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Jürgen Voigt

Publications and source records attributed to Jürgen Voigt.

12 recordsLinked to original sources
arXiv2023

Couplings and primitives on topological spaces

For an open covering $\mathcal U$ of a topological space and a mapping $d\colon\mathcal I\to\mathbb{K}$, where $\mathcal I:=\bigl\{(U,V)\in\mathcal U\times\mathcal U;\ U\cap V\ne\varnothing\bigr\}$, we present a context for the existence of a mapping $C\colon\mathcal U\to\mathbb{K}$ satisfying $C_V-C_U=d_{UV}$ for all $(U,V)\in\mathcal I$. The result is applied to a Poincaré type theorem concerning distributional potentials. We also put the result into the context of algebraic topology.

math.FA↗
arXiv2023

On sequences of sectorial forms converging `from above'

We present a form convergence theorem for sequences of sectorial forms and their associated semigroups in a complex Hilbert space. Roughly speaking, the approximating forms $a_n$ are all `bounded below' by the limiting form $a$, but in contrast to the previous literature there is no monotonicity hypothesis on the sequence. Moreover, the forms are not supposed to be closed or densely defined. For a sectorial form one obtains an associated linear relation, whose negative generates a degenerate strongly continuous semigroup of linear operators. Our hypotheses on the sequence of forms imply strong resolvent convergence of the associated linear relations, which in turn implies convergence of the corresponding semigroups. The result is illustrated by two examples, one of them closely related to the Galerkin method of numerical analysis.

math.FA↗
arXiv2022

On the existence of distributional potentials

We present proofs for the existence of distributional potentials $F\in{\mathcal D}'(Ω)$ for distributional vector fields $G\in{\mathcal D}'(Ω)^n$, i.e. $\operatorname{grad} F=G$, where $Ω$ is an open subset of ${\mathbb R}^n$. The hypothesis in these proofs is the compatibility condition $\partial_jG_k=\partial_kG_j$ for all $j,k\in\{1,\dots,n\}$, if $Ω$ is simply connected, and a stronger condition in the general case. A key ingredient of our treatment is the use of the Bogovskii formula, assigning vector fields $v\in{\mathcal D}(Ω)^n$ with $\operatorname{div} v=φ$ to functions $φ\in{\mathcal D}(Ω)$ with $\int φ(x)\,\mathrm{d}x=0$. The results are applied to properties of Hilbert spaces of functions occurring in the treatment of the Stokes operator and the Navier--Stokes equations.

math.AP↗
arXiv2021

On Hausdorff measure and an inequality due to Maz'ya

We give an "elementary" proof of an inequality due to Maz'ya. As a prerequisite we prove an approximation property for the Hausdorff measure. We also comment on the relations between Maz'ya's inequality, the isoperimetric inequality and the Sobolev inequality.

math.FA↗
arXiv2017

Holomorphic families of forms, operators and $C_0$-semigroups

If $z\mapsto a_z$ is a holomorphic function with values in the sectorial forms in a Hilbert space, then the associated operator valued function $z\mapsto A_z$ is resolvent holomorphic. We give a proof of this result of Kato, on the basis of the Lax-Milgram lemma. We also show that the $C_0$-semigroups $T_z$ generated by $-A_z$ depend holomorphically on $z$.

math.FA↗
arXiv2016

Bands in $L_p$-spaces

For a general measure space $(Ω,μ)$, it is shown that for every band $M$ in $L_p(μ)$ there exists a decomposition $μ=μ'+μ^{\prime\prime}$ such that $M=L_p(μ')=\{f\in L_p(μ);f=0\ μ^{\prime\prime}\text{-a.e.}\}$. The theory is illustrated by an example, with an application to absorption semigroups.

math.FA↗
arXiv2016

Dirichlet forms for singular diffusion on graphs

We describe operators driving the time evolution of singular diffusion on finite graphs whose vertices are allowed to carry masses. The operators are defined by the method of quadratic forms on suitable Hilbert spaces. The model also covers quantum graphs and discrete Laplace operators.

math.FA↗
arXiv2015

Bound States for Nano-Tubes with a Dislocation

As a model for an interface in solid state physics, we consider two real-valued potentials $V^{(1)}$ and $V^{(2)}$ on the cylinder or tube $S=\mathbb R \times (\mathbb R/\mathbb Z)$ where we assume that there exists an interval $(a_0,b_0)$ which is free of spectrum of $-Δ+V^{(k)}$ for $k=1,2$. We are then interested in the spectrum of $H_t = -Δ+ V_t$, for $t \in \mathbb R$, where $V_t(x,y) = V^{(1)}(x,y)$, for $x > 0$, and $V_t(x,y) = V^{(2)}(x+t,y)$, for $x < 0$. While the essential spectrum of $H_t$ is independent of $t$, we show that discrete spectrum, related to the interface at $x = 0$, is created in the interval $(a_0, b_0)$ at suitable values of the parameter $t$, provided $-Δ+ V^{(2)}$ has some essential spectrum in $(-\infty, a_0]$. We do not require $V^{(1)}$ or $V^{(2)}$ to be periodic. We furthermore show that the discrete eigenvalues of $H_t$ are Lipschitz continuous functions of $t$ if the potential $V^{(2)}$ is locally of bounded variation.

math-ph↗
arXiv2013

Boundary systems and (skew-)self-adjoint operators on infinite metric graphs

We generalize the notion of Lagrangian subspaces to self-orthogonal subspaces with respect to a (skew-)symmetric form, thus characterizing (skew-)self-adjoint and unitary operators by means of self-ortho-gonal subspaces. By orthogonality preserving mappings, these characterizations can be transferred to abstract boundary value spaces of (skew-)symmetric operators. Introducing the notion of boundary systems we then present a unified treatment of different versions of boundary triples and related concepts treated in the literature. The application of the abstract results yields a description of all (skew-)self-adjoint realizations of Laplace and first derivative operators on graphs.

math.FA↗