Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces

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• 作者：Andrea Mantile^a ; ^{andrea.mantile@univ-reims.fr" class="auth_mail" title="E-mail the corresponding author} ; Andrea Posilicano^b ; ^{andrea.posilicano@uninsubria.it" class="auth_mail" title="E-mail the corresponding author} ; Mourad Sini^c ; ^{mourad.sini@oeaw.ac.at" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35J15 ; 35J25 ; 47B25 ; 47F05
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 July 2016
• 年：2016
• 卷：261
• 期：1
• 页码：1-55
• 全文大小：661 K

文摘

The theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic differential operator on Rⁿ${\mathbb{R}}^n$ with linear boundary conditions on (a relatively open part of) a compact hypersurface. Our approach allows to obtain Kreĭn-like resolvent formulae where the reference operator coincides with the “free” operator with domain H²(Rⁿ)$H^2({\mathbb{R}}^n)$; this provides an useful tool for the scattering problem from a hypersurface. Concrete examples of this construction are developed in connection with the standard boundary conditions, Dirichlet, Neumann, Robin, δ   and δ^′${\delta }^{\prime }$-type, assigned either on a (n−1)$(n-1)$ dimensional compact boundary Γ=∂Ω$\mathrm{\Gamma }=\partial \mathrm{\Omega }$ or on a relatively open part Σ⊂Γ$\mathrm{\Sigma }\subset \mathrm{\Gamma }$. Schatten–von Neumann estimates for the difference of the powers of resolvents of the free and the perturbed operators are also proven; these give existence and completeness of the wave operators of the associated scattering systems.