Wolfgang Arendt

Wolfgang Arendt
Ulm University | UULM · Institute of Applied Analysis

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183
Publications
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8,266
Citations
Citations since 2016
42 Research Items
3313 Citations
20162017201820192020202120220100200300400500
20162017201820192020202120220100200300400500
20162017201820192020202120220100200300400500
20162017201820192020202120220100200300400500

Publications

Publications (183)
Preprint
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Given a densely defined skew-symmetric operators A 0 on a real or complex Hilbert space V , we parametrize all m-dissipative extensions in terms of contractions $\Phi$ : H-$\rightarrow$ H + , where Hand H + are Hilbert spaces associated with a boundary quadruple. Such an extension generates a unitary C 0-group if and only if $\Phi$ is a unitary ope...
Preprint
We use form methods to define suitable realisations of the Laplacian on a domain $\Omega$ with Wentzell boundary conditions, i.e. such that $\partial_\nu u + \beta u + \Delta u = 0$ holds in a suitable sense on the boundary of $\Omega$. For those realisations, we study their semigroup generation properties. Using the approximative trace, we give a...
Preprint
Full-text available
The Representation Theorem of Lions (RTL) is a version of the Lax--Milgram Theorem where completeness of one of the spaces is not complete. In this paper, RTL is deduced from an operator-theoretical version on normed space. The main point of the paper is a theory of derivations, based on RTL, for which well-posedness is proved. One application conc...
Preprint
The paper makes use of recent results in the theory of Banach lattices and positive operators to deal with abstract semilinear equations. The aim is to work with minimal or no regularity conditions on the boundary of the domains, where the usual arguments based on maximum principles do not apply. A key result is an application of Kato's inequality...
Article
We use form methods to define suitable realisations of the Laplacian on a domain \begin{document}$ \Omega $\end{document} with Wentzell boundary conditions, i.e. such that \begin{document}$ \partial_ { \rm{{n}}} u + \beta u + \Delta u = 0 $\end{document} holds in a suitable sense on the boundary of \begin{document}$ \Omega $\end{document}. For thos...
Article
In the first part of the article perturbation of a closed form by a weakly continuous form is studied. This notion of weakly continuous perturbation is very handy and, as is shown in the article, leads to a new semigroup whose difference with the given semigroup consists of compact operators. We apply the results to elliptic operators on the Hardy...
Article
Full-text available
The notion Perron–Frobenius theory usually refers to the interaction between three properties of operator semigroups: positivity, spectrum and long-time behaviour. These interactions gives rise to a profound theory with plenty of applications. By a brief walk-through of the field and with many examples, we highlight two aspects of the subject, both...
Article
Full-text available
We consider elliptic operators with measurable coefficients and Robin boundary conditions on a bounded domain {\Omega\subset\mathbb{R}^{d}} and show that the first eigenfunction v satisfies {v(x)\geq\delta>0} for all {x\in\overline{\Omega}} , even if the boundary {\partial\Omega} is only Lipschitz continuous. Under such weak regularity assumptions...
Preprint
Full-text available
The notion \emph{Perron-Frobenius theory} usually refers to the interaction between three properties of operator semigroups: positivity, spectrum and long-time behaviour. These interactions gives rise to a profound theory with plenty of applications. By a brief walk-through of the field and with many examples, we highlight two aspects of the subjec...
Preprint
All the known counterexamples to Kac' famous question "can one hear the shape of a drum", i.e., does isospectrality of two Laplacians on domains imply that the domains are congruent, consist of pairs of domains composed of copies of isometric building blocks arranged in different ways, such that the unitary operator intertwining the Laplacians acts...
Article
Full-text available
Let \(\Omega \subset \mathbb {R}^d\) be open. We investigate conditions under which an operator T on \(L_2(\Omega )\) has a continuous kernel \(K \in C({{\overline{\Omega }}} \times \overline{\Omega })\). In the centre of our interest is the condition \(T L_2(\Omega ) \subset C({{\overline{\Omega }}})\), which one knows for many semigroups generate...
