Bounded Extremal and Cauchy–Laplace Problems on the Sphere and Shell

Abstract

In this work, we develop a theory of approximating general vector fields on subsets of the sphere in ℝn by harmonic gradients from the Hardy space Hp of the ball, 1<p<∞. 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.