Abstract
The Cauchy problem for the Laplace operator $\sum\limits_{k = 1}^\infty {\frac{{\left| {\hat f(n_k )} \right|}}{k}} \leqslant const\left\| f \right\|1$ is modified by replacing the Laplace equation by an asymptotic estimate of the form $\begin{gathered} \Delta u(x,y) = 0, \hfill \\ u(x,0) = f(x),\frac{{\partial u}}{{\partial y}}(x,0) = g(x) \hfill \\ \end{gathered} $ with a given majorant h, satisfying h(+0)=0. This asymptotic 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 data f, g, and this smoothness is strictly controlled by h. 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.
Funding Statement
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.
Citation
Evsey Dyn'kin. "An asymptotic Cauchy problem for the Laplace equation." Ark. Mat. 34 (2) 245 - 264, October 1996. https://doi.org/10.1007/BF02559547
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