Preprint
Full-text available
We consider elliptic operators with Robin boundary conditions on a bounded domain $\Omega \subset \mathbb{R}^d$ and show that the first eigenfunction $v$ fulfils $v(x) \ge \delta > 0$ for all $x \in \bar{\Omega}$, even if the boundary $\partial \Omega$ is only Lipschitz continuous and the differential operator has merely bounded and measurable coef...
Preprint
In this paper we study the conforming Galerkin approximation of the problem: find u $\in$ U such that a(u, v) = L, v for all v $\in$ V, where U and V are Hilbert or Banach spaces, a is a continuous bilinear or sesquilinear form and L $\in$ V a given data. The approximate solution is sought in a finite dimensional subspace of U, and test functions a...
Preprint
In the first part we show that a vector-valued almost separably valued function $f$ is holomorphic (harmonic) if and only if it is dominated by an $L^1_\mathrm{loc}$ function and there exists a separating set $W\subset X'$ such that $\langle f,x'\rangle$ is holomorphic (harmonic) for all $x'\in W$. This improves a known result which requires $f$ to...
Preprint
Let $\Omega \subset {\bf R}^d$ be open. We investigate conditions under which an operator $T$ on $L_2(\Omega)$ has a continuous kernel $K \in C(\overline \Omega \times \overline \Omega)$. In the centre of our interest is the condition $T L_2(\Omega) \subset C(\overline \Omega)$, which one knows for many semigroups generated by elliptic operators. T...
Preprint
The nonlinear semigroup generated by the subdifferential of a convex lower semicontinuous function $\varphi$ has a smoothing effect, discovered by H. Br\'ezis, which implies maximal regularity for the evolution equation. We use this and Schaefer's fixed point theorem to solve the evolution equation perturbed by a Nemytskii-operator of sublinear gro...
Preprint
Let $\varphi:\mathbb{D} \to \mathbb{D}$ be a holomorphic map with a fixed point $\alpha\in\mathbb{D}$ such that $0\leq |\varphi'(\alpha)|<1$. We show that the spectrum of the composition operator $C_\varphi$ on the Fr\'echet space $ \textrm{Hol}(\mathbb{D})$ is $\{0\}\cup \{ \varphi'(\alpha)^n:n=0,1,\cdots\}$ and its essential spectrum is reduced t...
Article
Consider the Dirichlet problem with respect to an elliptic operator \[ A = - \sum_{k,l=1}^d \partial_k \, a_{kl} \, \partial_l - \sum_{k=1}^d \partial_k \, b_k + \sum_{k=1}^d c_k \, \partial_k + c_0 \] on a bounded Wiener regular open set $\Omega \subset R^d$, where $a_{kl}, c_k \in L_\infty(\Omega,R)$ and $b_k,c_0 \in L_\infty(\Omega,C)$. Suppose...
Article
Full-text available
Let $A$ be the generator of a $C_0$-semigroup $T$ on a Banach space of analytic functions on the open unit disc. If $T$ consists of composition operators, then there exists a holomorphic function $G:{\mathbb D}\to{\mathbb C}$ such that $Af=Gf'$ with maximal domain. The aim of the paper is the study of the reciprocal implication.
Article
We consider a semilinear problemu′(t)+A(t)u(t)=f(u(t)),u(0)=u0, where A(t) is associated with a non-autonomous form a(t,⋅,⋅). Using an invariance principle for closed, convex sets in the underlying Hilbert space we find conditions for global solutions. This can be applied to reaction diffusion systems on L2(Ω)N. Our point is that the forms a(t,⋅,⋅)...
Chapter
In diesem Kapitel leiten wir für eine ganze Reihe von partiellen Differenzialgleichungen explizite Lösungen her. Die hier betrachteten Gleichungen repräsentieren interessante Modelle wie die schwingende Saite, die Wärmeleitung oder die Bewertung von Optionen. Darüber hinaus sind es Prototypen von partiellen Differenzialgleichungen, die wir in Kapit...
Chapter
Die Spektralzerlegung des Laplace-Operators mit verschiedenen Randbedingungen gibt uns genaue Information über die Lösbarkeit der Gleichung.
Chapter
Um die wesentlichen Punkte der mathematischen Theorie möglichst elementar darzustellen, hatten wir Sobolev-Räume zunächst nur im eindimensionalen Fall eingeführt; genauer gesagt für Funktionen, die auf einem offenen Intervall im R definiert sind. In einer Raumdimension sind H¹-Funktionen automatisch stetig und man kann sie durch ein unbestimmtes In...
