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Harnack inequality and Green function for a certain class of degenerate elliptic differential operators. (English) Zbl 0770.35023

The author considers degenerate linear elliptic operators of the form \[ Lu=-\sum^ n_{i,j=1}D_ i(a_{i,j}(x)D_ ju), \] where the \(n\times n\) matrix \((a_{i,j})\) is assumed to be measurable, symmetric, and satisfying the condition \[ \nu(x)\sum^ n_{i=1}\lambda^ 2_ i(x)\xi^ 2_ i\leq\sum^ n_{i,j=1}a_{i,j}(x)\xi_ i\xi_ j \leq w(x)\sum^ n_{i=1}\lambda^ 2_ i(x)\xi^ 2_ i \] for suitable functions \(\nu,w,\lambda_ i\). By means of the functions \(\lambda_ i\) \((i=1,\dots,n)\) a quasidistance \(\delta\) on \(\mathbb{R}^ n\) is constructed, and for a \(\delta\)-ball \(S\) the Sobolev space \(H(S)\), defined as the completion of \(Lip(S)\) with respect to the norm \[ \| u\|=\left(\int_ S\sum^ n_{i,j=1}a_{i,j}(x)D_ iuD_ judx+\int_ Su^ 2w(x)dx \right)^{1/2}, \] is considered.
The Harnack inequality the author obtains is the following: Let \(Q_ 0\) be a \(\delta\)-ball and let \(u\in H(Q_ 0)\) be a nonnegative solution of \(Lu=0\). If \(Q\) is the \(\delta\)-ball with radius 1/4 of the radius of \(Q_ 0\), then \(\sup_ Qu\leq K\inf_ Qu\) where the constant \(K\) depends only on \(\nu,w,\lambda_ i\). The paper contains also several estimates on the Green function associated to the operator \(L\).
Reviewer: G.Buttazzo (Pisa)

MSC:

35J70 Degenerate elliptic equations
35B45 A priori estimates in context of PDEs
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