A variational approach to strongly damped wave equations

We discuss a Hilbert space method that allows to prove analytical well-posedness of a class of linear strongly damped wave equations. The main technical tool is a perturbation lemma for sesquilinear forms, which seems to be new. In most common linear cases we can furthermore apply a recent result due to Crouzeix--Haase, thus extending several known results and obtaining optimal analyticity angle.Comment: This is an extended version of an article appeared in \emph{Functional Analysis and Evolution Equations -- The G\"unter Lumer Volume}, edited by H. Amann et al., Birkh\"auser, Basel, 2008. In the latest submission to arXiv only some typos have been fixe

Maximal Regularity for Non-Autonomous Second Order Cauchy Problems

We consider non-autonomous wave equations \left\{ \begin{aligned} &\ddot u(t) + \B(t)\dot u(t) + \A(t)u(t) = f(t) \quad t\text{-a.e.}\\ &u(0)=u_0,\, \dot u(0) = u_1. \end{aligned} \right. where the operators \A(t) and \B(t) are associated with time-dependent sesquilinear forms \fra(t,.,.) and \frb defined on a Hilbert space $H$ with the same domain $V$. The initial values satisfy $u_0 \in V$ and $u_1 \in H$. We prove well-posedness and maximal regularity for the solution both in the spaces $V'$ and $H$. We apply the results to non-autonomous Robin-boundary conditions and also use maximal regularity to solve a quasilinear problem.Aux frontières de l'analyse Harmoniqu

Dirichlet-to-Neumann and elliptic operators on C 1+κ -domains: Poisson and Gaussian bounds

We prove Poisson upper bounds for the heat kernel of the Dirichlet-to-Neumann operator with variable Hölder coefficients when the underlying domain is bounded and has a C 1+κ-boundary for some κ > 0. We also prove a number of other results such as gradient estimates for heat kernels and Green functions G of elliptic operators with possibly complex-valued coefficients. We establish Hölder continuity of ∇ x ∇ y G up to the boundary. These results are used to prove L p-estimates for commutators of Dirichlet-to-Neumann operators on the boundary of C 1+κ-domains. Such estimates are the keystone in our approach for the Poisson bounds

Remarks on the Cwikel-Lieb-Rozenblum and Lieb-Thirring Estimates for Schrödinger Operators on Riemannian Manifolds

Let M be a general complete Riemannian manifold and consider a Schrödinger operator − + V on L2(M). We prove Cwikel-Lieb-Rozenblum as well as Lieb-Thirring type estimates for − + V . These estimates are given in terms of the potential and the heat kernel of the Laplacian on the manifold. Some of our results hold also for Schrödinger operators with complex-valued potentials