Uniqueness and free interpolation for logarithmic potentials and the Cauchy problem for the Laplace equation in \(\mathbb{R}^ 2\)
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 Hitherto unpublished results of the individual authors from the period 1989-94 are brought together in the paper.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.
boundary value problems for harmonic functions on the unit disc
scientific article; zbMATH DE number 785233
scientific article