Bounded extremal and Cauchy-Laplace problems on the sphere and shell (Q967577)

Autors' abstract: In this work, we develop a theory of approximating general vector fields on subsets of the sphere in \(\mathbb{R}^n\) by harmonic gradients from the Hardy space \(H^p\) of the ball, \(1<p<\infty\). This theory is constructive for \(p=2\), enabling us to solve approximate recovery problems for harmonic functions from incomplete boundary values. An application is given to Dirichlet-Neumann inverse problems for \(n=3\), which are of practical importance in medical engineering. The method is illustrated by two numerical examples.