Chapter
In Hilbert-Räumen spielen geometrische und analytische Eigenschaften in vielfacher und wunderbarerWeise zusammen. Das zeigen wir in dieser elementaren und ausführlichen Einführung. Höhepunkt ist der Satz von Riesz-Fréchet, der die stetigen Linearformen auf einem Hilbert-Raum beschreibt. Wichtig für uns ist es, dass wir diesen Satz als einen Existen...
Chapter
Sobolev-Räume über einem Intervall haben einen besonderen Reiz: Sie bestehen aus Funktionen, die sich als unbestimmtes Integral von integrierbaren Funktionen schreiben lassen. Zahlreiche Eigenschaften und Rechenregeln, wie z.B. das partielle Integrieren, sind gültig und geben uns eine Differenzial- und Integralrechnung an die Hand, die fast genauso...
Chapter
Wir haben die Herleitung einer ganzen Reihe von partiellen Differenzialgleichungen aus Naturvorgängen bzw. den Wirtschaftswissenschaften kennen gelernt. Aus der Vielfalt der Gleichungen kann man bereits vermuten, dass es keine einheitliche mathematische Theorie und keine einheitliche Lösungsstrategie für partielle Differenzialgleichungen gibt bzw....
Chapter
Maple ® ist ein von der Firma Maplesoft (auch in Kooperation mit Hochschulen und Forschungseinrichtungen) entwickeltes und vertriebenes Programm. Es handelt sich um ein Computeralgebra-System, also ein Programm, das mathematische Umformungen nach gegebenen Regeln exakt durchführt, d.h. keine Rundungsfehler begeht, wie etwa bei numerischen Approxima...
Chapter
Partielle Differenzialgleichungen beschreiben zahlreiche Vorgänge in der Natur, der Technik, der Medizin oder der Wirtschaft. In diesem ersten Kapitel wollen wir für einige prominente Beispiele die Herleitung von partiellen Differenzialgleichungen mit Hilfe von Naturgesetzen und mathematischen Tatsachen beschreiben. Eine solche Herleitung nennt man...
Chapter
Wir haben die Modellierung, einfache Lösungsverfahren und die mathematische Theorie einer ganzen Reihe von partiellen Differenzialgleichungen kennen gelernt. All diese Untersuchungen sind „analytisch“, d.h., wir können sie auf dem Papier ausführen und die konstruierten Lösungen erfüllen die jeweiligen partiellen Differenzialgleichungen „exakt“. Zwa...
Book
Dieses Lehrbuch gibt eine Einführung in die partiellen Differenzialgleichungen. Wir beginnen mit einigen ganz konkreten Beispielen aus den Natur-, Ingenieur und Wirtschaftswissenschaften. Danach werden elementare Lösungsmethoden dargestellt, z.B. für die Black-Scholes-Gleichung aus der Finanzmathematik. Schließlich wird die analytische Untersuchung...
Chapter
In diesem Kapitel untersuchen wir zunächst elliptische partielle Differenzialgleichungen mit Neumann-Randbedingungen.
Article
Let $\Omega \subset {\bf R}^d$ be an open bounded set with Lipschitz boundary $\Gamma$. Let $D_V$ be the Dirichlet-to-Neumann operator with respect to a purely second-order symmetric divergence form operator with real Lipschitz continuous coefficients and a positive potential $V$. We show that the semigroup generated by $-D_V$ leaves $C(\Gamma)$ in...
Article
Full-text available
We show an interesting relation between ultracontractivity and Weyl asymptotics. Then both properties are studied for their behaviour with respect to perturbation. The results are used to establish Weyl’s law for the Dirichlet-to-Neumann operator associated with -Δ+V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage...
Article
This is a survey on recent progress concerning maximal regularity of non-autonomous equations governed by time-dependent forms on a Hilbert space. It also contains two new results showing the limits of the theory.
