71 research outputs found
Some remarks on the Krein--von Neumann extension of different Laplacians
We discuss the Krein--von Neumann extensions of three Laplacian-type
operators -- on discrete graphs, quantum graphs, and domains. In passing we
present a class of one-dimensional elliptic operators such that for any infinitely many elements of the class have -dimensional null
space.Comment: 13 page
Operator matrices as generators of cosine operator functions
We introduce an abstract setting that allows to discuss wave equations with
time-dependent boundary conditions by means of operator matrices. We show that
such problems are well-posed if and only if certain perturbations of the same
problems with homogeneous, time-independent boundary conditions are well-posed.
As applications we discuss two wave equations in and in
equipped with dynamical and acoustic-like boundary conditions,
respectively
Parabolic theory of the discrete p-Laplace operator
We study the discrete version of the -Laplacian. Based on its variational
properties we discuss some features of the associated parabolic problem. Our
approach allows us in turn to obtain interesting information about positivity
and comparison principles as well as compatibility with the symmetries of the
graph. We conclude briefly discussing the variational properties of a handful
of nonlinear generalized Laplacians appearing in different parabolic equations.Comment: 35 pages several corrections and enhancements in comparison to the v
Gaussian estimates for a heat equation on a network
We consider a diffusion problem on a network on whose nodes we impose
Dirichlet and generalized, non-local Kirchhoff-type conditions. We prove
well-posedness of the associated initial value problem, and we exploit the
theory of sub-Markovian and ultracontractive semigroups in order to obtain
upper Gaussian estimates for the integral kernel. We conclude that the same
diffusion problem is governed by an analytic semigroup acting on all -type
spaces as well as on suitable spaces of continuous functions. Stability and
spectral issues are also discussed. As an application we discuss a system of
semilinear equations on a network related to potential transmission problems
arising in neurobiology.Comment: In comparison with the already published version of this paper (Netw.
Het. Media 2 (2007), 55-79), a small gap in the proof of Proposition 3.2 has
been fille
On moments-preserving cosine families and semigroups in
We use the newly developed Kelvin's method of images \cite{kosinusy,kelvin}
to show existence of a unique cosine family generated by a restriction of the
Laplace operator in , that preserves the first two moments. We
characterize the domain of its generator by specifying its boundary conditions.
Also, we show that it enjoys inherent symmetry properties, and in particular
that it leaves the subspaces of odd and even functions invariant. Furthermore,
we provide information on long-time behavior of the related semigroup.Comment: 20 pages, 2 figure
- …