On the shape of the positivity region for a free boundary problem describing cell polarization

Technical Summary: On the shape of the positivity region for a free boundary problem describing cell polarization

Problem Statement
This paper investigates a mass-constrained free boundary problem modeling cell polarization, specifically in the regime where the total mass of signaling proteins ($M$) is small. The model, derived from a bulk-surface reaction-diffusion system (as established in [15]), reduces to an obstacle-type problem for the density of active signaling proteins $u$ on the cell membrane $\Gamma = \partial \Omega \subset \mathbb{R}^3$. The governing equations are:
$$\begin{cases} -\Delta_\Gamma u = -(1-g)\xi + \alpha g & \text{on } \Gamma, \\ \xi \in [0,1], \quad u\xi = 1 & \text{on } \{u > 0\}, \\ u \ge 0, \quad \int_\Gamma u \, dS = M, \end{cases}$$
where $\Delta_\Gamma$ is the Laplace-Beltrami operator, $g$ is a normalized chemical signal, and $\alpha$ is a Lagrange multiplier associated with the mass constraint. The primary objective is to characterize the asymptotic behavior of the solution $(u_M, \xi_M, \alpha_M)$ and, crucially, the geometry of the free boundary $\partial \{u_M > 0\}$ as $M \to 0$.

Methodology
The authors employ a local analysis around the maximum points of the signal function $g$. The methodology relies on a blow-up technique where the solution is rescaled near a local maximum $p$ of $g$. The scaling depends on the homogeneity degree $\gamma$ of the leading order term in the Taylor expansion of $g_{\text{max}} - g$ near $p$.

1. Rescaling: For a maximum point $p$ where $g(x) \approx g_{\text{max}} - f(x)$ with $f$ being $\gamma$-homogeneous, the authors define a rescaled variable $y$ and a rescaled function $U_m$. The mass $m$ in the neighborhood of $p$ scales as $m \sim M^{\frac{\gamma+4}{\gamma+2}}$ (in the generic case).
2. Limit Problem: The rescaled functions converge to a global solution $U$ of a limiting obstacle problem in $\mathbb{R}^2$:
   $$\begin{cases} -\Delta U = \left( \alpha g_{\text{max}} - \frac{1}{g_{\text{max}}} f(y) \right) \mathbf{1}_{\{U>0\}} & \text{in } \mathbb{R}^2, \\ U \ge 0, \quad \int_{\mathbb{R}^2} U \, dx = 1. \end{cases}$$
3. Analysis of Limit Solutions: The paper establishes the existence and uniqueness of solutions to this limit problem for various forms of $f(y)$. It utilizes comparison principles, non-degeneracy estimates (Lemma 2.4), and regularity results for free boundaries (Proposition 2.5) to prove convergence in $C^{1,\delta}$ and Hausdorff distance for the interfaces.
4. Degenerate Cases: The authors extend the analysis to cases where $g$ has degenerate maxima (where the Hessian is singular or zero), requiring different scaling laws and limit operators (e.g., degenerate elliptic operators or operators with non-trivial zero sets in the forcing term).

Key Contributions and Results

• Generic Case (Morse Functions):
  For a generic signal $g$ with non-degenerate maxima (Morse functions), the authors prove that as $M \to 0$, the mass concentrates at the global maxima of $g$. Specifically, $u_M$ converges to a sum of Dirac deltas centered at these maxima, with weights $\lambda_i$ determined by the eigenvalues of the Hessian $D^2g(p_i)$.

  • Shape of the Free Boundary: Near each maximum, the free boundary $\partial \{u_M > 0\}$ converges to an ellipse. The paper provides an explicit formula for the coefficients of this ellipse based on the eigenvalues of the Hessian.
  • Mass Distribution: The relative mass concentrated at each maximum is explicitly calculated (Theorem 4.1), showing that it depends on the trace and determinant of the Hessian at that point.

• Degenerate Maxima (Theorem 2):
  The paper generalizes the result to maxima where $g_{\text{max}} - g$ behaves like a positive $\gamma$-homogeneous function $f$. The limit profile $U$ is the unique solution to the obstacle problem with a forcing term involving $f$. The free boundary shape depends on the specific homogeneity of $f$.

• Specific Degenerate Examples:
  The authors analyze two specific cases of degenerate maxima to illustrate the variety of behaviors:

  1. Anisotropic Degeneracy (Theorem 3): For $g(x) \approx g_{\text{max}} - ax_1^4 - bx_2^2$, the limit operator becomes degenerate ($-\partial_{22}U$). The free boundary converges to a specific non-elliptical shape defined by $y_2^2/3 + y_1^4 = \text{const}$.
  2. Non-Trivial Zero Set (Theorem 4): For $g(x) \approx g_{\text{max}} - ax_1^4 - bx_1^2 x_2^2 - cx_2^6$, the forcing term in the limit problem has a non-trivial zero set. Despite the potential for multiple scaling regimes, the authors prove a unique scaling limit exists, leading to a solution of $-\Delta U = (\alpha - x_1^4 - x_1^2 x_2^2)\mathbf{1}_{\{U>0\}}$.

• Regularity:
  The paper proves that for sufficiently small mass, the free boundary is a $C^1$ curve near the maximum points. This is achieved by showing the non-degeneracy of the right-hand side of the limit equation near the free boundary, allowing the application of Caffarelli's regularity results.

Significance and Claims
The paper claims to provide a complete classification of the asymptotic behavior of the free boundary for small mass in the generic case of Morse functions, explicitly identifying the limit shape as an ellipse. This extends previous physical literature observations (e.g., [10, 9]) regarding ellipsoidal shapes in obstacle and Stefan problems to a rigorous mathematical setting with explicit coefficients.

Furthermore, the work highlights that the geometry of the free boundary is not universally elliptical; it is dictated by the local homogeneity of the signal $g$ at its maxima. The authors demonstrate that even in degenerate cases, a unique limit profile exists, though the scaling laws and the resulting shapes (e.g., those defined by quartic or mixed-degree polynomials) differ significantly from the generic elliptical case. The results serve as a classification of bounded solutions to specific elliptic obstacle problems with polynomial forcing terms.

The paper does not propose new experimental applications but rather offers a rigorous mathematical framework for understanding the spatial organization of signaling proteins in the limit of low total protein mass, linking the microscopic signal landscape directly to the macroscopic shape of the polarized region.