An asymptotic Cauchy problem for the Laplace equation

Evsey Dyn'kin Department of Mathematics, Technion

TBD mathscidoc:1701.332858

Arkiv for Matematik, 34, (2), 245-264, 1995.5
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.
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  title={An asymptotic Cauchy problem for the Laplace equation},
  author={Evsey Dyn'kin},
  booktitle={Arkiv for Matematik},
Evsey Dyn'kin. An asymptotic Cauchy problem for the Laplace equation. 1995. Vol. 34. In Arkiv for Matematik. pp.245-264.
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