A sufficient condition for the harmonicity of functions of two variables

Abstract

As is well known, one can weaken the continuity assumption in the claim that any continuous function u(x, y) satisfying the Laplace equation is harmonic. Tolstov relocated this assumption by the boundedness condition and, later on, the author of the present paper relocated it by the summability condition. The summability condition cannot be substantially weakened here. In the present paper, a generalization of the Laplace equation is studied. Assume that, at every point of a domain, the sum of second derivatives (treated in the Peano sense) of a function along a pair of orthogonal lines passing through the point vanishes, where the directions of the lines in the pair depend on the point in general. It is proved that the summability of the function is sufficient for its harmonicity. One cannot get rid of the orthogonality assumption for the above lines.