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

Abstract

AbstractThis article investigates nonlocal, quasilinear generalizations of the classical biharmonic operator $$(-\Delta )^2$$ ( - Δ ) 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 $$H^{t,p}$$ H t , p for any $$t\in {\mathbb R}$$ t ∈ R , $$1 \le p < \infty $$ 1 ≤ p < ∞ and $$s \in {\mathbb R}_+ {\setminus } {\mathbb N}$$ s ∈ R + \ N : If $$u\in H^{t,p}({\mathbb R}^n)$$ u ∈ H t , p ( R n ) satisfies $$(-\Delta )^su=u=0$$ ( - Δ ) s u = u = 0 in a nonempty open set V, then $$u\equiv 0$$ u ≡ 0 in $${\mathbb R}^n$$ 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.