Partial data inverse problems for semilinear elliptic equations with gradient nonlinearities

TL;DR

This paper proves the density of certain gradient scalar products in $L^1( abla ext{harmonic functions})$ and applies it to solve inverse boundary problems for semilinear elliptic PDEs with quadratic gradient nonlinearities.

Contribution

It establishes a density result for scalar products of harmonic gradients and uses this to address partial data inverse problems for semilinear elliptic equations.

Findings

Density of scalar products in $L^1( abla ext{harmonic functions})$

Solution of partial data inverse boundary problems for semilinear elliptic PDEs with quadratic gradient terms

Advancement in inverse problems with gradient nonlinearities

Abstract

We show that the linear span of the set of scalar products of gradients of harmonic functions on a bounded smooth domain $\Omega\subset \mathbb{R}^n$ which vanish on a closed proper subset of the boundary is dense in $L^1(\Omega)$. We apply this density result to solve some partial data inverse boundary problems for a class of semilinear elliptic PDE with quadratic gradient terms.