$G$-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb R^3$

Teresa Alberico; Costantino Capozzoli; Luigi D'Onofrio; Roberta Schiattarella

doi:10.3934/dcdss.2019009

Article Contents

2019, Volume 12, Issue 2: 129-137. Doi: 10.3934/dcdss.2019009

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$G$-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb R^3$

1.
Dipartimento di Matematica e Applicazioni, Università di Napoli "Federico Ⅱ", 80126 Napoli, Italy
2.
Università degli Studi di Napoli "Parthenope", Via Parisi 13, 80100 Napoli, Italy

* Corresponding author: Luigi D'Onofrio
* Corresponding author: Luigi D'Onofrio

Received: July 31, 2017

Revised: December 31, 2017

Published: April 2019

The third and fourth authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The third author has been also supported by "Sostegno alla Ricerca Locale " Università degli studi di Napoli "Parthenope ". The fourth author has been partially supported by FIRB 2013 project "Geometrical and qualitative aspects of PDE's"

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Abstract

The aim of this paper is to prove a reverse Hölder inequality for nonnegative adjoint solutions for elliptic operator in non divergence form in $\mathbb R^3$. As an application we generalize a Theorem due to Sirazhudinov and Zhikov [24] and, under suitable assumptions, we characterize the $G$-limit of a sequence of elliptic operator.The operator $N$

$N[v] = \sum\limits_{i,j = 1}^3 {\frac{{{\partial ^2}({a_{ij}}v)}}{{\partial {x_i}\partial {x_j}}}} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right)$

arises naturally as the formal adjoint of the operator in "non divergence form"

$L[u] = \sum\limits_{i,j = 1}^3 {{a_{i,j}}} (x)\frac{{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}} = Tr(A{D^2}u).\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 2 \right)$

The reason to study the solutions of the adjoint operator is that they are not only important for the solvability of $Lu = f$ but for the properties of the Green's function for $L$. There is a long literature in this context, see for example Sÿogren [22], Bauman [2], Fabes and Stroock [12], Fabes, Garofalo, Marĺn-Malavé, and Salsa [11], Escauriaza and Kenig [10], and Escauriaza [9].

Keywords:

Elliptic operators,
G-convergence.

Mathematics Subject Classification: Primary: 35B40; Secondary: 35R05.

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References

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