Article
We consider Sobolev spaces with values in Banach spaces as they are frequently useful in applied problems. Given two Banach spaces $X\neq\{0\}$ and $Y$, each Lipschitz continuous mapping $F:X\rightarrow Y$ gives rise to a mapping $u\mapsto F\circ u$ from $W^{1,p}(\Omega,X)$ to $W^{1,p}(\Omega,Y)$ if and only if $Y$ has the Radon-Nikodym Property. B...
Article
Full-text available
We investigate a second order elliptic differential operator $A_{\beta, \mu}$ on a bounded, open set $\Omega\subset\mathbb{R}^{d}$ with Lipschitz boundary subject to a nonlocal boundary condition of Robin type. More precisely we have $\beta\in L^{\infty}(\partial\Omega)$ and $\mu\colon \partial \Omega \to \mathscr{M}(\overline{\Omega})$, and bounda...
Article
In the very influential paper \cite{CS07} Caffarelli and Silvestre studied regularity of $(-\Delta)^s$, $0<s<1$, by identifying fractional powers with a certain Dirichlet-to-Neumann operator. Stinga and Torrea \cite{ST10} and Gal\'e, Miana and Stinga \cite{GMS13} gave several more abstract versions of this extension procedure. The purpose of this p...
Article
Full-text available
Holomorphic and harmonic functions with values in a Banach space are investigated. Following an approach given in a joint article with Nikolski [4] it is shown that for bounded functions with values in a Banach space it suffices that the composition with functionals in a separating subspace of the dual space be holomorphic to deduce holomorphy. Ano...
Article
We define two versions of the Dirichlet-to-Neumann operator on exterior domains and study convergence properties when the domain is truncated.
Book
This proceedings volume originates from a conference held in Herrnhut in June 2013. It provides unique insights into the power of abstract methods and techniques in dealing successfully with numerous applications stemming from classical analysis and mathematical physics. The book features diverse topics in the area of operator semigroups, including...
Chapter
We reconsider the contour argument and proof by transfinite induction of the ABLV-Theorem given in [AB88]. But here we use the method to prove a Tauberian Theorem for Laplace transforms which has the ABVLTheorem about stability of a semigroup as corollary and also gives quantitative estimates. It is interesting that considering countable spectrum l...
Article
We consider a non-autonomous Cauchy problem involving linear operators associated with time-dependent forms $a(t;.,.):V\times V\to {\mathbb{C}}$ where $V$ and $H$ are Hilbert spaces such that $V$ is continuously embedded in $H$. We prove $H$-maximal regularity under a new regularity condition on the form $a$ with respect to time; namely H\"older co...
Article
Full-text available
We consider second order differential operators $A_\mu$ on a bounded, Dirichlet regular set $\Omega \subset \mathbb{R}^d$, subject to the nonlocal boundary conditions \[ u(z) = \int_\Omega u(x)\, \mu (z, dx)\quad \mbox{for} z \in \partial \Omega. \] Here the function $\mu : \partial\Omega \to \mathscr{M}^+(\Omega)$ is $\sigma (\mathscr{M} (\Omega),...
Article
Full-text available
We study diffusion in a network which is governed by non-autonomous Kirchhoff conditions at the vertices of the graph. Also the diffusion coefficients may depend on time. We prove at first a result on existence and uniqueness using form methods. Our main results concern the long-term behavior of the solution. In the case when the conductivity and t...
Article
We show that to each symmetric elliptic operator of the form \[ \mathcal{A} = - \sum \partial_k \, a_{kl} \, \partial_l + c \] on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ one can associate a self-adjoint Dirichlet-to-Neumann operator on $L_2(\partial \Omega)$, which may be multi-valued if 0 is in the Dirichlet spectrum of $\mathcal{...
Article
We reexamine the proofs of isospectrality of the counterexample domains to Kac' question `Can one hear the shape of a drum?' from an analytical viewpoint. We reformulate isospectrality in a more abstract setting as the existence of a similarity transform intertwining two operators associated with elliptic forms, and give several equivalent characte...
Article
Full-text available
We consider a non-autonomous evolutionary problem \[ \dot{u} (t)+\A(t)u(t)=f(t), \quad u(0)=u_0 \] where the operator $\A(t):V\to V^\prime$ is associated with a form $\fra(t,.,.):V\times V \to \R$ and $u_0\in V$. Our main concern is to prove well-posedness with maximal regularity which means the following. Given a Hilbert space $H$ such that $V$ is...
