Analyticity of the semigroup generated by the operator of a viscous compressible fluid

The linearized initial‐boundary value problem describing small motions of the viscous, barotropic compressible fluid in a bounded vessel is studied under various boundary conditions (Dirichlet, Neumann and intermediate). It is shown that the corresponding operator generates an analytic semigroup in the space L₂. Copyright © 1999 John Wiley & Sons, Ltd.     

Shape-preserving properties and asymptotic behaviour of the semigroup generated by the Black-Scholes operator

The paper is devoted to a careful analysis of the shape-preserving properties of the strongly continuous semigroup generated by a particular second-order differential operator, with particular emphasis on the preservation of higher order convexity and Lipschitz classes. In addition, the asymptotic behaviour of the semigroup is investigated as well. The operator considered is of interest, since it is a unidimensional Black-Scholes operator so that our results provide qualitative information on the solutions of classical problems in option pricing theory in Mathematical Finance. The paper is dedicated to Professor Luigi Albano on the occasion of his 70th birthday.     

Dirichlet-to-Neumann semigroup acts as a magnifying glass

The first aim of this paper is to illustrate numerically that the Dirichlet-to-Neumann semigroup represented by P. Lax acts as a magnifying glass. In this perspective, we used the finite element method for discretizing of the correspondent boundary dynamical system using the implicit and explicit Euler schemes. We prove by using the Chernoff’s Theorem that the implicit and explicit Euler methods converge to the exact solution and we use the (P1)-finite elements to illustrate this convergence through a FreeFem++ implementation which provides a movie available online. In the Dirichlet-to-Neumann semigroup represented by P. Lax the conductivity \(\gamma \) is the identity matrix \(I_n\) , but for a different conductivity \(\gamma \) , the authors of Cornean et al. (J Inverse Ill-posed Prob 12:111–134, 2006) supplied an estimation of the operator norm of the difference between the Dirichlet-to-Neumann operator \(\Lambda _\gamma \) and \(\Lambda _1\) , when \(\gamma =\beta I_n\) and \(\beta =1\) near the boundary \(\partial \Omega \) (see Lemma 2.1). We will use this result to estimate the accuracy between the correspondent Dirichlet-to-Neumann semigroup and the Lax semigroup, for \(f\in H^{1/2}(\partial \Omega )\) .     

Analytic semigroups generated by Dirichlet-to-Neumann operators on manifolds

Semigroup Forum - We consider the Dirichlet-to-Neumann operator associated to a strictly elliptic operator on the space $$mathrm {C}(partial M)$$ of continuous functions on the boundary...     

On explicit representation and approximations of Dirichlet-to-Neumann semigroup

In his book (Functional Analysis, Wiley, New York, 2002), P. Lax constructs an explicit representation of the Dirichlet-to-Neumann semigroup, when the matrix of electrical conductivity is the identity matrix and the domain of the problem in question is the unit ball in ? ⁿ . We investigate some representations of Dirichlet-to-Neumann semigroup for a bounded domain. We show that such a nice explicit representation as in Lax book, is not possible for any domain except Euclidean balls. It is interesting that the treatment in dimension 2 is completely different than other dimensions. Finally, we present a natural and probably the simplest numerical scheme to calculate this semigroup in full generality by using Chernoff’s theorem.     

The Dirichlet-to-Neumann operator on rough domains

We consider a bounded connected open set Ω⊂Rd whose boundary Γ has a finite (d−1)-dimensional Hausdorff measure. Then we define the Dirichlet-to-Neumann operator D₀ on L₂(Γ) by form methods. The operator −D₀ is self-adjoint and generates a contractive C₀-semigroup S=(St)_t>0 on L₂(Γ). We show that the asymptotic behaviour of St as t→∞ is related to properties of the trace of functions in H¹(Ω) which Ω may or may not have.     

Analysis of the heat kernel of the Dirichlet-to-Neumann operator

We prove Poisson upper bounds for the kernel _(Kt)t>0${(K_t)}_{t>0}$ of the semigroup generated by the Dirichlet-to-Neumann operator if the underlying domain is bounded and has a C^∞$C^{\infty }$-boundary. We also prove Poisson bounds for _Kz$K_z$ for all z in the right half-plane and for all its derivatives.     

Approximation of the semigroup generated by the Robin Laplacian in terms of the Gaussian semigroup

We consider the Laplacian ΔR subject to Robin boundary conditions on the space , where Ω is a smooth, bounded, open subset of RN. It is known that ΔR generates an analytic contraction semigroup. We show how this semigroup can be obtained from the Gaussian semigroup on C₀(RN) via a Trotter formula. As the main ingredient, we construct a positive, contractive, linear extension operator Eβ from to C₀(RN) which maps an operator core for ΔR into the domain of the generator of the Gaussian semigroup.     

On the semigroup generated by the renormalized Nelson Hamiltonian

We consider the renormalized Nelson model at a fixed total momentum P: $H_{\mathrm{ren}}(P)$; the Hamiltonian $H_{\mathrm{ren}}(P)$ is defined through an infinite energy renormalization. We prove that $e^{?\beta H_{\mathrm{ren}}(P)}$ is positivity improving for all $P\in {\mathbb{R}}^3$ and $\beta >0$ in the Fock representation.     

On the first eigenvalue of the Dirichlet-to-Neumann operator on forms

We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical Dirichlet-to-Neumann (or Steklov) problem on functions. We derive a number of upper and lower bounds for the first eigenvalue in several contexts: many of these estimates will be sharp, and for some of them we characterize equality. We also relate these new eigenvalues with those of other operators, like the Hodge Laplacian or the biharmonic Steklov operator.     

The semigroup generated by certain operators on the congruence lattice of a clifford semigroup

Communicated by N. R. Reilly     

A compact quantum semigroup generated by an isometry

In this paper we construct a compact quantum semigroup structure on a Toeplitz algebra. We prove the existence of a subalgebra in the dual algebra isomorphic to the algebra of regular Borel measures on a circle with the convolution product. We also prove the existence of Haar functionals in the dual algebra and in the mentioned subalgebra. We show that this compact quantum semigroup contains a dense subalgebra with the structure of a weak Hopf algebra.     

Dirichlet-to-Neumann semigroup with respect to a general second order eigenvalue problem

In this paper we study a Dirichlet-to-Neumann operator with respect to a second order elliptic operator with measurable coefficients, including first order terms, namely, the operator on \(L^2(\partial \Omega )\) given by \(\varphi \mapsto \partial _{\nu }u\) where u is a weak solution of

$$\begin{aligned} \left\{ \begin{aligned}&-\mathrm{div}\, (a\nabla u) +b\cdot \nabla u -\mathrm{div}\, (cu)+du =\lambda u \ \ \text {on}\ \Omega ,\\&u|_{\partial \Omega } =\varphi . \end{aligned} \right. \end{aligned}$$

Under suitable assumptions on the matrix-valued function a, on the vector fields b and c, and on the function d, we investigate positivity, sub-Markovianity, irreducibility and domination properties of the associated Dirichlet-to-Neumann semigroups.

     

Construction of some quantum stochastic operator cocycles by the semigroup method

A new method for the construction of Fock-adapted quantum stochastic operator cocycles is outlined, and its use is illustrated by application to a number of examples arising in physics and probability. The construction uses the Trotter-Kato theorem and a recent characterisation of such cocycles in terms of an associated family of contraction semigroups. In celebration of Kalyan Sinha’s sixtieth birthday