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 $\epsilon >0$ 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 ${\mathbf{A}}_{\epsilon }(x)={\text{Id}}_2$, the identity matrix, for $x\notin {\Omega }_{\epsilon }:=\{x=(x_1,x_2)\in {\mathbb{R}}^2:|x_2|<\epsilon \}$ while $${\mathbf{A}}_{\epsilon }(x):=\left(\begin{array}{cc}{a}_{\epsilon }^{-}& 0\\ 0& a_{\epsilon }^{\mid }\end{array}\right)$$ with two positive constants $a_{\epsilon }^{-},a_{\epsilon }^{\mid }$ for $x\in {\Omega }_{\epsilon }$. There exists a diffusion process $X^{\epsilon }$ on ${\mathbb{R}}^2$ associated to this heat equation in the sense that $u^{\epsilon }(t,x):={\mathbf{E}}^x{u}^{\epsilon }(0,X_t^{\epsilon })$ is its unique weak solution. Note that ${\Omega }_{\epsilon }$ collapses to the $x_1$-axis, a barrier of zero volume, as $\epsilon \downarrow 0$. The main purpose of this paper is to derive all possible limiting process $X$ of $X^{\epsilon }$ as $\epsilon \downarrow 0$. In addition, the limiting flux $u$ of the solution $u^{\epsilon }$ as $\epsilon \downarrow 0$ and all possible boundary conditions satisfied by $u$ will be also characterized.

References in corpus (2)

• The snapping out Brownian motion
• Regular subspaces of Dirichlet forms