2023, Denk, Robert, Ploß, David, Rau, Sophia, Seiler, Jörg

2021, Denk, Robert, Kunze, Markus, Ploß, David

2020-10, Denk, Robert, Saal, Jürgen

We consider a fluid–structure interaction model for an incompressible fluid where the elastic response of the free boundary is given by a damped Kirchhoff plate model. Utilizing the Newton polygon approach, we first prove maximal regularity in L^p-Sobolev spaces for a linearized version. Based on this, we show existence and uniqueness of the strong solution of the nonlinear system for small data.

2019, Denk, Robert, Shibata, Yoshihiro

2021-11-30, Barraza Martínez, Bienvenido, Denk, Robert, González Ospino, Jonathan, Hernández Monzón, Jairo, Rau, Sophia

2021, Anop, Anna, Denk, Robert, Murach, Aleksandr

We investigate regular elliptic boundary-value problems in boun\-ded domains and show the Fredholm property for the related operators in an extended scale formed by inner product Sobolev spaces (of arbitrary real orders) and corresponding interpolation Hilbert spaces. In particular, we can deal with boundary data with arbitrary low regularity. In addition, we show interpolation properties for the extended scale, embedding results, and global and local a priori estimates for solutions to the problems under investigation. The results are applied to elliptic problems with homogeneous right-hand side and to elliptic problems with rough boundary data in Nikolskii spaces, which allows us to treat some cases of white noise on the boundary.

2020, Denk, Robert, Kupper, Michael, Nendel, Max

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 (Aλ)λ∈Λ of linear Lévy processes which guarantees the existence of a nonlinear Lévy process such that the corresponding nonlinear Markovian convolution semigroup is a viscosity solution of the fully nonlinear PDE ∂tu=supλ∈ΛAλu . The results are illustrated with several examples.

2021-11-26, Denk, Robert, Giga, Yoshikazu, Kozono, Hideo, Saal, Jürgen, Simonett, Gieri, Titi, Edriss

2021, Denk, Robert, Kupper, Michael, Nendel, Max

In this paper, we investigate convex semigroups on Banach lattices with order continuous norm, having L^p-spaces in mind as a typical application. We show that the basic results from linear C₀-semigroup theory extend to the convex case. We prove that the generator of a convex C₀-semigroup is closed and uniquely determines the semigroup whenever the domain is dense. Moreover, the domain of the generator is invariant under the semigroup, a result that leads to the well-posedness of the related Cauchy problem. In a last step, we provide conditions for the existence and strong continuity of semigroup envelopes for families of C₀-semigroups. The results are discussed in several examples such as semilinear heat equations and nonlinear integro-differential equations.

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.