Off-diagonal bounds for the Dirichlet-to-Neumann operator

Let $\Omega$ be a bounded domain of $\mathbb{R}^{n+1}$ with $n \ge 1$. We assume that the boundary $\Gamma$ of $\Omega$ is Lipschitz. Consider the Dirichlet-to-Neumann operator $N_0$ associated with a system in divergence form of size $m$ with real symmetric and H\''older continuous coefficients. We prove $L^p(\Gamma)\to L^q(\Gamma)$ off-diagonal bounds of the form$$\| 1_F e^{-t N_0} 1_E f \|_q \lesssim (t \wedge 1)^{\frac{n}{q}-\frac{n}{p}} \left( 1 + \frac{dist(E,F)}{t} \right)^{-1} \| 1_E f \|_p$$for all measurable subsets $E$ and $F$ of $\Gamma$. If $\Gamma$ is $C^{1+ \kappa}$ for some $\kappa > 0$ and $m=1$, we obtain a sharp estimate in the sense that $\left( 1 + \frac{dist(E,F)}{t} \right)^{-1}$ can be replaced by$\left( 1 + \frac{dist(E,F)}{t} \right)^{-(1 + \frac{n}{p} - \frac{n}{q})}$. Such bounds are also valid for complex time. For $n=1$, we apply our off-diagonal bounds to prove that the Dirichlet-to-Neumann operator associated with a system generates an analytic semigroup on $L^p(\Gamma)$ for all $p \in (1, \infty)$. In addition, the corresponding evolution problem has $L^q(L^p)$-maximal regularity

Maximal regularity for non-autonomous evolution equations

We consider the maximal regularity problem for non-autonomous evolution equations of the form $u(t) + A(t) u(t) = f(t)$ with initial data $u(0) = u\_0$ . Each operator $A(t)$ is associated with a sesquilinear form $a(t; *, *)$ on a Hilbert space $H$ . We assume that these forms all have the same domain and satisfy some regularity assumption with respect to t (e.g., piecewise $\alpha$-H{\"o}lder continuous for some \alpha\textgreater{} 1/2). We prove maximal Lp-regularity for all initial values in the real-interpolation space $(H, D(A(0)))\_{1/p,p}$ . The particular case where $p = 2$ improves previously known results and gives a positive answer to a question of J.L. Lions [11] on the set of allowed initial data $u\_0$ .Comment: 19 pages. To appear in Math. An

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

Localization on a quantum graph with a random potential on the edges

We prove spectral and dynamical localization on a cubic-lattice quantum graph with a random potential. We use multiscale analysis and show how to obtain the necessary estimates in analogy to the well-studied case of random Schroedinger operators.Comment: LaTeX2e, 18 page

Interpolation Theorems for Self-adjoint Operators

We prove a complex and a real interpolation theorems on Besov spaces and Triebel-Lizorkin spaces associated with a selfadjoint operator $L$, without assuming the gradient estimate for its spectral kernel. The result applies to the cases where $L$ is a uniformly elliptic operator or a Schr\"odinger operator with electro-magnetic potential.Comment: 8 pages. Submitte