# Unique continuation for a non bi-Laplacian fourth order elliptic operator

## TL;DR

This paper establishes unique continuation principles for a non-Laplacian fourth order elliptic operator using Carleman estimates, including stability and strong unique continuation in two dimensions, expanding understanding beyond classical Laplacian-based operators.

## Contribution

It introduces Carleman estimates for a non-Laplacian fourth order elliptic operator, proving unique continuation and related stability results.

## Key findings

- Proved unique continuation for a non-Laplacian fourth order elliptic operator.
- Derived Carleman estimates as a key tool.
- Established strong unique continuation in 2D.

## Abstract

This paper discusses the unique continuation principal of the solutions of the following perturbed fourth order elliptic differential operator $\mathcal{L}_{A,q}u=0$, where \[ \mathcal{L}_{A,q}(x,D)\ =\ \sum_{j=1}^nD^4_{x_j} + \sum_{j=1}^n A_jD_{x_j} + q, \qquad (A, q) \in W^{1,\infty}(\Omega,\mathbb{C}^n) \times L^{\infty}(\Omega,\mathbb{C}) \] whose principal term is not given by some integer power of the Laplacian operator. We derive some suitable Carleman estimates which is the main tool to prove the unique continuation principle. As a by-product, we also deduce some stability estimate and prove the strong unique continuation principle in $2$-dimension.