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Shape derivative for the Dirichlet-to-Neumann operator on a manifold and application to cellular protrusion

F Noisette

TL;DR

This work derives a shape-derivative formula for the Dirichlet-to-Neumann operator on compact manifolds and applies it to a free-boundary model of cell protrusions, unifying geometric analysis with a biologically motivated PDE. The authors fix the moving domain via an admissible diffeomorphism, obtain an explicit derivative formula using Alinhac’s good unknown, and linearize the boundary evolution around a flat state. They establish well-posedness of the linearized equation in H^1 through L^2 and H^1 Garding-type energy estimates and a Leray scheme, under a Rayleigh–Taylor-type condition. By generalizing to arbitrary d-dimensional manifolds, the results provide a rigorous basis for stability analysis and pave the way for quasi-linear extensions of the full model with potential applications to understanding cellular protrusion dynamics.

Abstract

We establish a shape-derivative formula for the Dirichlet-to-Neumann operator on a compact manifold. Then, we apply this formula to obtain the well-posedness in H 1 under a specific Rayleigh-Taylor condition to an equation describing cell protrusions. This equation is a generalisation of the theoretical part of [9] to any -2D as well as 3D-surfaces.

Shape derivative for the Dirichlet-to-Neumann operator on a manifold and application to cellular protrusion

TL;DR

This work derives a shape-derivative formula for the Dirichlet-to-Neumann operator on compact manifolds and applies it to a free-boundary model of cell protrusions, unifying geometric analysis with a biologically motivated PDE. The authors fix the moving domain via an admissible diffeomorphism, obtain an explicit derivative formula using Alinhac’s good unknown, and linearize the boundary evolution around a flat state. They establish well-posedness of the linearized equation in H^1 through L^2 and H^1 Garding-type energy estimates and a Leray scheme, under a Rayleigh–Taylor-type condition. By generalizing to arbitrary d-dimensional manifolds, the results provide a rigorous basis for stability analysis and pave the way for quasi-linear extensions of the full model with potential applications to understanding cellular protrusion dynamics.

Abstract

We establish a shape-derivative formula for the Dirichlet-to-Neumann operator on a compact manifold. Then, we apply this formula to obtain the well-posedness in H 1 under a specific Rayleigh-Taylor condition to an equation describing cell protrusions. This equation is a generalisation of the theoretical part of [9] to any -2D as well as 3D-surfaces.

Paper Structure

This paper contains 19 sections, 21 theorems, 152 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The biological context
  3. Description of the Chapter and sketch of the proofs
  4. Shape derivative of the Dirichlet-to-Neumann operator
  5. Reminders of differential geometry
  6. Construction of an admissible diffeomorphism
  7. Definition of the Dirichlet-to-Neumann operator
  8. Shape derivative of the Dirichlet-to-Neumann operator
  9. Linearisation of the equation
  10. Reminder on the biological model
  11. Derivation of the evolution equation for the boundary.
  12. Linearization of the equation \ref{['e:prot-equa']} around $\rho = 0$.
  13. Well posedness of the linearised equation in $H^1$
  14. Regularising properties for $\mathrm{Id}-\mathcal{A}_0$
  15. Gärding type $L^2$ estimates
  16. ...and 4 more sections

Key Result

Proposition 2.1

Let $f,g\in C^1(\mathcal{T}_0,\mathbb{R})$ and $\mathbf{v},\mathbf{w}\in C^1(\mathcal{T}_0,\mathbb{R}^{d+1})$, we have the following:

Figures (3)

  • Figure 1: Schematic description of the biological model.
  • Figure 2: Parametrization of the cell over a reference cell $\Omega_0$: In black the boundary $\Gamma_0$ of the reference cell, the dotted line represent the boundaries of the tubular neighborhood $\mathcal{T}_0$ and the red line represent $\Gamma(\rho)$, the boundary of $\Omega$
  • Figure 3: Interior and exterior domains

Theorems & Definitions (56)

  • Definition 2.1
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Theorem 2.2: Stokes
  • Definition 2.3
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.4
  • proof
  • ...and 46 more