arXiv subjects

Delio Mugnolo

Publications and source records attributed to Delio Mugnolo.

At least 73 records · Page 4Linked to original sources

arXiv2014

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 and relies upon the method of Schur complements, furthermore 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.

math.SP↗

arXiv2014

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.

math-ph↗

arXiv2014

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 $n\in \mathbb N$ infinitely many elements of the class have $n$-dimensional null space.

math.FA↗

arXiv2014

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.

math.AP↗

arXiv2013

Asymptotics of semigroups generated by operator matrices

We survey some known results about operator semigroup generated by operator matrices with diagonal or coupled domain. These abstract results are applied to the characterization of well-/ill-posedness for a class of evolution equations with dynamic boundary conditions on domains or metric graphs. In particular, our ill-posedness results on the heat equation with general Wentzell-type boundary conditions complement those previously obtained by, among others, Bandle-von Below-Reichel and Vitillaro-Vázquez.

math.FA↗

arXiv2013

The heat equation under linear conditions on the moments in higher dimensions

We consider the heat equation on the $N$-dimensional cube $(0,1)^N$ and impose different classes of integral conditions, instead of usual boundary ones. Well-posedness results for the heat equation under the condition that the moments of order 0 and 1 are conserved had been known so far only in the case of N=1 -- for which such conditions can be easily interpreted as conservation of mass and barycenter. In this paper we show that in the case of general $N$ the heat equation with such integral conditions is still well-posed, upon suitably relax the notion of solution. Existence and uniqueness of solutions with general initial data in a suitable space of distibutions over $(0,1)^N$ are proved by introducing two appropriate realizations of the Laplacian and checking by form methods that they generate analytic semigroups. The solution thus obtained does however solve the heat equation only in a certain distributional sense. However, it turns out that one of these realizations is tightly related to a well-known object of operator theory, the Krein--von Neumann extension of the Laplacian. This connection also establishes well-posedness in a classical sense, as long as the initial data are $L^2$-functions.

math.FA↗

arXiv2013

Quantum graphs with mixed dynamics: the transport/diffusion case

We introduce a class of partial differential equations on metric graphs associated with mixed evolution: on some edges we consider diffusion processes, on other ones transport phenomena. This yields a system of equations with possibly nonlocal couplings at the boundary. We provide sufficient conditions for these to be governed by a contractive semigroup on a Hilbert space naturally associated with the system. We show that our setting is also adequate to discuss specific systems of diffusion equations with boundary delays.

math.AP↗

arXiv2013

Construction of exact travelling waves for the Benjamin-Bona-Mahony equation on networks

We are interested in the existence of travelling waves for the Benjamin-Bona-Mahony equation on a network. First we construct an explicit wave, defined in $\mathbb{R}$. Then, we use this wave to derive some conditions on the coefficients appearing in the equations and on the geometry of the network to ensure the existence of travelling waves on the network.

math.AP↗

arXiv2013

Parabolic theory of the discrete p-Laplace operator

We study the discrete version of the $p$-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.

math.CO↗

arXiv2013

Diffusion processes on an interval under linear moment conditions

We discuss a class of diffusion-type partial differential equations on a bounded interval and discuss the possibility of replacing the boundary conditions by certain linear conditions on the moments of order 0 (the total mass) and of another arbitrarily chosen order n. Each choice of n induces the addition of a certain potential in the equation, the case of zero potential arising exactly in the special case of n=1 corresponding to a condition on the barycenter. In the linear case we exploit smoothing properties and perturbation theory of analytic semigroups to obtain well-posedness for the classical heat equation (with said conditions on the moments). Long time behavior is studied for both the linear heat equation with potential and certain nonlinear equations of porous medium or fast diffusion type. In particular, we prove polynomial decay in the porous medium range and exponential decay in the fast diffusion range, respectively.

math.AP↗

arXiv2012

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

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 $C[0,1]$, 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.

math.FA↗

arXiv2012

Convergence of sectorial operators on varying Hilbert space

Convergence of operators acting on a given Hilbert space is an old and well studied topic in operator theory. The idea of introducing a related notion for operators acting on arying spaces is natural. However, it seems that the first results in this direction have been obtained only recently, to the best of our knowledge. Here we consider sectorial operators on scales of Hilbert spaces. We define a notion of convergence that generalises convergence of the resolvents in operator norm to the case when the operators act on different spaces and show that this kind of convergence is compatible with the functional calculus of the operator and moreover implies convergence of the spectrum. Finally, we present examples for which this convergence can be checked, including convergence of coefficients of parabolic problems. Convergence of a manifold (roughly speaking consisting of thin tubes) towards the manifold's skeleton graph plays a prominent role, being our main application.

math.FA↗

arXiv2012

Variational and Semigroup Methods for Waves and Diffusion in Networks

We study diffusion and wave equations in networks. Combining semigroup and variational methods we obtain well-posedness and many nice properties of the solutions in general L^p -context. Following earlier articles of other authors, we discuss how the spectrum of the generator can be connected to the structure of the network. We conclude by describing asymptotic behavior of solutions to the diffusion problem.

math.AP↗

arXiv2012

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

We consider a large class of self-adjoint elliptic problem associated with the second derivative acting on a space of vector-valued functions. We present 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 latter seems to apply particularly well to a specific boundary condition, sometimes dubbed "anti-Kirchhoff" in the literature, that arise 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 either case our results yield, among other, some results on the symmetry of the spectrum.

math.SP↗

arXiv2011

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 generalises 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.

math.FA↗

arXiv2011

Towards a gauge theory for evolution equations on vector-valued spaces

We investigate symmetry properties of vector-valued diffusion and Schrödinger equations. For a separable Hilbert space $H$ we characterize the subspaces of $L^2(Ω, H)$ that are local (i.e., defined pointwise) and discuss the issue of their invariance under the time evolution of the differential equation. In this context, the possibility of a connection between our results and the theory of gauge symmetries in mathematical physics is explored.

math-ph↗

arXiv2011

Gradient Systems on Networks

We consider a class of linear differential operators acting on vector-valued function spaces with general coupled boundary conditions. Unlike in the more usual case of so-called quantum graphs, the boundary conditions can be nonlinear. After introducing a suitable Lyapunov function we prove well-posedness and invariance results for the corresponding nonlinear diffusion problem.

math.AP↗

arXiv2010

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 ${\bf R}^n$. We prove a well-posedness result and develop a spectral theory which also allows to prove a conjecture proposed in (Gal-Goldstein-Goldstein, J. Evol. Equations 3 (2004), 623-636). Concrete problems are also discussed.

math.AP↗