Integration by parts and Pohozaev identities for space-dependent fractional-order operators

详细信息    查看全文

• 作者：Gerd Grubb ^{grubb@math.ku.dk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Fractional Laplacian ; Nonlocal Dirichlet problem ; Pohozaev identity ; Pseudodifferential operator ; a-Transmission property ; Symbol factorization
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 August 2016
• 年：2016
• 卷：261
• 期：3
• 页码：1835-1879
• 全文大小：616 K

文摘

Consider a classical elliptic pseudodifferential operator P   on Rⁿ${\mathbb{R}}^n$ of order 2a  (0<a<1)$(0<a<1)$ with even symbol. For example, P=A(x,D)^a$P=A{(x,D)}^a$ where f1011d6f72b4b0c66f5b" title="Click to view the MathML source">A(x,D)$A(x,D)$ is a second-order strongly elliptic differential operator; the fractional Laplacian (−Δ)^a${(-\mathrm{\Delta })}^a$ is a particular case. For solutions u   of the Dirichlet problem on a bounded smooth subset Ω⊂Rⁿ$\mathrm{\Omega }\subset {\mathbb{R}}^n$, we show an integration-by-parts formula with a boundary integral involving (d^−au)|_∂Ω$(d^{-a}u){|}_{\partial \mathrm{\Omega }}$, where $d(x)=\mathrm{dist}(x,\partial \mathrm{\Omega })$. This extends recent results of Ros-Oton, Serra and Valdinoci, to operators that are x  -dependent, nonsymmetric, and have lower-order parts. We also generalize their formula of Pohozaev-type, that can be used to prove unique continuation properties, and nonexistence of nontrivial solutions of semilinear problems. An illustration is given with P=(−Δ+m²)^a$P={(-\mathrm{\Delta }+m^2)}^a$. The basic step in our analysis is a factorization of P  , 70f2800876c1a52c864a35e74683" title="Click to view the MathML source">P∼P⁻P⁺$P\sim P^{-}{P}^{+}$, where we set up a calculus for the generalized pseudodifferential operators P^±$P^{\pm }$ that come out of the construction.