arXiv subjects

Robert Denk

Publications and source records attributed to Robert Denk.

31 records · Page 2Linked to original sources

arXiv2019

A semigroup approach to nonlinear Lévy processes

We study the relation between Lévy processes under nonlinear expectations, nonlinear semigroups and fully nonlinear PDEs. First, we establish a one-to-one relation between nonlinear Lévy processes and nonlinear Markovian convolution semigroups. Second, we provide a condition on a family of infinitesimal generators $(A_λ)_{λ\inΛ}$ of linear Lévy processes which guarantees the existence of a nonlinear Lévy processes such that the corresponding nonlinear Markovian convolution semigroup is a viscosity solution of the fully nonlinear PDE $\partial_t u=\sup_{λ\in Λ} A_λu$. The results are illustrated with several examples.

math.PR↗

arXiv2018

Regularity and asymptotic behaviour for a damped plate-membrane transmission problem

We consider a transmission problem where a structurally damped plate equation is coupled with a damped or undamped wave equation by transmission conditions. We show that exponential stability holds in the damped-damped situation and polynomial stability (but no exponential stability) holds in the damped-undamped case. Additionally, we show that the solutions first defined by the weak formulation, in fact have higher Sobolev space regularity.

math.AP↗

arXiv2018

Generation of semigroups for the thermoelastic plate equation with free boundary conditions

We consider the linear thermoelastic plate equations with free boundary conditions in uniform $C^4$-domains, which includes the half-space, bounded and exterior domains. We show that the corresponding operator generates an analytic semigroup in $L^p$-spaces for all $p\in(1,\infty)$ and has maximal $L^q$-$L^p$-regularity on finite time intervals. On bounded $C^4$-domains, we obtain exponential stability.

math.AP↗

arXiv2018

Dispersive mixed-order systems in $L^p$-Sobolev spaces and application to the thermoelastic plate equation

We study dispersive mixed-order systems of pseudodifferential operators in the setting of $L^p$-Sobolev spaces. Under the weak condition of quasi-hyperbolicity, these operators generate a semigroup in the space of tempered distributions. However, if the basic space is a tuple of $L^p$-Sobolev spaces, a strongly continuous semigroup is in many cases only generated if $p=2$ or $n=1$. The results are applied to the linear thermoelastic plate equation inertial term and with Fourier's or Maxwell-Cattaneo's law of heat conduction.

math.AP↗

arXiv2017

An elliptic boundary problem acting on generalized Sobolev spaces

We consider an elliptic boundary problem over a bounded region $Ω$ in $\mathbb{R}^n$ and acting on the generalized Sobolev space $W^{0,χ}_p(Ω)$ for $1 < p < \infty$. We note that similar problems for $Ω$ either a bounded region in $\mathbb{R}^n$ or a closed manifold acting on $W^{0,χ}_2(Ω)$, called Hörmander space, have been the subject of investigation by various authors. Then in this paper we will, under the assumption of parameter-ellipticity, establish results pertaining to the existence and uniqueness of solutions of the boundary problem. Furthermore, under the further assumption that the boundary conditions are null, we will establish results pertaining to the spectral properties of the Banach space operator induced by the boundary problem, and in particular, to the angular and asymptotic distribution of its eigenvalues.

math.AP↗

arXiv2017

Mapping properties for operator-valued pseudodifferential operators on toroidal Besov spaces

In this paper, we consider pseudodifferential operators on the torus with operator-valued symbols and prove continuity properties on vector-valued toroidal Besov spaces, without assumptions on the underlying Banach spaces. The symbols are of limited smoothness with respect to $x$ and satisfy a finite number of estimates on the discrete derivatives. The proof of the main result is based on a description of the operator as a convolution operator with a kernel representation which is related to the dyadic decomposition appearing in the definition of the Besov space.

math.AP↗

arXiv2017

Kolmogorov type and general extension results for nonlinear expectations

We provide extension procedures for nonlinear expectations to the space of all bounded measurable functions. We first discuss a maximal extension for convex expectations which have a representation in terms of finitely additive measures. One of the main results of this paper is an extension procedure for convex expectations which are continuous from above and therefore admit a representation in terms of countably additive measures. This can be seen as a nonlinear version of the Daniell-Stone theorem. From this, we deduce a robust Kolmogorov extension theorem which is then used to extend nonlinear kernels to an infinite dimensional path space. We then apply this theorem to construct nonlinear Markov processes with a given family of nonlinear transition kernels.

