Differential and Integral Equations

$G$-convergence for non-divergence second order elliptic operators in the plane

Teresa Alberico, Costantino Capozzoli, and Luigi D'Onofrio

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The central theme of this paper is the study of $G$-convergence of elliptic operators in the plane. We consider the operator $$ \mathcal{M}[u]=\text{Tr}(A(z) D^2u)=a_{11}(z)u_{xx}+2a_{12}(z)u_{xy}+a_{22}(z)u_{yy} $$ and its formal adjoint $$ \mathcal{N}[v]=D^2(A(w)v)= (a_{11}(w)v)_{xx} + 2(a_{12}(w)v)_{xy}+ (a_{22}(w)v)_{yy}, $$ where $u\in W^{2,p}$ and $v\in L^p$, with $p>1$, and $A$ is a symmetric uniformly bounded elliptic matrix such that $\text{det}A=1$ almost everywhere. We generalize a theorem due to Sirazhudinov--Zhikov, which is a counterpart of the Div-Curl lemma for elliptic operators in non-divergence form. As an application, under suitable assumptions, we characterize the $G$-limit of a sequence of elliptic operators. In the last section we consider elliptic matrices whose coefficients are also in $VMO$; this leads us to extend our result to any exponent $p\in (1,2)$.

Article information

Source
Differential Integral Equations, Volume 26, Number 9/10 (2013), 1127-1138.

Dates
First available in Project Euclid: 3 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1372858566

Mathematical Reviews number (MathSciNet)
MR3100081

Zentralblatt MATH identifier
1299.35085

Subjects
Primary: 35B40: Asymptotic behavior of solutions 35R05: Partial differential equations with discontinuous coefficients or data

Citation

Alberico, Teresa; Capozzoli, Costantino; D'Onofrio, Luigi. $G$-convergence for non-divergence second order elliptic operators in the plane. Differential Integral Equations 26 (2013), no. 9/10, 1127--1138. https://projecteuclid.org/euclid.die/1372858566


Export citation