Uniqueness and free interpolation for logarithmic potentials and the Cauchy problem for the Laplace equation in \(\mathbb{R}^ 2\)
type: Item
instance of: scholarly article
title: Uniqueness and free interpolation for logarithmic potentials and the Cauchy problem for the Laplace equation in \(\mathbb{R}^ 2\)
zbMATH Open document ID: 0833.31001
DOI: 10.1007/BF01895831
author: M. Giesecke
author: V. P. Havin
author: A. B. Aleksandrov
author: Jean Bourgain
author: Yu. Ya. Vymenets
published in: Geometric and Functional Analysis. GAFA
publication date: 1996-03-12T00:00:00Z
full work available at URL: https://eudml.org/doc/58200
review text: The paper contains a deep study of several classes of subsets of the unit circle which arise naturally in the theory of boundary value problems for harmonic functions on the unit disc. Hitherto unpublished results of the individual authors from the period 1989-94 are brought together in the paper. Here we give two definitions and then state two sample results. Let \(D\) denote the open unit disc, let \(T= \partial D\), and let \(H^1= \{u\in C^1 (D\cup T)\): \(u\) is harmonic on \(D\}\). (In this review all functions are real-valued.) Also let \(D_n u\) denote the outward normal derivative of a function \(u\in C^1 (D\cup T)\). Definitions: Let \(E\) be a closed subset of \(T\). (i) \(E\) is called a uniqueness set if the only function \(u\in H^1\) satisfying \(u|E= D_n u|E=0\) is the zero function. (ii) \(E\) is called solvable if \(\{(u|E, D_n u|E)\): \(u\in H^1\}= C^1 (E)\times C(E)\) (where \(C^1 (E)\) is the set of restrictions to \(E\) of functions in \(C^1 (T)\)). Sample results: (i) A closed subset \(E\) of \(T\) is solvable if and only if \(E\) is porous (that is, there exists \(c>0\) such that every arc \(I\subset T\) contains an arc \(J\), disjoint from \(E\), with \(\text{length}(J)\geq c\text{ length}(I)\)). (ii) If \(E\) is a closed, infinite subset of \(T\), then there exists a countable set \(e\subset T\setminus E\) such that \(E\) contains all the limit points of \(e\) and \(e\cup E\) is a uniqueness set.
Mathematics Subject Classification ID: 31A25
Mathematics Subject Classification ID: 35R25
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 785233
zbMATH Keywords: ill-posed
zbMATH Keywords: uniqueness
zbMATH Keywords: Cauchy problem
zbMATH Keywords: boundary value problems for harmonic functions on the unit disc
reviewed by: D. H. Armitage
MaRDI profile type: Publication
cites work: The interpolating sets for \(A^ \infty\)
cites work: Q3924466
cites work: Q5338374
cites work: Q3881103
cites work: Sets of uniqueness for functions regular in the unit circle
cites work: Q3922907
cites work: Q3961103
cites work: Q4314473
cites work: Q5727906
cites work: A Criterion for the Nonporosity of a General Cantor Set
cites work: Q3240967
cites work: Real functions
cites work: Porous Sets and Quasisymmetric Maps
cites work: Inner functions and multipliers of Cauchy type integrals
OpenAlex ID: W2321522050
description: scientific article; zbMATH DE number 785233