math.PR↗

arXiv2017

Exponential stability for a coupled system of damped-undamped plate equations

We consider the transmission problem for a coupled system of undamped and structurally damped plate equations in two sufficiently smooth and bounded subdomains. It is shown that, independently of the size of the damped part, the damping is strong enough to produce uniform exponential decay of the energy of the coupled system.

math.AP↗

arXiv2015

Generation of semigroups for vector-valued pseudodifferential operators on the torus

We consider toroidal pseudodifferential operators with operator-valued symbols, their mapping properties and the generation of analytic semigroups on vector-valued Besov and Sobolev spaces. We show that a parabolic toroiodal pseudodifferential operator generates an analytic semigroup both on Banach space-valued Besov spaces and on Banach space-valued Sobolev spaces. Here, no condition on the Banach space is imposed. For the proof of the Sobolev space result, we establish a uniform estimate on the kernel which is given as an infinite parameter-dependent sum. An application to abstract non-autonomous periodic pseudodifferential Cauchy problems gives the existence and uniqueness of classical solutions for such problems.

math.AP↗

arXiv2014

Maximal $L_p$-regularity of non-local boundary value problems

We investigate the $\mathcal R$-boundedness of operator families belonging to the Boutet de Monvel calculus. In particular, we show that weakly and strongly parameter-dependent Green operators of nonpositive order are $\mathcal R$-bounded. Such operators appear as resolvents of non-local (pseudodifferential) boundary value problems. As a consequence, we obtain maximal $L_p$-regularity for such boundary value problems. An example is given by the reduced Stokes equation in waveguides.

math.AP↗

arXiv2013

$L^p$-estimates for a transmission problem of mixed elliptic-parabolic type

We consider the situation when an elliptic problem in a subdomain $Ω_1$ of an $n$-dimensional bounded domain $Ω$ is coupled via inhomogeneous canonical transmission conditions to a parabolic problem in $Ω\setminusΩ_1$. In particular, we can treat elliptic-parabolic equations in bounded domains with discontinuous coefficients. Using Fourier multiplier techniques, we prove an a priori estimate for strong solutions to the equations in $L^p$-Sobolev spaces.

math.AP↗

arXiv2011

Discrete Fourier multipliers and cylindrical boundary value problems

We consider operator-valued boundary value problems in $(0,2π)^n$ with periodic or, more generally, $ν$-periodic boundary conditions. Using the concept of discrete vector-valued Fourier multipliers, we give equivalent conditions for the unique solvability of the boundary value problem. As an application, we study vector-valued parabolic initial boundary value problems in cylindrical domains $(0,2π)^n\times V$ with $ν$-periodic boundary conditions in the cylindrical directions. We show that under suitable assumptions on the coefficients, we obtain maximal $L^q$-regularity for such problems.

math.AP↗

arXiv2006

R-boundedness, pseudodifferential operators, and maximal regularity for some classes of partial differential operators

It is shown that an elliptic scattering operator $A$ on a compact manifold with boundary with coefficients in the bounded operators of a bundle of Banach spaces of class (HT) and Pisier's property $(α)$ has maximal regularity (up to a spectral shift), provided that the spectrum of the principal symbol of $A$ on the scattering cotangent bundle of the manifold avoids the right half-plane. This is deduced directly from a Seeley theorem, i.e. the resolvent is represented in terms of pseudodifferential operators with R-bounded symbols, thus showing by an iteration argument the R-boundedness of $λ(A-λ)^{-1}$ for $\Re(λ) \geq 0$. To this end, elements of a symbolic and operator calculus of pseudodifferential operators with R-bounded symbols are introduced. The significance of this method for proving maximal regularity results for partial differential operators is underscored by considering also a more elementary situation of anisotropic elliptic operators on $R^d$ with operator valued coefficients.

math.AP↗