On a stiff problem in two-dimensional space

Abstract

In this paper we will study a stiff problem in two-dimensional space and especially its probabilistic counterpart. Roughly speaking, the heat equation with a parameter $\varepsilon>0$ is under consideration: \[ \partial_t u^\varepsilon(t,x)=\frac{1}{2}\nabla \cdot \left(\mathbf{A}_\varepsilon(x)\nabla u^\varepsilon(t,x) \right),\quad t\geq 0, x\in \mathbb{R}^2, \] where $\mathbf{A}_\varepsilon(x)=\text{Id}_2$, the identity matrix, for $x\notin \Omega_\varepsilon:=\{x=(x_1,x_2)\in \mathbb{R}^2: |x_2|<\varepsilon\}$ while $$\mathbf{A}_\varepsilon(x):=\begin{pmatrix} a_\varepsilon^- & 0 \\ 0 & a^\shortmid_\varepsilon \end{pmatrix}$$ with two positive constants $a^-_\varepsilon, a^\shortmid_\varepsilon$ for $x\in \Omega_\varepsilon$. There exists a diffusion process $X^\varepsilon$ on $\mathbb{R}^2$ associated to this heat equation in the sense that $u^\varepsilon(t,x):=\mathbf{E}^xu^\varepsilon(0,X_t^\varepsilon)$ is its unique weak solution. Note that $\Omega_\varepsilon$ collapses to the $x_1$-axis, a barrier of zero volume, as $\varepsilon\downarrow 0$. The main purpose of this paper is to derive all possible limiting process $X$ of $X^\varepsilon$ as $\varepsilon\downarrow 0$. In addition, the limiting flux $u$ of the solution $u^\varepsilon$ as $\varepsilon\downarrow 0 $ and all possible boundary conditions satisfied by $u$ will be also characterized.