96 research outputs found
Off-diagonal bounds for the Dirichlet-to-Neumann operator
Let be a bounded domain of with . We
assume that the boundary of is Lipschitz. Consider the
Dirichlet-to-Neumann operator associated with a system in divergence form
of size with real symmetric and H\''older continuous coefficients. We prove
off-diagonal bounds of the formfor all measurable subsets
and of . If is for some and
, we obtain a sharp estimate in the sense that can be replaced by. Such bounds
are also valid for complex time. For , we apply our off-diagonal bounds to
prove that the Dirichlet-to-Neumann operator associated with a system generates
an analytic semigroup on for all . In
addition, the corresponding evolution problem has -maximal
regularity
Maximal regularity for non-autonomous evolution equations
We consider the maximal regularity problem for non-autonomous evolution
equations of the form with initial data
. Each operator is associated with a sesquilinear form on a
Hilbert space . We assume that these forms all have the same domain and
satisfy some regularity assumption with respect to t (e.g., piecewise
-H{\"o}lder continuous for some \alpha\textgreater{} 1/2). We prove
maximal Lp-regularity for all initial values in the real-interpolation space
. The particular case where improves previously
known results and gives a positive answer to a question of J.L. Lions [11] on
the set of allowed initial data .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
Well-Posedness and Symmetries of Strongly Coupled Network Equations
We consider a diffusion process on the edges of a finite network and allow
for feedback effects between different, possibly non-adjacent edges. This
generalizes the setting that is common in the literature, where the only
considered interactions take place at the boundary, i. e., in the nodes of the
network. We discuss well-posedness of the associated initial value problem as
well as contractivity and positivity properties of its solutions. Finally, we
discuss qualitative properties that can be formulated in terms of invariance of
linear subspaces of the state space, i. e., of symmetries of the associated
physical system. Applications to a neurobiological model as well as to a system
of linear Schroedinger equations on a quantum graph are discussed.Comment: 25 pages. Corrected typos and minor change
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 , without
assuming the gradient estimate for its spectral kernel. The result applies to
the cases where is a uniformly elliptic operator or a Schr\"odinger
operator with electro-magnetic potential.Comment: 8 pages. Submitte
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