The Asymptotic Analysis of Some PDE and Steklov Eigenvalue Problems with Partially Reactive Patches in 3-D

TL;DR

The paper develops a unified, rigorous three-term asymptotic framework for analyzing diffusion with many small partially reactive patches on a 3D spherical boundary, addressing MFRT, splitting probabilities, and spectral Steklov-type problems (SDN and SN). The approach combines matched asymptotics with local Steklov spectral data and geodesic coordinates, yielding explicit formulas that separate leading, intermediate, and global interaction effects via patch capacitances $C_i(\kappa_i)$, monopole coefficients $E_i(\kappa_i)$, and the surface Neumann Green’s function. Key contributions include dimensionless three-term expansions valid for arbitrary patch shapes and reactivities, a homogenization formula for effective reactivity with many patches, and detailed nonresonant and near-resonant analyses for the SDN and SN eigenvalue problems, validated by numerical comparisons. The results illuminate how patch geometry, activity, and arrangement govern diffusion-controlled processes and spectral properties, with potential applications to reaction kinetics, transport in porous media, and surface-geometry-driven trapping phenomena.

Abstract

We consider steady-state diffusion in a three-dimensional bounded domain with a smooth reflecting boundary that is partially covered by small partially reactive patches. By using the method of matched asymptotic expansions, we investigate the competition of these patches for a diffusing particle and the crucial role of surface reactions on these targets. After a brief overview of former contributions to this field, we first illustrate our approach by considering the classical problems of the mean first-reaction time (MFRT) and the splitting probability for partially reactive patches characterized by a Robin boundary condition. For a spherical domain, we derive a three-term asymptotic expansion for the MFRT and splitting probabilities in the small-patch limit. This expansion is valid for arbitrary reactivities, and also accounts for the effect of the spatial configuration of patches on the boundary. Secondly, we consider more intricate surface reactions modeled by mixed Steklov-Neumann or Steklov-Neumann-Dirichlet problems. We provide the first derivation of the asymptotic behavior of the eigenvalues and eigenfunctions for these spectral problems in the small-patch limit for a spherical domain. Extensions of these asymptotic results to arbitrary domains and their physical applications are discussed.

Figures (12)

• Figure 1: (a): Sketch of a Brownian trajectory in the unit sphere in ${\mathbb{R}}^{3}$ with partially reactive patches ${\partial\Omega}^{{\varepsilon}}_1, \ldots, {\partial\Omega}^{{\varepsilon}}_N$ on the boundary. (b): Geodesic normal coordinates $(\xi_1,\xi_2,\xi_3)^T$ centered at ${\bf{x}}_i\in \partial\Omega$, with the geodesics (orange and blue curves) indicated.
• Figure 1: The reactive capacitance $C_i(\kappa_i)$ for a circular patch ${\Gamma}_i$ of unit radius ($a_i = 1$), as computed from (\ref{['eq:Cmu_def0']}). Filled circles presents the poles $\{-\mu_{ki}\}$, all located on the negative axis, at which $C_i(\kappa_i)$ diverges. The dash-dotted horizontal line indicates the asymptotic limit $C_i(\infty)=2/\pi$.
• Figure 1: Dimensionless volume-averaged MFRT $\overline{u}$ from (\ref{['mfpt_b:main_res_2']}) to two identical circular patches of angle $\epsilon$ (with $a_1 = a_2 = 1$), located at the north and south poles of the unit sphere, with $\kappa_1 = \kappa_2 = \infty$ (squares) or $\kappa_1 = \kappa_2 = 1$ (circles). The curves present the asymptotic formula (\ref{['mfpt_b:main_res_2']}) with ${\varepsilon} = 2\sin(\epsilon/2)$, whereas symbols indicate the FEM solution with the maximal meshsize $h_{\rm max} = 0.0025$.
• Figure 1: The volume-averaged splitting probability $\overline{u}$ to the first circular patch of reactivity $\kappa_1 = \kappa$ in the presence of the second circular patch of infinite reactivity ($\kappa_2 = \infty$). Two patches are located at the north and south poles of the unit sphere. Two configurations are considered: (i) patches of equal radii ($a_1 = a_2 = 1$, ${\varepsilon} = 0.2$) and (ii) patches of different radii ($a_1 = 1$, $a_2 = 0.5$, ${\varepsilon} = 0.4$). The curves present the asymptotic formula (\ref{['split_b:main_res_2']}), while the symbols indicate a FEM solution with the maximal meshsize $h_{\rm max} = 0.0025$.
• Figure 1: Asymptotic behavior of the first two SN eigenvalues (that correspond to axially-symmetric eigenfunctions) for a single circular patch of radius ${\varepsilon}$ on the unit sphere. Each curve shows the difference between the numerical value $\sigma^{(k)}_{\rm num}$, computed using the methodology in Appendix \ref{['appf:numer']}, and its asymptotic value $\sigma^{(k)}_{\rm asy}$ given in (\ref{['sn:example_1']}). This difference is shown as a function of ${\varepsilon}^2$ to highlight the order of the error estimate in (\ref{['sn:example_1']}).

Theorems & Definitions (24)

• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Proof 1
• Proposition 1
• Proposition 2
• Proposition 3
• Remark 1
• Remark 2
• Remark 3