Limit behaviour of a class of nonlinear elliptic problems in infinite cylinders

Nicolas Bruyère

Abstract

We study the asymptotic behaviour of the solution of nonlinear monotone elliptic problem \begin{equation*} -{\operatorname{div}}[a(x,Du_l)] = \mu \mbox{ on } \ ( -l,l)^q \times {\omega} \end{equation*} with homogeneous Cauchy-Dirichlet boundary conditions, where ${\omega}$ is a bounded, open, connected subset of $\mathbb R^{N-q}$ with second member in $L^{1}{\omega} +W^{-1,p'}(\omega)$, using the framework of renormalized solutions. Assuming specific dependence of the operator $a$ with respect to the variable $(x_1,x_2) \in ( -l,l ) ^q \times {\omega}$ and that $\mu=\mu(x_2)$, we show the convergence of $u_l$, in an appropriate sense, toward the solution of the same problem posed in $\omega$.

Article information

Source
Adv. Differential Equations Volume 12, Number 10 (2007), 1081-1114.

Dates
First available in Project Euclid: 29 April 2013

Mathematical Reviews number (MathSciNet)
MR2362264

Zentralblatt MATH identifier
1158.35042

Citation

Bruyère, Nicolas. Limit behaviour of a class of nonlinear elliptic problems in infinite cylinders. Adv. Differential Equations 12 (2007), no. 10, 1081--1114.https://projecteuclid.org/euclid.ade/1367241159