Triplets of closely embedded Hilbert spaces
Author
Cojuhari, P.
Gheondea, A.
Date
2014Source Title
Integral Equations and Operator Theory
Print ISSN
0378-620X
Electronic ISSN
1420-8989
Publisher
Springer
Volume
81
Issue
1
Pages
1 - 33
Language
English
Type
ArticleItem Usage Stats
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Abstract
We obtain a general concept of triplet of Hilbert spaces with closed (unbounded) embeddings instead of continuous (bounded) ones. We provide a model and an abstract theorem as well for a triplet of closely embedded Hilbert spaces associated to positive selfadjoint operator H, that is called the Hamiltonian of the system, which is supposed to be one-to-one but may not have a bounded inverse. Existence and uniqueness results, as well as left-right symmetry, for these triplets of closely embedded Hilbert spaces are obtained. We motivate this abstract theory by a diversity of problems coming from homogeneous or weighted Sobolev spaces, Hilbert spaces of holomorphic functions, and weighted L2 spaces. An application to weak solutions for a Dirichlet problem associated to a class of degenerate elliptic partial differential equations is presented. In this way, we propose a general method of proving the existence of weak solutions that avoids coercivity conditions and Poincaré–Sobolev type inequalities. © 2014, Springer Basel.
Keywords
Closed embeddingDegenerate elliptic operators
Dirichlet problem
Hamiltonian
kernel operator
Rigged Hilbert spaces
Triplet of Hilbert spaces
Weak solutions