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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:261

Article

Mantile, Andrea¹  Posilicano, Andrea²  Sini, Mourad³ 

[1] Univ Reims, CNRS, FR3399, Math Lab, BP 1039, F-51687 Reims, France

[2] Univ Insubria, DiSAT Sez Matemat, Via Valleggio 11, I-22100 Como, Italy

[3] Austrian Acad Sci, RICAM, Altenbergerstr 69, A-4040 Linz, Austria

关键词: Elliptic operators;    Boundary conditions;    Krein's resolvent formulae;    Self-adjoint extensions;   

DOI  :  10.1016/j.jde.2015.11.026

来源: Elsevier

【 摘 要 】

The theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic differential operator on R-n with linear boundary conditions on (a relatively open part of) a compact hypersurface. Our approach allows to obtain Krein-like resolvent formulae where the reference operator coincides with the free operator with domain H-2(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, delta and delta'-type, assigned either on a (n - 1) dimensional compact boundary Gamma = partial derivative Omega or on a relatively open part Sigma subset of 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. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free