论文标题
分数$ p \，$ - BIHARMONIC SYSTEMS：最佳Poincaré常数，独特的延续和反面问题
The fractional $p\,$-biharmonic systems: optimal Poincaré constants, unique continuation and inverse problems
论文作者
论文摘要
本文调查了经典Biharmonic operator $（-δ）^2 $的非本地，完全非线性的概括。这些分数$ p $ -biharmonic运算符自然出现在贝塞尔电位空间中最佳分数庞加莱常数的变化表征。我们研究各向异性分数$ p $ -biharmonic Systems的基本问题：对相关内部源和外部价值问题的薄弱解决方案的存在和唯一性，独特的延续性能（UCP），单调性关系，单调关系以及外部迪里奇特对 - 尼古兰的映射的相反问题。此外，我们在所有贝塞尔潜在空间中向分数拉普拉斯的UCP展示$ h^{t，p} $的任何$ t \ in \ mathbb {r} $，$ 1 \ leq p <\ leq p <\ inftty $ and $ s \ in \ in \ mathbb in \ mathbb {r} h^{t，p}（\ mathbb {r}^n）$满足$（ - δ）^su = u = u = 0 $ in nonepply Open Set set $ v $，然后$ \ equiv 0 $ in $ \ mathbb {r}^n $。然后，使用分数拉普拉斯的这种属性来获得分数$ p $ biharmonic系统的UCP，并在分析相关的反问题中起着核心作用。我们的证明使用变异方法和Caffarelli-Silvestre扩展。
This article investigates nonlocal, fully nonlinear generalizations of the classical biharmonic operator $(-Δ)^2$. 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 (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 $(-Δ)^su=u=0$ 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.
