Triplets of Closely Embedded Hilbert Spaces

TL;DR

This paper introduces a new framework for triplets of Hilbert spaces with unbounded embeddings, based on a positive selfadjoint operator, and applies it to solve degenerate elliptic PDEs without traditional coercivity assumptions.

Contribution

It develops a general theory of triplets of closely embedded Hilbert spaces using unbounded operators, extending classical concepts and providing new tools for PDE analysis.

Findings

Established existence and uniqueness of such triplets.

Demonstrated symmetry properties of the triplets.

Applied the theory to weak solutions of degenerate elliptic PDEs.

Abstract

We obtain a general concept of triplet of Hilbert spaces with closed (unbounded) embeddings instead of continuous (bounded) ones. The construction starts with a 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, and for which a model is obtained. From this model we get the abstract concept and show that its basic properties are the same with those of the model. 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 $L^2$ 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 Poincar\'e-Sobolev type inequalities.