# Monotonicity and enclosure methods for the p-Laplace equation

## TL;DR

This paper demonstrates that the convex hull of a monotone perturbation in a p-conductivity equation can be uniquely identified using the nonlinear Dirichlet-Neumann map, employing monotonicity and enclosure methods without smoothness assumptions.

## Contribution

It introduces two independent, constructive proofs for the unique determination of the convex hull in the p-conductivity problem, expanding the applicability of inverse methods.

## Key findings

- Convex hull can be determined from the nonlinear Dirichlet-Neumann operator.
- Two independent proofs: monotonicity method and enclosure method.
- No smoothness or jump conditions needed on the conductivity perturbation.

## Abstract

We show that the convex hull of a monotone perturbation of a homogeneous background conductivity in the $p$-conductivity equation is determined by knowledge of the nonlinear Dirichlet-Neumann operator. We give two independent proofs, one of which is based on the monotonicity method and the other on the enclosure method. Our results are constructive and require no jump or smoothness properties on the conductivity perturbation or its support.