Abstract
The Cauchy problem for the Laplace operator
is modified by replacing the Laplace equation by an asymptotic estimate of the form
with a given majoranth, satisfyingh(+0)=0. Thisasymptotic Cauchy problem only requires that the Laplacian decay to zero at the initial submanifold. It turns out that this problem has a solution for smooth enough Cauchy dataf, g, and this smoothness is strictly controlled byh. This gives a new approach to the study of smooth function spaces and harmonic functions with growth restrictions. As an application, a Levinson-type normality theorem for harmonic functions is proved.
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This research was supported by the fund for the promotion of research at the Technion and by the Technion V.P.R. fund—Tragovnik research fund.
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Dyn'kin, E. An asymptotic Cauchy problem for the Laplace equation. Ark. Mat. 34, 245–264 (1996). https://doi.org/10.1007/BF02559547
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DOI: https://doi.org/10.1007/BF02559547