A density property for tensor products of gradients of harmonic functions and applications

TL;DR

This paper proves that tensor products of three or more gradients of harmonic functions are dense in continuous functions on bounded domains in dimensions three and higher, with applications to inverse boundary value problems.

Contribution

It establishes a density property for tensor products of gradients of harmonic functions and applies this to prove uniqueness in inverse boundary value problems.

Findings

Tensor products of three or more gradients are dense in C(Ω̄).

The density result aids in solving inverse boundary value problems.

Discussion of the two-gradient case and its relation to the Calderón problem.

Abstract

We show that tensor products of $k$ gradients of harmonic functions, with $k$ at least three, are dense in $C(\overline{\Omega})$, for any bounded domain $\Omega$ in dimension 3 or higher. The bulk of the argument consists in showing that any smooth compactly supported $k$-tensor that is $L^2$-orthogonal to all such products must be zero. This is done by using a Gaussian quasi-mode based construction of harmonic functions in the orthogonality relation. We then demonstrate the usefulness of this result by using it to prove uniqueness in the inverse boundary value problem for a coupled quasilinear elliptic system. The paper ends with a discussion of the corresponding property for products of two gradients of harmonic functions, and the connection of this property with the linearized anisotropic Calder\'on problem.