Fundamental solutions of boundary problems and resolvents of differential operators

The main objects of our considerations are differential operators generated by a formally selfadjoint differential expression of an even order. The coefficients of this expression are operator valued functions defined on the interval [0, bi (b ≤ ∞) with values in the set of all linear bounded operators in a separable Hilbert space H. Our approach is based on the concept of a decomposing D-boundary triplet, which enables to describe various properties of (regular and singular) differential operators immediately in terms of boundary conditions

On the 3D steady flow of a second grade fluid past an obstacle

We study steady flow of a second grade fluid past an obstacle in three space dimensions. We prove existence of solution in weighted Lebesgue spaces with anisotropic weights and thus existence of the wake region behind the obstacle. We use properties of the fundamental Oseen tensor together with results achieved in \cite{Koch} and properties of solutions to steady transport equation to get up to arbitrarily small \ep the same decay as the Oseen fundamental solution

Condensation of Silica Nanoparticles on a Phospholipid Membrane

The structure of the transient layer at the interface between air and the aqueous solution of silica nanoparticles with the size distribution of particles that has been determined from small-angle scattering has been studied by the X-ray reflectometry method. The reconstructed depth profile of the polarizability of the substance indicates the presence of a structure consisting of several layers of nanoparticles with the thickness that is more than twice as large as the thickness of the previously described structure. The adsorption of 1,2-distearoyl-sn-glycero-3-phosphocholine molecules at the hydrosol/air interface is accompanied by the condensation of anion silica nanoparticles at the interface. This phenomenon can be qualitatively explained by the formation of the positive surface potential due to the penetration and accumulation of Na+ cations in the phospholipid membrane.Comment: 7 pages, 5 figure

Generalized boundary triples, I. Some classes of isometric and unitary boundary pairs and realization problems for subclasses of Nevanlinna functions

© 2020 The Authors. Mathematische Nachrichten published by Wiley‐VCH Verlag GmbH & Co. KGaA. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.fi=vertaisarvioitu|en=peerReviewed

Pseudospectral functions of various dimensions for symmetric systems with the maximal deficiency index

We consider first-order symmetric system Jy′ −A(t)y = λ∆(t)y with n×n-matrix coefficients defined on an interval [a, b) with the regular endpoint a. It is assumed that the deficiency indices N± of the system satisfies N− ≤ N+ = n. The main result is a parametrization of all pseudospectral functions σ(•) of any possible dimension nσ ≤ n by means of a Nevanlinna parameter τ = {C₀ (λ), C₁ (λ)}

Weyl solutions and j-selfadjointness for Dirac operators

We consider a non-selfadjoint Dirac-type differential expression D(Q)y:=Jn [Formula presented] +Q(x)y, with a non-selfadjoint potential matrix Q∈Lloc 1(I,Cn×n) and a signature matrix Jn=−Jn −1=−Jn ⁎∈Cn×n. Here I denotes either the line R or the half-line R+. With this differential expression one associates in L2(I,Cn) the (closed) maximal and minimal operators Dmax(Q) and Dmin(Q), respectively. One of our main results for the whole line case states that Dmax(Q)=Dmin(Q) in L2(R,Cn). Moreover, we show that if the minimal operator Dmin(Q) in L2(R,Cn) is j-symmetric with respect to an appropriate involution j, then it is j-selfadjoint. Similar results are valid in the case of the semiaxis R+. In particular, we show that if n=2p and the minimal operator Dmin +(Q) in L2(R+,C2p) is j-symmetric, then there exists a 2p×p-Weyl-type matrix solution Ψ(z,⋅)∈L2(R+,C2p×p) of the equation Dmax +(Q)Ψ(z,⋅)=zΨ(z,⋅). A similar result is valid for the expression (0.1) whenever there exists a proper extension A˜ with dim(domA˜/domDmin +(Q))=p and nonempty resolvent set. In particular, it holds if a potential matrix Q has a bounded imaginary part. This leads to the existence of a unique Weyl function for the expression (0.1). The main results are proven by means of a reduction to the self-adjoint case by using the technique of dual pairs of operators. The differential expression (0.1) is of significance as it appears in the Lax formulation of the vector-valued nonlinear Schrödinger equation. © 2019 Elsevier Inc