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

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 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 (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 $t\in \mathbb{R}$, $1\le p<\infty$ and $s\in {\mathbb{R}}_{+}\setminus \mathbb{N}$: If $u\in H^{t,p}({\mathbb{R}}^n)$ satisfies $(-\Delta {)}^{s}u=u=0$ 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.

arXiv comment31 pages

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