove an existence result for a free boundary problem inspired by the modelization of accretive growth. The growth process is formulated through a level-set approach, leading to a boundary-value problem for a Hamilton-Jacobi equation within a prescribed constraining set. Existence, variational representability, and regularity of solutions to the growth subproblem are investigated. The full system arises from coupling the growth dynamics with an elliptic equation for the activation field. Existence of solutions to the fully coupled free boundary problems is obtained via an iterative procedure.
Paper Structure (10 sections, 7 theorems, 134 equations, 3 figures)

This paper contains 10 sections, 7 theorems, 134 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Accretive growth
  3. Boundary conditions
  4. Results
  5. Connections with previous studies
  6. Structure of the paper
  7. Setting and main result
  8. Assumptions
  9. Some remarks on the Hamilton--Jacobi subproblem
  10. Free boundary problem: Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 2.1

Under assumptions eq:H05--eq:H06 there exist $v \in C(\overline \Omega)$ and $u:Q_v \to {\mathbb R}$ measurable with $u(\cdot,t) \in H^1_{\Gamma}(V(t))$ for a.e. $t >0$ such that and $v$ is a viscosity solution to eq:02, namely, for any $\varphi \in C^1({\mathbb R}^n)$, we have that at any local minimum (maximum, respectively) point $x\in \Omega\setminus \overline {V_0}$ of $\varphi - v$. Moreov

Figures (3)

  • Figure 1: Setting and notation.
  • Figure 2: The curve $\gamma$ is obtained by concatenating $\gamma_1$ and $\gamma_2$. The twisted cone around $\gamma_1$ is described by \ref{['eq:elisa1']}, whereas the one around $\gamma_2$ follows from $V_0$ being a John domain.
  • Figure 3: Construction of the curve $\widetilde{\gamma}$ (Step 2). The curve $\gamma$ touches the boundary $\partial \Omega$. Within the $\varepsilon_0$-neighborhood of $\partial \Omega$ (indicated by the dashed curve), the curve $\widetilde{\gamma}$ is obtained by displacing $\gamma$ in the inward normal direction. This allows to fit a twisted cone around $\widetilde{\gamma}$ contained in $V(t)$.

Theorems & Definitions (16)

  • Theorem 2.1: Existence for the free boundary problem
  • Lemma 3.1: Properties of $\sigma$
  • proof
  • Proposition 3.2: Existence and representation in the unconstrained case
  • proof
  • Proposition 3.3: Existence and representation in the constrained case
  • Remark 3.4: Alternative assumptions
  • Remark 3.5: Uniqueness
  • proof : Proof of Proposition \ref{['prop:representation']}
  • Remark 3.6: Alternative assumptions, continued
  • ...and 6 more