Ultracontractivity and Gaussian bounds for evolution families associated with nonautonomous forms

We develop a variational approach in order to study qualitative properties of nonautonomous parabolic equations. Based on the method of product integrals, we discuss invariance properties and ultracontractivity of evolution families in Hilbert space. Our main results give sufficient conditions for the heat kernel of the evolution family to satisfy Gaussian-type bounds. Along the way, we study examples of nonautonomous equations on graphs, metric graphs, and domains.     

On the Spectral Gap of a Quantum Graph

We consider the problem of finding universal bounds of “isoperimetric” or “isodiametric” type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We investigate which combinations of parameters are necessary to obtain non-trivial upper and lower bounds and obtain a number of sharp estimates in terms of these parameters. We also show that, in contrast to the Laplacian matrix on a combinatorial graph, no bound depending only on the diameter is possible. As a special case of our results on metric graphs, we deduce estimates for the normalised Laplacian matrix on combinatorial graphs which, surprisingly, are sometimes sharper than the ones obtained by purely combinatorial methods in the graph theoretical literature.     

Convergence of operator semigroups associated with generalised elliptic forms

In a recent article, Arendt and ter Elst have shown that every sectorial form is in a natural way associated with the generator of an analytic strongly continuous semigroup, even if the form fails to be closable. As an intermediate step, they have introduced so-called j?elliptic forms, which generalise the concept of elliptic forms in the sense of Lions. We push their analysis forward in that we discuss some perturbation and convergence results for semigroups associated with j?elliptic forms. In particular, we study convergence with respect to the trace norm or other Schatten norms. We apply our results to Laplace operators and Dirichlet-to-Neumann-type operators.     

Laplacians on quantum hypergraphs

We introduce quantum hypergraphs, in analogy with the theory of quantum graphs developed over the last 15 years by many authors. We emphasize some problems that arise when one tries to define a Laplacian on a hypergraph. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Abstract wave equations with acoustic boundary conditions

We define an abstract setting to treat wave equations equipped with time‐dependent acoustic boundary conditions on bounded domains of R ⁿ . We prove a well‐posedness result and develop a spectral theory which also allows to prove a conjecture proposed in [13]. Concrete problems are also discussed. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Well-posedness and spectral properties of heat and wave equations with non-local conditions

We consider the one-dimensional heat and wave equations but – instead of boundary conditions – we impose on the solution certain non-local, integral constraints. An appropriate Hilbert setting leads to an integration-by-parts formula in Sobolev spaces of negative order and eventually allows us to use semigroup theory leading to analytic well-posedness, hence sharpening regularity results previously obtained by other authors. In doing so we introduce a parametrization of such integral conditions that includes known cases but also shows the connection with more usual boundary conditions, like periodic ones. In the self-adjoint case, we even obtain eigenvalue asymptotics of so-called Weyl?s type.     

Dynamic and generalized Wentzell node conditions for network equations

Motivated by a neurobiological problem, we discuss a class of diffusion problems on a network. The celebrated Rall lumped soma model for the spread of electrical potential in a dendritical tree prescribes that the common cable equation must be coupled with particular dynamic conditions in some nodes (the cell bodies, or somata). We discuss the extension of this model to the case of a whole network of neurons, where the ramification nodes can be either active (with excitatory time‐dependent boundary conditions) or passive (where no dynamics take place, i.e. only Kirchhoff laws are imposed). While well‐posedness of the system has already been obtained in previous works, using abstract tools based on variational methods and semigroup theory we are able to prove several qualitative properties, including asymptotic behaviour, regularity of solutions, and monotonicity of the semigroups in dependence on the physical coefficients. Copyright © 2006 John Wiley & Sons, Ltd.     

A new Gershgorin‐type result for the localisation of the spectrum of matrices

We present a Gershgorin's type result on the localisation of the spectrum of a matrix. Our method is elementary as it relies upon the method of Schur complements, but it outperforms the one based on the Cassini ovals of Ostrovski and Brauer. Furthermore, it yields estimates that hold without major differences in the cases of both scalar and operator matrices. Several refinements of known results are obtained.     

The spectrum of the Hilbert space valued second derivative with general self-adjoint boundary conditions

We consider a large class of self-adjoint elliptic problems associated with the second derivative acting on a space of vector-valued functions. We present and survey several results that can be obtained by means of two different approaches to the study of the associated eigenvalues problems. The first, more general one allows to replace a secular equation (which is well known in some special cases) by an abstract rank condition. The second one, though available in general, seems to apply particularly well to a specific boundary condition, the sometimes dubbed anti-Kirchhoff condition in the literature, that arises in the theory of differential operators on graphs; it also permits to discuss interesting and more direct connections between the spectrum of the differential operator and some graph theoretical quantities, in particular some results on the symmetry of the spectrum in either case.     

On moments-preserving cosine families and semigroups in C[0, 1]

We use the newly developed Lord Kelvin’s method of images (Bobrowski in J Evol Equ 10(3):663–675, 2010; Semigroup Forum 81(3):435–445, 2010) to show existence of a unique cosine family generated by a restriction of the Laplace operator in C[0, 1] 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.