The fractional $p\,$-biharmonic systems: optimal Poincar\'e constants, unique continuation and inverse problems

TL;DR

This paper explores fractional $p$-biharmonic operators, establishing existence, uniqueness, and unique continuation properties, and addresses inverse problems related to these nonlocal, nonlinear operators using variational methods and extension techniques.

Contribution

It introduces new results on unique continuation and inverse problems for fractional $p$-biharmonic systems, extending understanding of nonlocal nonlinear PDEs.

Findings

UCP for fractional Laplacian in all Bessel potential spaces

Existence and uniqueness of solutions for fractional $p$-biharmonic systems

Development of inverse problem framework for exterior Dirichlet-to-Neumann maps

Abstract

This article investigates nonlocal, fully nonlinear generalizations of the classical biharmonic operator $(-\Delta)^2$. These fractional $p$-biharmonic operators appear naturally in the variational characterization of the optimal fractional Poincar\'e constants in Bessel potential spaces. We study the following basic questions for anisotropic fractional $p$-biharmonic systems: existence and uniqueness of weak solutions to the associated interior source and exterior value problems, unique continuation properties (UCP), monotonicity relations, and inverse problems for the exterior Dirichlet-to-Neumann maps. Furthermore, we show the UCP for the fractional Laplacian in all Bessel potential spaces $H^{t,p}$ for any $t\in \mathbb{R}$, $1 \leq p < \infty$ and $s \in \mathbb{R}_+ \setminus \mathbb{N}$: If $u\in H^{t,p}(\mathbb{R}^n)$ satisfies $(-\Delta)^su=u=0$ in a nonempty open set $V$, then $u\equiv 0$ in $\mathbb{R}^n$. This property of the fractional Laplacian is then used to obtain a UCP for the fractional $p$-biharmonic systems and plays a central role in the analysis of the associated inverse problems. Our proofs use variational methods and the Caffarelli-Silvestre extension.