# Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations

## TL;DR

This paper develops a method to determine the Taylor series of the nonlinear term in semilinear elliptic equations from partial boundary data, enabling simultaneous recovery of boundary features and internal coefficients.

## Contribution

It introduces a higher order linearization technique to solve inverse boundary value problems for semilinear elliptic equations with partial data, advancing understanding beyond linear cases.

## Key findings

- Determines the Taylor series of the nonlinear term from partial boundary data.
- Enables simultaneous detection of cavities or boundary parts and coefficients.
- Extends partial data inverse problem solutions to certain semilinear equations.

## Abstract

We study various partial data inverse boundary value problems for the semilinear elliptic equation $\Delta u+ a(x,u)=0$ in a domain in $\mathbb R^n$ by using the higher order linearization technique introduced in [LLS 19, FO19]. We show that the Dirichlet-to-Neumann map of the above equation determines the Taylor series of $a(x,z)$ at $z=0$ under general assumptions on $a(x,z)$. The determination of the Taylor series can be done in parallel with the detection of an unknown cavity inside the domain or an unknown part of the boundary of the domain. The method relies on the solution of the linearized partial data Calder\'on problem [FKSU09], and implies the solution of partial data problems for certain semilinear equations $\Delta u+ a(x,u) = 0$ also proved in [KU19]. The results that we prove are in contrast to the analogous inverse problems for the linear Schr\"odinger equation. There recovering an unknown cavity (or part of the boundary) and the potential simultaneously are long-standing open problems, and the solution to the Calder\'on problem with partial data is known only in special cases when $n \geq 3$.