Title: Partial Differential Equations and Applied Mathematics Seminar (2016.6.10)

Author: CMAC

Date: 2016-06-08 (19:40)

Date/Time: June 10th, Fri., 11:00 ~ AM

Location: ASTC #616B Yonsei University

Speaker: Prof. Michel Chipot

Affiliation: Institute for Mathematics University of Zurich, Switzerland

Title: Asymptotic issues in cylinders

Abstract:

We would like to present some results on the asymptotic behavior of different problems set in cylindrical domains of the type $l\omega_{1}\times \omega_{2}$ when $l\rightarrow \infty$. For $i=1,2$, $\omega_{i}$ are two bounded open subsets in $\mathbb{R}^{d_{i}}$. To fix the ideas on a simple example consider for instance $\omega_{1}=\omega_{2}=(-1,1)$ and ${u_{l}}$ the solution to $-\Delta u_{l}=f$ in $\Omega_{l}=(-l,l)\times (-1,1)$, $u_{l}=0$ on $\partial \Omega_{l}$. It is more or less clear that, when $l\rightarrow \infty$, ${u_{l}}$ will converge toward $u_{\infty}$ solution to $-\Delta u_{\infty}=f$ in $\Omega_{\infty}=(-\infty,\infty)\times (-1,1)$, $u_{\infty}=0$ on $\partial \Omega_{\infty}$. However this problem has infinitely many solutions since for every integer 𝑘 $exp(k\pi x_{1})sin(k\pi x_{2})$ is solution of the corresponding homogeneous problem. Our goal is to explain the selection process of the solution for different problems of this type when $l\rightarrow \infty$