Density results of biharmonic functions on symmetric tensor fields and their applications to inverse problems

TL;DR

This paper establishes density results for products of biharmonic functions on symmetric tensor fields and applies these to solve a partial data inverse problem involving nonlinear anisotropic perturbations in higher dimensions.

Contribution

It introduces new density results for biharmonic functions on tensor fields and applies them to a novel inverse problem with nonlinear perturbations, extending previous linear and vector field results.

Findings

Density of products of biharmonic functions in tensor fields is proven.

Partial data inverse problem for nonlinear perturbations is solved in dimensions two and higher.

Results extend inverse problem solutions beyond linear and vector field cases.

Abstract

In this article we discuss density of products of biharmonic functions vanishing on an arbitrarily small part of the boundary. We prove that one can use three or more such biharmonic functions to construct a dense subset of smooth symmetric tensor fields up to order three, in a bounded domain. Furthermore, as an application of the density results, in dimension two or higher, we solve a partial data inverse problem for a biharmonic operator with nonlinear anisotropic third and lower order perturbations. For the inverse problem, we take the Dirichlet data to be supported in an arbitrarily small open set of the boundary and measure the Neumann data on the same set. Note that the analogous problem for linear perturbations are still unknown. So far, partial data problems recovering nonlinear perturbations were studied only up to vector fields. The full data analogues of the inverse problem has recently been studied for three or higher dimensions.