2019, Denk, Robert, Hummel, Felix Benjamin

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 with and without inertial term and with Fourier's or Maxwell-Cattaneo's law of heat conduction.

2018, Barraza Martínez, Bienvenido, Denk, Robert, Hernández Monzón, Jairo, Nendel, Max

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.

2017-10-05T10:47:25Z, Denk, Robert, Faierman, Melvin

We consider an elliptic boundary problem over a bounded region Ω in Rⁿ and acting on the generalized Sobolev space W⁰,χ_p(Ω) for 1n or a closed manifold acting on W⁰,χ₂(Ω), called H\"{o}rmander 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.

2016-08, Barraza Martínez, Bienvenido, Denk, Robert, Hernández Monzón, Jairo, Nau, Tobias

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. Here, we restrict ourselves to pseudodifferential operators with x-independent symbols (Fourier multipliers). We show that a parabolic toroidal pseudodifferential operator generates an analytic semigroup on the Besov space B^s_pq(Tⁿ,E) and on the Sobolev space W^k_p(Tⁿ,E), where E is an arbitrary Banach space, 1≤p,q≤∞, s∈R and k∈N₀. 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.

2019, Denk, Robert, Shibata, Yoshihiro

2018, Denk, Robert, Kammerlander, Felix

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.

2017-03, Denk, Robert, Shibata, Yoshihiro

2019, Barraza Martínez, Bienvenido, Denk, Robert, Hernández Monzón, Jairo, Kammerlander, Felix, Nendel, Max

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.

2018, Denk, Robert, Kupper, Michael, Nendel, Max

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 article 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.

2017, Bothe, Dieter, Denk, Robert, Hieber, Matthias, Schnaubelt, Roland, Simonett, Gieri, Wilke, Mathias, Zacher, Rico