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

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.

2016, Denk, Robert, Shibata, Yoshihiro

We consider the linear thermoelastic plate equations with free boundary conditions in the L_p in time and L_q in space setting. We obtain unique solvability with optimal regularity for the inhomogeneous problem in a uniform C⁴-domain, which includes the cases of a bounded domain and of an exterior domain with C⁴-boundary. Moreover, we prove uniform a priori-estimates for the solution. The proof is based on the existence of R-bounded solution operators of the corresponding generalized resolvent problem which is shown with the help of an operator-valued Fourier multiplier theorem due to Weis.

2013, Denk, Robert, Seiler, Jörg

We investigate the 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 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.

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

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.

2014, Denk, Robert, Schnaubelt, Roland

We investigate sectoriality and maximal regularity in L^p-L^q-Sobolev spaces for the structurally damped plate equation with Dirichlet-Neumann (clamped) boundary conditions. We obtain unique solutions with optimal regularity for the inhomogeneous problem in the whole space, in the half-space, and in bounded domains of class C⁴.



It turns out that the first-order system related to the scalar equation on Rⁿ is sectorial only after a shift in the operator. On the half-space one has to include zero boundary conditions in the underlying function space in order to obtain sectoriality of the shifted operator and maximal regularity for the case of homogeneous boundary conditions. We further show that the semigroup solving the problem on bounded domains is exponentially stable.

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

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.

2014, Denk, Robert, Seger, Tim

We consider the situation when an elliptic problem in a subdomain Ω₁ of an n-dimensional bounded domain Ω is coupled via inhomogeneous canonical transmission conditions to a parabolic problem in Ω∖Ω₁. 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.