## Differential and Integral Equations

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

#### 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

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