On a property of harmonic functions (Q1346949)

Let \(\Gamma\) be a piecewise smooth hypersurface in \(\mathbb{R}^ n\) which divides \(\mathbb{R}^ n\) into two domains. Then \(\Gamma\) has the so-called ``linear property'' if for each pair \((\Phi_ -, \Phi_ +)\) of harmonic functions \(\Phi_ \pm\in C^ 1 (\mathbb{R}^ n_ \pm \cup \Gamma)\) the identities \(\partial_ \nu \Phi_ -= \partial \nu \Phi_ +\) on \(\Gamma\) (\(\nu\) is the unit normal), \(\lim_{| x|\to \infty} \text{grad } \Phi_ \pm (x)=0\) imply \(\text{Grad } \Phi_ -+ \text{Grad } \Phi_ + =0\) on \(\Gamma\) (\(\text{Grad } \Phi\) denotes the tangential projection of \(\text{grad } \Phi\) on \(\Gamma\)). The goal of the paper is to characterize all hypersurfaces in \(\mathbb{R}^ 2\) having the above property. Using complex methods the author proves that the only hypersurfaces possessing the linear property are circles and straight lines. An application of these results is given to the description of an ideal flow through a porous surface.