Linear Property in R^{n}

Ivan Keglević, Hagen, Germany

Keywords: Laplace equation, Harmonic functions, Boundary value problems AMS Subject classification: 35J05, 31C05, 35J25

Abstract: The paper characterizes all hypersurfaces Γ ⊂ R^{n} which have the the property that for all functions which are harmonic in R^{n} \ Γ, lie in C^{∞}(\overline{⊗}) for each Ω ⊂ R^{n} \ Γ and have the same normal derivative on each side of Γ the tangential components of derivative add up to 0 in each point of Γ. It is shown that in R^{2} the set Γ can only be a subset of a circle or a subset of a straight line. For n ≥ 3 the only sets Γ ⊂ R^{n} are contained in a hyperplane. For Γ with this property is the problem of the flow of an ideal fluid through the porous Γ linear, otherwise is the problem strongly non-linear. The approach is to use a standard double layer ansatz over the set Γ and to derive from the condition on the tangential derivatives a local condition, from which then the above results follow by standard geometric considerations.