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Positive solutions of elliptic equations in nondivergence form and their adjoints. (English) Zbl 0557.35033

Consider a uniformly elliptic second order operator L with only continuous coefficients in the principal part which is not in divergence form. The main result of the paper under review is a ”comparison theorem” for positive L-harmonic functions u in a Lipschitz domain \(\Omega\) stating (roughly) that these may approach zero on a portion of \(\partial \Omega\) only at a ”fixed rate” independent of u. (Similar results - which have applications in potential theory - for more regular operators have been obtained by Hunt, Wheeden and Ancona.) Using standard tools of the theory of L-harmonic functions (maximum principle, Harnack inequality etc.) the comparison theorem is shown to follow from an estimate for Green’s function G. Thus the second part of the paper is devoted to the study of G and of positive solutions for \(L^*v=O\). Due to possibly pathological behavior of these much care has to be exercised. One of the ideas is to work with ”normalized” functions \(\tilde v(\)y):\(= v(y)/G(x,y)\) instead of v.
Reviewer: N.Weck

MSC:

35J15 Second-order elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35B50 Maximum principles in context of PDEs
31B25 Boundary behavior of harmonic functions in higher dimensions
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