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September/October 2013 $G$-convergence for non-divergence second order elliptic operators in the plane
Teresa Alberico, Costantino Capozzoli, Luigi D'Onofrio
Differential Integral Equations 26(9/10): 1127-1138 (September/October 2013). DOI: 10.57262/die/1372858566


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 $p\in (1,2)$.


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Teresa Alberico. Costantino Capozzoli. Luigi D'Onofrio. "$G$-convergence for non-divergence second order elliptic operators in the plane." Differential Integral Equations 26 (9/10) 1127 - 1138, September/October 2013.


Published: September/October 2013
First available in Project Euclid: 3 July 2013

zbMATH: 1299.35085
MathSciNet: MR3100081
Digital Object Identifier: 10.57262/die/1372858566

Primary: 35B40 , 35R05

Rights: Copyright © 2013 Khayyam Publishing, Inc.

Vol.26 • No. 9/10 • September/October 2013
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