Article
We consider a non-autonomous form $\fra:[0,T]\times V\times V \to \C$ where $V$ is a Hilbert space which is densely and continuously embedded in another Hilbert space $H$. Denote by $\A(t) \in \L(V,V')$ the associated operator. Given $f \in L^2(0,T, V')$, one knows that for each $u_0 \in H$ there is a unique solution $u\in H^1(0,T;V')\cap L^2(0,T;V...
Article
Full-text available
If Ω is any compact Lipschitz domain, possibly in a Riemannian manifold, with boundary Γ = ∂Ω, the Dirichlet-to-Neumann operator Dλ is defined on L2(Γ) for any real λ. We prove a close relationship between the eigenvalues of D λ and those of the Robin Laplacian Δμ i.e. the Laplacian with Robin boundary conditions ∂vu = μu. This is used to give anot...
Article
We provide a short proof for the theorem that two compact Riemannian manifolds are isomorphic if and only there exists an order isomorphism which intertwines between the heat semigroups on the manifolds.
Article
Full-text available
Form methods give a very efficient tool to solve evolutionary problems on Hilbert space. They were developed by T. Kato [Kat] and, in slightly different language by J.L. Lions. In this expository article we give an introduction based on [AE2]. The main point in our approach is that the notion of closability is not needed anymore. The new setting is...
Chapter
Full-text available
In this chapter the emphasis of the discussion shifts from Laplace integrals \(\hat f \)(λ) and \( \mathop {dF}\limits^ \wedge \) (λ) to the Laplace transform \( L:f \mapsto \hat f \) and to the Laplace-Stieltjes transform \( {L_S}:F \mapsto \mathop {dF}\limits^ \wedge \) . The Laplace transform is considered first as an operator acting on L ∞ (ℝ+,...
Chapter
Full-text available
In this chapter we study systematically well-posedness of the Cauchy problem. Given a closed operator A on a Banach space X we will see in Section 3.1 that the abstract Cauchy problem {lcr u¢(t) = Au(t) (t ³ 0), u(0) = x, \left\{\begin{array}{lcr} u^\prime(t) = Au(t) \,\,(t\geq0), \\ u(0) = x, \end{array}\right. is mildly well-posed (i.e., for ea...
Article
The first three sections of this chapter are of a preliminary nature. There, we collect properties of the Bochner integral of functions of a real variable with values in a Banach space X. We then concentrate on the basic properties of the Laplace integral $$\hat f(\lambda ): = \int_0^\infty {{e^{ - \lambda t}}} f(t)dt: = \mathop {\lim }\limits_{\ta...
Article
Frequently, convergence of a function f : \mathbbR+ ® X f : \mathbb{R}_+ \rightarrow X for t ® ¥t \rightarrow \infty implies convergence of an average of this function. Assertions of this type are called Abelian theorems. A theorem is called Tauberian if, conversely, convergence of the function is deduced from the convergence of an average.
Article
In this chapter, we give various results concerning the long-time asymptotic behaviour of mild solutions of homogeneous and inhomogeneous Cauchy problems on \(\mathbb {R}_+\) (see Section 3.1 for the definitions and basic properties). For the most part, we shall assume that the homogeneous problem is well-posed, so that the operator A generates a C...
Chapter
In this chapter we consider differential operators with constant coefficients, and more generally pseudo-differential operators on Lp(ℝn). The realization Opp (a) in Lp(ℝn) of such an operator is translation invariant. We assume that the “symbol” a satisfies certain smoothness and growth assumptions. In particular, when a is a polynomial, then Opp...
Chapter
In this chapter we consider the Laplacian on spaces of continuous functions. If Ω ⊂ ℝn is an open, bounded set with boundary ∂Ω which is Dirichlet regular, we will show that the Laplacian generates a holomorphic semigroup on the space $${C_0}(\Omega )\,: = \left\{ {u\, \in (\bar \Omega )} \right\}:\,u{|_{\partial \Omega }}\, = 0.$$ Furthermore, usi...
