An asymptotic Cauchy problem for the Laplace equation
type: Item
instance of: scholarly article
title: An asymptotic Cauchy problem for the Laplace equation
zbMATH Open document ID: 0865.35037
DOI: 10.1007/BF02559547
author: Evsey Dyn'kin
published in: Arkiv för Matematik
publication date: 1996-12-12T00:00:00Z
review text: The author studies the asymptotic Cauchy problem for the Laplace equation, namely, to find a smooth \(u(x,y)\) with \((x,y)\in\mathbb{R}^{d+1}\) verifying \[ u(x,y)= h(y)\quad\text{for}\quad y\neq 0\quad\text{and}\quad u(x,0)=f(x),\;{\partial u\over \partial y} (x,0)=g(x) \tag{P} \] (with \(x\in\mathbb{R}^d\)) where \(h\) is an increasing function with \(h(0)=0\). A function \(u\) continuously differentiable in \(\mathbb{R}^{d+1}\), uniformly bounded together with its gradient and twice differentiable outside \(\mathbb{R}^d\) is called admissible. Then, the author proves that if \(h\) is a regular majorant (i.e. satisfying some technical conditions), then there exists an asymptotic solution of the Cauchy problem. This solution has all derivatives bounded. Conversely, if the Cauchy problem has an admissible solution \(u\), then necessarily the Cauchy data belong to a certain space \(M_n\). In particular, the asymptotic Cauchy problem for (P) has an admissible solution \(u\) of compact support iff \(f,g\in C_0(\mathbb{R}^d)\). As applications, the author obtains integral representations for certain classes of harmonic functions, condition for harmonic extension across linear boundary, and a Levinson type theorem for harmonic functions.
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 944332
zbMATH Keywords: quasianalytic functions
zbMATH Keywords: asymptotic solution
zbMATH Keywords: admissible solution
zbMATH Keywords: integral representations
zbMATH Keywords: harmonic extension
zbMATH Keywords: Levinson type theorem
reviewed by: Gheorghe Gussi
MaRDI profile type: Publication
cites work: Q4023465
cites work: Analytic continuation across a linear boundary
cites work: On the existence of a largest subharmonic minorant of a given function
cites work: Uniform Boundedness in Families Related to Subharmonic Functions
cites work: The pseudoanalytic extension
cites work: Q3671563
full work available at URL: https://doi.org/10.1007/bf02559547
OpenAlex ID: W1555335379
description: scientific article; zbMATH DE number 944332