The fractional p,-biharmonic systems: optimal Poincaré constants, unique continuation and inverse problems

Abstract

This article investigates nonlocal, quasilinear generalizations of the classical biharmonic operator (-Delta )^2. These fractional p -biharmonic operators appear naturally in the variational characterization of the optimal fractional Poincaré 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, 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 tin {mathbb R}, 1 le p < infty and s in {mathbb R}_+ {setminus } {mathbb N}: If uin H^{t,p}({mathbb R}^n) satisfies (-Delta )^su=u=0 in a nonempty open set V, then uequiv 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.