Chapter
In this chapter we study the wave equation $${u_{tt}}\, = \,\Delta u$$ on an open subset Ω of ℝn . We will use the theory of cosine functions and work on L 2 (Ω). We first consider the Laplacian with Dirichlet boundary conditions. This is a selfadjoint operator and well-posedness is a consequence of the spectral theorem. A further aim is to replace...
Article
Let ! ! RN be an open and bounded set and let m :! " (0,# ) be measurable and locally bounded. We study a natural realization of the oper- ator m$ in C0(!) := ˘ u % C(!) : u|! ! =0 ¯ . If ! is Dirichlet regular, then the operator generates a positive contraction semigroup on C0(!) whenever 1 m % L p loc(!) for some p> N 2 . If m(x) does not go fast...
Article
Full-text available
Let Ω ⊂ ℝn be a bounded open set satisfying the uniform exterior cone condition. Let A be a uniformly elliptic operator given by Au = ni,j=1 aij∂iju + nj=1 bj∂ju + cu where aji = aij ∈ C(Ω) and bj, c ∈ L∞(Ω), c ≤ 0. We show that the realization A0 of A in C0(Ω) := {u ∈ C(Ω¯) : u|∂Ω = 0} given by D(A0) := {u ∈ C0(Ω) ∩ W2,nloc (Ω) : Au ∈ C0(Ω)} A0u :...
Article
We consider the quasilinear parabolic equation u(t) - beta(t, x, u, del u) Delta u = f(t, x, u, del u) in a cylindrical domain, together with initial-boundary conditions, where the quasilinearity operates on the diffusion coefficient of the Laplacian. Under suitable conditions we prove global existence of a solution in the energy space. Our proof d...
Article
We consider a bounded connected open set $\Omega \subset {\rm R}^d$ whose boundary $\Gamma$ has a finite $(d-1)$-dimensional Hausdorff measure. Then we define the Dirichlet-to-Neumann operator $D_0$ on $L_2(\Gamma)$ by form methods. The operator $-D_0$ is self-adjoint and generates a contractive $C_0$-semigroup $S = (S_t)_{t > 0}$ on $L_2(\Gamma)$....
Article
Maple® ist ein von der Firma Maplesoft (auch in Kooperation mit Hochschulen und Forschungseinrichtungen) entwickeltes und vertriebenes Programm. Es handelt sich umein Computeralgebra-System, also ein Programm, dasmathematische Umformungen nach gegebenen Regeln exakt durchführt, d.h. keine Rundungsfehler begeht, wie etwa bei numerischen Approximatio...
Article
Bisectorial operators play an important role since exactly these operators lead to a well-posed equation on the entire line. The most simple example of a bisectorial operator is obtained by taking the direct sum of an invertible generator of a bounded holomorphic semigroup and the negative part of such an operator. Our main result in this paper sho...
Article
Die Spektralzerlegung des Laplace-Operatorsmit verschiedenen Randbedingungen gibt uns genaue Information über die Lösbarkeit der Gleichung λu - ∆u = ƒ.
Article
Um die wesentlichen Punkte der mathematischen Theorie möglichst elementar darzustellen, hatten wir Sobolev-Räume zunächst nur im eindimensionalen Fall eingeführt; genauer gesagt für Funktionen, die auf einem offenen Intervall im R definiert sind. In einer Raumdimension sind H1-Funktionen automatisch stetig und man kann sie durch ein unbestimmtes In...
Article
In diesem Kapitel untersuchen wir zunächst elliptische partielle Differenzialgleichungenmit Neumann-Randbedingungen \frac¶u¶v = 0. \frac{{\partial u}}{{\partial v}} = 0.\ Diese bedürfen etwas anderer Techniken, als wir sie bei Dirichlet-Randbedingungen gesehen haben. Wir beginnen damit, C1-Gebiete zu definieren und den Satz von Gauß zu beweisen....
Article
In diesem Kapitel leiten wir für eine ganze Reihe von partiellen Differenzialgleichungen explizite Lösungen her. Die hier betrachteten Gleichungen repräsentieren interessante Modelle wie die schwingende Saite, die Wärmeleitung oder die Bewertung von Optionen. Darüber hinaus sind es Prototypen von partiellen Differenzialgleichungen, die wir in Kapit...
