Boundary triplets for skew-symmetric operators and the generation of strongly continuous semigroups

We give a self-contained and streamlined exposition of a generation theorem for C0-semigroups based on the method of boundary triplets. We apply this theorem to port-Hamiltonian systems where we discuss recent results appearing in stability and control theory. We give detailed proofs and require only a basic knowledge of operator and semigroup theory.Comment: 19 page

A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains

In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\geq \varepsilon I_{\mathcal{H}}$ for some $\varepsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for $H_{K,\Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-\Delta+V$ (in short, the perturbed Krein Laplacian) defined on $C^\infty_0(\Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $\Omega\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$.Comment: 68 pages. arXiv admin note: extreme text overlap with arXiv:0907.144