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A Harnack inequality for degenerate parabolic equations. (English) Zbl 0546.35035
The Harnack inequality for parabolic equations is shown to hold for positive weak solutions of equations of the form \((\partial /\partial x_ i)(a_{ij}(x,t)\partial u/\partial x_ j)=\partial u/\partial t\) which are degenerate in the following way: there are positive constants \(\lambda\),\(\Lambda\), and a non-negative function \(\omega\) on \(R^ n\) such that \[ \lambda\omega (x)|\xi |^ 2\leq a_{ij}(x,t)\xi_ i\xi_ j\leq\Lambda \omega (x)|\xi |^ 2 \] for almost every (x,t) and every \(\xi\). The function \(\omega\) is assumed to belong to a class of functions studied by B. Muckenhoupt [Trans. Am. Math. Soc. 165, 207-226 (1976; Zbl 0236.26016)]. The proof is based on a method used earlier by J. Moser [Commun. Pure Appl. Math. 24, 727-740 (1971; Zbl 0227.35016)].
Reviewer: N.A.Watson

MSC:
35K65 Degenerate parabolic equations
35B45 A priori estimates in context of PDEs
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References:
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