An asymptotic analysis of the mean first passage time for narrow escape problems. II: The sphere (Q2786356)

As in [\textit{S. Pillay, M. J. Ward, A. Peirce} and \textit{T. Kolokolnikov}, Multiscale Model. Simul. 8, No. 3, 803--835 (2010; Zbl 1203.35023)], the authors consider the mean first passage time (MFPT) \(v(x)\), solution of the Dirichlet-Neumann problem: NEWLINENEWLINE\[NEWLINE \Delta v=-\tfrac{1}{D},\quad x\in\Omega,\tag{1}NEWLINE\]NEWLINE NEWLINENEWLINE\[NEWLINEv=0 \quad\text{on }\partial\Omega_a,\qquad \partial_nv=0 \quad \text{on }\partial\Omega_r.\tag{2}NEWLINE\]NEWLINE NEWLINEHere, \(\Omega=\{x\in\mathbb R^3\); \(|x|<1\}\) is the open unit-ball and its boundary \(\partial\Omega\) is the unit-sphere. It is assumed that \(\partial \Omega=\partial\Omega_r\cup\partial\Omega_a\), where \(\partial \Omega_a=\bigcup _{j=1}^{N}\partial\Omega_{\varepsilon_ j}\). The small absorbing circular windows \(\partial \Omega_{\varepsilon_j}\) are of aera \(|\partial\Omega_{\varepsilon_j}|\sim\pi\varepsilon^2a_j^2\) and \(\partial \Omega_{\varepsilon_j} \to x_j, |x_j|=1,\) as \(\varepsilon \to 0\), with \(|x_i-x_j|={\mathcal O}(1)\) for \(i\neq j\). Let NEWLINENEWLINE\[NEWLINE{\mathcal H}(x_1,\dots, x_N)={\mathcal H}_C+\frac{1}{2}{\mathcal H}_L-\frac{1}{2}\sum_{i=1}^N\sum_{j=i+1}^N \log(2+|x_i-x_j|)NEWLINE\]NEWLINE NEWLINEbe the discrete energy-like, where \({\mathcal H}_C\) (resp. \( {\mathcal H}_L\)) is the Coulomb energy (resp., the logarithmic energy). The aim of the paper is to obtain a three terms asymptotic expansions for \(v(x)\), for the average MFPT \(\bar v\), and for the principal eigenvalue \(\lambda\) of the Laplacian in \(\Omega\), with the boundary conditions (2). For instance, for \(N\) windows which have a common radius \(\varepsilon\ll1\), the three terms asymptotic expansion for \(\bar v\) is NEWLINENEWLINE\[NEWLINE\begin{multlined}\bar v=\frac{|\Omega|}{4\varepsilon DN} \Bigg(1+\frac{\varepsilon}{\pi}\log \bigg(\frac{2}{\varepsilon}\bigg)+\frac{\varepsilon}{\pi} \bigg(-\frac{9N}{5}+2(N-2)\log 2+\frac{3}{2}\bigg)\\ +\frac{4}{N}{\mathcal H}(x_1,\dots,x_N))+{\mathcal O}(\varepsilon^2\log\varepsilon)\Bigg). \end{multlined}NEWLINE\]NEWLINE NEWLINEIngredients of the proofs are the spherical coordinates, the surface Neumann Green function for \(\Omega\), and the method of matched asymptotic expansions. In the last section of the paper, various numerical methods are used to compute \({\mathcal H}\), \(\bar v\), \(\lambda\), and the optimal arrangements of the centers \(x_1,\dots,x_N\) of the windows which minimize \({\mathcal H}\) for different values of \(N\) and \(\varepsilon\). Some open problems are mentioned.