On a stiff problem in two-dimensional space

TL;DR

This paper investigates the limiting behavior of a diffusion process and its associated heat equation in two-dimensional space as a thin barrier collapses to a line, characterizing the possible limits and boundary conditions.

Contribution

It provides a comprehensive analysis of the limit processes and boundary conditions for a family of diffusions with anisotropic coefficients collapsing onto a line.

Findings

Characterizes all possible limiting diffusions as the barrier shrinks to zero

Describes the boundary conditions satisfied by the limiting heat equation

Identifies the flux conditions at the collapsing barrier

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.