Competition of small targets in planar domains: from Dirichlet to Robin and Steklov boundary condition

Abstract

We consider steady-state diffusion in a bounded planar domain with multiple small targets on a smooth boundary. Using the method of matched asymptotic expansions, we investigate the competition of these targets for a diffusing particle and the crucial role of surface reactions on the targets. We start from the classical problem of splitting probabilities for perfectly reactive targets with Dirichlet boundary condition and improve some earlier results. We discuss how this approach can be generalized to partially reactive targets characterized by a Robin boundary condition. In particular, we show how partial reactivity reduces the effective size of the target. In addition, 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-target limit. Finally, we show how our asymptotic approach can be extended to interior targets in the bulk and to exterior problems where diffusion occurs in an unbounded planar domain outside a compact set. Direct applications of these results to diffusion-controlled reactions are discussed.

Figures (14)

• Figure 1: Illustration of a bounded domain $\Omega \subset{\mathbb R}^2$ with a smooth boundary $\partial\Omega$ split into three absorbing patches $\Gamma_{\varepsilon_i}$ of length $2\varepsilon_i$ (in red and blue), and the remaining reflecting part $\partial\Omega_0$ (gray dashed line). For a particle starting from a point $\bm{x}\in\Omega$, the splitting probability $S_1(\bm{x})$ is the probability of hitting the blue patch $\Gamma_{\varepsilon_1}$ first.
• Figure 2: Splitting probability $S_1(\bm{x})$, given by Eq. (\ref{['eq:Sk_two']}), for the unit disk with two Dirichlet patches of length $2\varepsilon_1 = 0.2$ (red) and $2\varepsilon_2 = 0.4$ (blue). Note that $S_1(\bm{x})$ was capped by $0$ and $1$, i.e., we plotted $\max\{0, \min\{1, S_1(\bm{x})\}\}$.
• Figure 3: (a) Function ${\mathcal{C}}(\mu)$ from Eq. (\ref{['eq:Cmu_spectral']}), in which the infinite series is truncated either to 50 terms (solid line) or to 10 terms (crosses), to highlight the accuracy of both truncations. Filled circles indicate the values $-\mu_{2k}$, at which ${\mathcal{C}}(\mu)$ diverges. Dash-dotted line outlines the asymptotic limit $\ln(2)$ of ${\mathcal{C}}(\mu)$ as $\mu\to \infty$. (b) Comparison of ${\mathcal{C}}(\mu)$ and its approximation (\ref{['eq:Cmu_approx']}), which is accurate over a broad range of $\mu$.
• Figure 4: The ratio of ${\mathcal{C}}(\mu)$ with its approximation (\ref{['eq:Cmu_approx']}) is very close to unity on the range $0<\mu<1$.
• Figure 5: Volume-averaged splitting probability $\overline{S}_1 = \chi_1$ for the unit disk, calculated from (\ref{['eq:A2chi_twoRobin_b']}), with two patches of equal length $2\varepsilon = 0.2$ located at boundary points $(\pm 1,0)$. Three curves correspond to three values of the reactivity parameter $q_2$ of the second patch. Symbols present the numerical solution of the BVP (\ref{['eq:Sj_1']}) with Robin boundary condition (\ref{['eq:Sk_RobinBC']}) by a finite-element method in Matlab PDEtool, with the maximal meshsize $0.02$.

Theorems & Definitions (1)

• Remark 1