Approximation by harmonic functions, and stability of the Dirichlet problem
Let \(K\) be a compact subset of \(\mathbb{R}^ d\), where \(d\geq 2\), and let \(K^ o\), \(K^ c\) and \(\partial K\) denote its interior, complement and boundary. A famous theorem of M. V. Keldysh (1941) may be stated as follows: Every function \(h:K \to \mathbb{R}\) which is continuous on \(K\) and harmonic on \(K^ 0\) can be uniformly approximated by functions that are harmonic on some neighbourhood of \(K\) if and only if the sets \(K^ c\) and \(K^{oc}\) are thin at the same points of \(\partial K\). V. P. Khavin (1968) proved a similar result for \(L^ 2\) approximation of holomorphic functions. Let \(K\) be a compact subset of \(C=\mathbb{R}^ 2\). Every function of class \(L^ 2(K)\) that is holomorphic on \(K^ o\) can be approximated in the \(L^ 2\)-norm by functions that are holomorphic on some neighbourhood of \(K\) if and only if the subset of \(\partial K\) where \(K^ c\) is thin has zero capacity. It was realized that the conditions on \(K\) in the theorems of Keldysh and Khavin are equivalent. Hence, on a compact subset of the plane, \(L^ \infty\)-approximation of harmonic functions (in Keldysh's sense) is possible if and only if \(L^ 2\)-approximation of holomorphic functions (in Khavin's sense) is possible, but it seems that no direct proof of this is known: ``... this paper has arisen out of a desire to understand the situation''. ``J. Deny (1950) gave a proof [of Keldysh's theorem] using duality and balayage. Khavin's proof used duality. It seems that the key to an understanding of the equivalence of the two approximation problems lies in an understanding of the dual problems. It turns out that this equivalence can be understood by means of H. Cartan's (1946) theory of balayage, based on projections in Hilbert space.'' The paper is a detailed expository article justifying and expanding the remarks quoted above. The presentation is almost self-contained. Large parts of classical potential theory are required (for example, a study of the Dirichlet problem and thin sets), and these are developed ab initio. The paper concludes with some interesting and helpful historical and bibliographical remarks.
scientific article; zbMATH DE number 428232