An asymptotic analysis of the mean first passage time for narrow escape problems. I: Two-dimensional domains (Q2786355)

Let \(\Omega\) be a smooth bounded domain in \(\mathbb R^2\) whose boundary \(\partial \Omega\) is the union \(\partial\Omega_r\cup\partial\Omega_a\), where \(\partial \Omega_a= \bigcup _{j=1}^{N}\partial\Omega_{\varepsilon_ j}\) is the union of N windows \(\partial\Omega_{ \varepsilon_ j}\) centered at \(x_j\in\partial\Omega\) and whose length is \(\varepsilon l_j\), \( l_j={\mathcal O}(1)\). It is assumed that the windows are well separated, i.e. \(|x_i-x_j|={\mathcal O}(1), i\not =j\). The mean first passage time (MFPT) \(v(x)\) is solution of the Dirichlet-Neumann problem NEWLINENEWLINE\[NEWLINE \Delta v=-\frac{1}{D},\quad x\in\Omega,\tag{1}NEWLINE\]NEWLINE NEWLINENEWLINE\[NEWLINEv=0 \quad\text{on }\partial\Omega_a,\quad \partial_nv=0 \quad\text{on }\partial\Omega_r,\tag{2}NEWLINE\]NEWLINE NEWLINEwhere \(D\) is a diffusion coefficient. The average MFPT \(\bar v\) is defined as the mean value of \(v\) on \(\Omega\). By means of matched asymptotic expansions, the authors study the asymptotic behavior of \(v\) and \(\bar v\), as \(\varepsilon \to 0\). The surface Neumann-Green's function \(G\) for the Laplacian is essentially involved in their investigation. For specific domains such as the unit disk or the unit square, associated functions \(G\) are analytically known. For instance, in the case of a unit disk, with \(N\) equally spaced windows of length \(2\varepsilon\), it leads to the formula NEWLINENEWLINE\[NEWLINE\bar v \sim \frac{1}{DN}\bigg(-\log\bigg(\frac{\varepsilon N}{2}\bigg)+ \frac{N}{8}\bigg)NEWLINE\]NEWLINE NEWLINEFor an arbitrary bounded two-dimensional domain with a sooth boundary, a boundary integral numerical scheme to calculate \(G\) is developed. As an application, a calculation of \(v\) and \(\bar v\) is presented when \(\Omega\) is an ellipse. In the last section of the paper, the first eigenvalue \(\lambda(\varepsilon)\) of the Laplacian in \(\Omega\), associated with the boundary conditions (2), is studied, when \(\partial\Omega_{\varepsilon_ j}\to x_j\) for \(j=1,\dots,N\). An expansion of \(\bar v\) in terms of the principal part \(\lambda *\) of \(\lambda\) is established.