Nondivergence elliptic equations with unbounded coefficients

Luigi Greco; Gioconda Moscariello; Teresa Radice

doi:10.3934/dcdsb.2009.11.131

Article Contents

2009, Volume 11, Issue 1: 131-143. Doi: 10.3934/dcdsb.2009.11.131

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Nondivergence elliptic equations with unbounded coefficients

1.
Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli, Via Cintia - 80126, Napoli
2.
Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli "Federico II", Via Cintia - 80126, Napoli

Received: November 30, 2007

Revised: December 31, 2007

Published: June 2009

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Abstract

We study the nonvariational equation

$ \sum_{i,j=1}^n a_{ij}(x)\,\frac{\partial^2 u}{\partial x_i\,\partial x_j}=f$

in domains of $r^n$. We assume that the coefficients $a_{ij}$ are in $BMO$ and the equation is elliptic, but not uniformly, and consider $f$ in $L^2(r^n)$, or even in the Zygmund class $L^2\log^\alpha L(r^n)$. We also solve Dirichlet problem.

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Mathematics Subject Classification: Primary: 35J15; Secondary: 35J99.

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