Article
Wir haben dieHerleitung einer ganzen Reihe von partiellenDifferenzialgleichungen aus Naturvorgängen kennen gelernt. Aus der Vielfalt der Gleichungen kann man bereits vermuten, dass es keine einheitliche mathematische Theorie und keine einheitliche Lösungsstrategie für partielle Differenzialgleichungen gibt. Es stellt sich aber die Frage, ob man par...
Article
In Hilbert-Räumen spielen geometrische und analytische Eigenschaften in vielfacher und wunderbarerWeise zusammen. Das zeigen wir in dieser elementaren und ausführlichen Einführung. Höhepunkt ist der Satz von Riesz-Fréchet, der die stetigen Linearformen auf einem Hilbert-Raum beschreibt. Wichtig für uns ist es, dass wir diesen Satz als einen Existen...
Article
Sobolev-Räume über einem Intervall haben einen besonderen Reiz: Sie bestehen aus Funktionen, die sich als unbestimmtes Integral von integrierbaren Funktionen schreiben lassen. Zahlreiche Eigenschaften und Rechenregeln, wie z.B. das partielle Integrieren, sind gültig und geben uns eine Differenzial- und Integralrechnung an die Hand, die fast genauso...
Article
Partielle Differenzialgleichungen beschreiben zahlreiche Vorgänge in der Natur, der Technik, derMedizin oder derWirtschaft. In diesem ersten Kapitelwollenwir für einige prominente Beispiele die Herleitung von partiellen Differenzialgleichungen mit Hilfe von Naturgesetzen und mathematischen Tatsachen beschreiben. Eine solche Herleitung nennt man (ma...
Book
Mathematical Analysis of Evolution, Information, and Complexity deals with the analysis of evolution, information and complexity. The time evolution of systems or processes is a central question in science, this text covers a broad range of problems including diffusion processes, neuronal networks, quantum theory and cosmology. Bringing together a...
Article
Full-text available
In the first part of the article we characterize automatic continuity of positive operators. As a corollary we consider complete norms for which a given cone E+ in an infinite dimensional Banach space E is closed and we obtain the following result: every two such norms are equivalent if and only if E+ Ç(-E+) = {0} and E+ - E+E_+ \cap (-E_+) = \{0\}...
Chapter
Weyl's law is in its simplest version a statement on the asymptotic growth of the eigenvalues of the Laplacian on bounded domains with Dirichlet and Neumann boundary conditions. In the typical applications in physics one deals either with the Helmholtz wave equation describing the vibrations of a string, a membrane (drum), a mass of air in a concer...
Article
Given a family A(t) of closed unbounded operators on a UMD Banach space X with common domain W, we investigate various properties of the operator D A := d dt − A(·) acting from W p per := {u ∈ W 1,p (0, 2π; X) ∩ L p (0, 2π; W) : u(0) = u(2π)} into X p := L p (0, 2π; X) when p ∈ (1, ∞). The primary focus is on the Fredholmness and index of D A , but...
Article
Full-text available
We demonstrate how the Laplace operator can be combined with a stochastic jump process to describe the evolution of a polymer network. Growth processes in polymer networks can proceed through the transfer of rather large oligomeric subunits from a pool of soluble molecules into the laments forming the network. In such a situation stochastic jump pr...
Article
If $a$ is a densely defined sectorial form in a Hilbert space which is possibly not closable, then we associate in a natural way a holomorphic semigroup generator with $a$. This allows us to remove in several theorems of semigroup theory the assumption that the form is closed or symmetric. Many examples are provided, ranging from complex sectorial...
Article
Keratin intermediate filament networks are part of the cytoskeleton in epithelial cells. They were found to regulate viscoelastic properties and motility of cancer cells. Due to unique biochemical properties of keratin polymers, the knowledge of the mechanisms controlling keratin network formation is incomplete. A combination of deterministic and s...
Article
Full-text available
We prove under a weak smoothness condition that two Riemannian manifold are isomorphic if and only there exists an order isomorphism which intertwines with the Dirichlet type heat semigroups on the manifolds.
Article
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The paper is concerned with properties of an ill-posed problem for the Helmholtz equation when Dirichlet and Neumann conditions are given only on a part Γ of the boundary ∂Ω. We present an equivalent formulation of this problem in terms of a moment problem defined on the part of the boundary where no boundary conditions are imposed. Using a weak de...
