Positive solutions of elliptic equations in nondivergence form and their adjoints
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.
scientific article; zbMATH DE number 3888152
scientific article