On a Property of Harmonic Functions

Abstract

If we divide the space ${\mathbb{R}}^n$ into two disjoint areas with one common hypersurface and define a harmonic function in each part of these areas such that their gradients vanish at infinity and the normal components of their gradients are equal on the hypersurface, then for some hypersurfaces such as a circle in ${\mathbb{R}}^2$ or a hyperplane in ${\mathbb{R}}^n$ the sum of the tangential components of the gradients is zero. We investigate for which hypersurfaces we have this property and prove that such hypersurfaces in ${\mathbb{R}}^2$ are only circles and straight lines. We also give an application of this property to an ideal plane flow through a porous surface.