Article
For Ω a bounded open set in ℝN we consider the space H01(Ω̄) = {u|Ω : u ∈ H1(ℝN): u(x) = 0 a.e. outside Ω̄}. The set Ω is called stable if H01(Ω) = H 01(Ω̄). Stability of Ω can be characterised by the convergence of the solutions of the Poisson equation -Δun = f in D(Ωn)′, un ∈ H01(Ωn) and also the Dirichlet Problem with respect to Ωn if Ωn converg...
Article
We consider a strictly elliptic operator Au = Sigma(ij) D-i(a(ij)D(j)u) - b . del u + div(c . u) - Vu, where 0 <= V is an element of Lloc infinity, aij is an element of Cb1(RN), b, c is an element of 1(RN, RN). If div b <= beta V, div c <= beta V, then generates a positive C-0-semigroup T in L-2(R-N). The semigroup satisfies pseudo-Gaussian estimat...
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Extending results of Davies and of Keicher on ℓp we show that the peripheral point spectrum of the generator of a positive bounded C0-semigroup of kernel operators on Lp is reduced to 0. It is shown that this implies convergence to an equilibrium if the semigroup is also irreducible and the fixed space non-trivial. The results are applied to ellipt...
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Let Omega subset of R-N be a bounded open set and phi is an element of C(partial derivative Omega). Assume that o has an extension Phi is an element of C((Omega) over bar) such that Delta Phi is an element of H-1(Omega). Then by the Riesz representation theorem there exists a unique u is an element of H-0(1)(Omega) suchthat -Delta u=Delta Phi in D(...
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Günter Lumer was an outstanding mathematician whose work has great influence on the research community in mathematical analysis and evolution equations. He was at the origin of the breath-taking development the theory of semigroups saw after the pioneering book of Hille and Phillips of 1957. This volume contains invited contributions presenting the...
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Let A:[0,τ]→L(D,X) be strongly measurable and bounded, where D, X are Banach spaces such that D↪X. We assume that the operator A(t) has maximal regularity for all t∈[0,τ]. Then we show under some additional hypothesis (viz. relative continuity) that the non-autonomous problem is well-posed in Lp; i.e. for all f∈Lp(0,τ;X) and all there exists a uniq...
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Let Ω ⊂ ℝN be (Wiener) regular. For λ > 0 and f ∈ L∞ (ℝN) there is a unique bounded, continuous function u: ℝN → ℝ solving λu - δu = f in D(Ω)′, u = 0 on ℝN \ Ω. (PΩ) Given open sets Ωn we introduce the notion of regular convergence of Ωn to Ω as n → ∞. It implies that the solutions un of (PΩn) converge (locally) uniformly to u on ℝN. Whereas L 2-c...
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Let A be a closed operator on a Banach space X. We study maximal Lp-regularity of the problems $ \begin{aligned} u^{\prime} (t) & = Au(t) + f(t)\quad {\hbox{and}}\\ u^{\prime\prime}(t) & = Au(t) + f(t) \end{aligned} $ \begin{aligned} u^{\prime} (t) & = Au(t) + f(t)\quad {\hbox{and}}\\ u^{\prime\prime}(t) & = Au(t) + f(t) \end{aligned} on the lin...
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Let aij ∈ Cb1(ℝN), i, j = 1,...,N be uniformly elliptic, and let b ∈ C1(ℝ N), V ∈ C(ℝN). p/div bIf ≤ V, then we construct a unique minimal positive semigroup generated by a restriction of the operator A defined by the expression Au = Σi,j=1N Di(aijDju) - Σi=1NbiDiu - Vu on L P (ℝN) with maximal domain. We give a criterion for Cc∞ (ℝN) to be a core...
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Let A be the generator of a cosine function on a Banach space X. In many cases, for example if X is a UMD-space, A+B generates a cosine function for each B∈L(D((ω−A)1/2),X). If A is unbounded and 12γ⩽1, then we show that there exists a rank-1 operator B∈L(D((ω−A)γ),X) such that A+B does not generate a cosine function. The proof depends on a modific...

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