# Reconstruction for the coefficients of a quasilinear elliptic partial differential equation

## TL;DR

This paper presents a method to reconstruct specific coefficients in a quasilinear elliptic PDE from boundary measurements, advancing inverse problem techniques for nonlinear PDEs.

## Contribution

It introduces a novel reconstruction approach for the coefficients gamma and b in a quasilinear elliptic equation using the Dirichlet to Neumann map.

## Key findings

- Reconstruction method successfully retrieves coefficients gamma and b.
- Method applicable to smooth bounded domains.
- Provides theoretical foundation for inverse coefficient problems in nonlinear PDEs.

## Abstract

In this paper we consider an inverse coefficients problem for a quasilinear elliptic equation of divergence form $\nabla\cdot\vec{C}(x,\nabla u(x))=0$, in a bounded smooth domain $\Omega$. We assume that $\overrightarrow{C}(x,\vec{p})=\gamma(x)\vec{p}+\vec{b}(x)|\vec{p}|^2+\mathcal{O}(|\vec{p}|^3)$, by expanding $\overrightarrow{C}(x,\vec{p})$ around $\vec{p}=0$. We give a reconstruction method for $\gamma$ and $\vec{b}$ from the Dirichlet to Neumann map defined on $\partial\Omega$.