On a stiff problem in two-dimensional space
A talk in the Bielefeld Stochastic Afternoon series by
Liping Li
| Abstract: | Please contact stochana@math.uni-bielefeld.de for Meeting-ID and Password. In this talk 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 for $x\in \Omega_\varepsilon$ the diagonal of $\mathbf{A}_\varepsilon(x)$ consists of two positive constants $a^{{-}}_\varepsilon,a^\mid_\varepsilon$. Note that $\Omega_\varepsilon$ collapses to the $x_1$-axis, a barrier of zero volume, as $\varepsilon\downarrow 0$. The main purpose is to derive all possible limiting process $X$ of the diffusion $X^\varepsilon$ associated to this heat equation as $\varepsilon\downarrow 0$. In addition, the limiting flux $u$ of the solution $u^\varepsilon$ to this heat equation as $\varepsilon\downarrow 0 $ and all possible boundary conditions satisfied by $u$ will be also characterized. This talk is based on a joint work with Dr. Wenjie Sun. |