Abstract
Let Γ be a non-singular real-analytic hypersurface in some domainU ⊂ ℝn and let Har0(U, Γ) denote the linear space of harmonic functions inU that vanish on Γ. We seek a condition onx 0,x 1 ∈U/Γ such that the reflection law (RL)u(x 0)+Ku(x 1)=0, ∀u∈Har0(U, Γ) holds for some constantK. This is equivalent to the class Har0 (U, Γ) not separating the pointsx 0,x 1. We find that in odd-dimensional spaces (RL)never holds unless Γ is a sphere or a hyperplane, in which case there is a well known reflection generalizing the celebrated Schwarz reflection principle in two variables. In even-dimensional spaces the situation is different. We find a necessary and sufficient condition (denoted the SSR—strong Study reflection—condition), which we described both analytically and geometrically, for (RL) to hold. This extends and complements previous work by e.g. P.R. Garabedian, H. Lewy, D. Khavinson and H. S. Shapiro.
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Ebenfelt, P., Khavinson, D. On point to point reflection of harmonic functions across real-analytic hypersurfaces in ℝn . J. Anal. Math. 68, 145–182 (1996). https://doi.org/10.1007/BF02790208
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DOI: https://doi.org/10.1007/BF02790208