# Delio MugnoloFernUniversität in Hagen · Fakultät für Mathematik und Informatik

Delio Mugnolo

Ph.D. in Mathematics, 2004

## About

119

Publications

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Introduction

Delio Mugnolo is a full professor at the Faculty for Mathematics and Computer Science, University of Hagen. He is currently working on spectral geometry of quantum graphs; evolution equation on networks; and gradient flows.

Additional affiliations

July 2014 - present

October 2008 - June 2014

October 2005 - September 2008

Education

October 2005 - January 2011

September 2001 - June 2000

September 1996 - December 2000

## Publications

Publications (119)

We develop the theory of torsional rigidity -- a quantity routinely considered for Dirichlet Laplacians on bounded planar domains -- for Laplacians on metric graphs. Using a variational characterization that goes back to P\'olya, we develop surgical principles that, in turn, allow us to prove isoperimetric-type inequalities: we can hence compare th...

We derive several upper bounds on the spectral gap of the Laplacian with standard or Dirichlet vertex conditions on compact metric graphs. In particular, we obtain estimates based on the length of a shortest cycle (girth), diameter, total length of the graph, as well as further metric quantities introduced here for the first time, such as the avoid...

We develop a comprehensive spectral geometric theory for two distinguished self-adjoint realisations of the Laplacian, the so-called Friedrichs and Neumann extensions, on infinite metric graphs. We present a new criterion to determine whether these extensions have compact resolvent or not, leading to concrete examples where this depends on the chos...

We prove first existence of a classical solution to a class of parabolic problems with unbounded coefficients on metric star graphs subject to Kirchhoff-type conditions. The result is applied to the Ornstein–Uhlenbeck and the harmonic oscillator operators on metric star graphs. We give an explicit formula for the associated Ornstein–Uhlenbeck semig...

A Correction to this paper has been published: 10.1007/s00028-021-00715-0

The aim of this paper is to study the wellposedness and L²‐regularity, firstly for a linear heat equation with dynamic boundary conditions by using the approach of sesquilinear forms, and secondly for its backward adjoint equation using the Galerkin approximation and the extension semigroup to a negative Sobolev space.

We investigate the relationship between one of the classical notions of boundaries for infinite graphs, graph ends, and self-adjoint extensions of the minimal Kirchhoff Laplacian on a metric graph. We introduce the notion of finite volume for ends of a metric graph and show that finite volume graph ends is the proper notion of a boundary for Markov...

We study diffusion-type equations supported on structures that are randomly varying in time. After settling the issue of well-posedness, we focus on the asymptotic behavior of solutions: our main result gives sufficient conditions for pathwise convergence in norm of the (random) propagator towards a (deterministic) steady state. We apply our findin...

We consider two $$C_0$$ C 0 -semigroups on function spaces or, more generally, Banach lattices and give necessary and sufficient conditions for the orbits of the first semigroup to dominate the orbits of the second semigroup for large times. As an important special case we consider an $$L^2$$ L 2 -space and self-adjoint operators A and B which gene...

We define and study two new kinds of “effective resistances” based on hubs-biased – hubs-repelling and hubs-attracting – models of navigating a graph/network. We prove that these effective resistances are squared Euclidean distances between the vertices of a graph. They can be expressed in terms of the Moore–Penrose pseudoinverse of the hubs-biased...

We prove first existence of a classical solution to a class of parabolic problems with unbounded coefficients on metric star graphs subject to Kirchhoff-type conditions. The result is applied to the Ornstein--Uhlenbeck and the harmonic oscillator operators on metric star graphs. We give an explicit formula for the associated Ornstein--Uhlenbeck sem...

Linear evolution equations are considered usually for the time variable being defined on an interval where typically initial conditions or time periodicity of solutions is required to single out certain solutions. Here, we would like to make a point of allowing time to be defined on a metric graph or network where on the branching points coupling c...

We establish metric graph counterparts of Pleijel’s theorem on the asymptotics of the number of nodal domains $$\nu _n$$ ν n of the n th eigenfunction(s) of a broad class of operators on compact metric graphs, including Schrödinger operators with $$L^1$$ L 1 -potentials and a variety of vertex conditions as well as the p -Laplacian with natural ver...

We study properties of spectral minimal partitions of metric graphs within the framework recently introduced in Kennedy et al. (Calc Var 60:6, 2021). We provide sharp lower and upper estimates for minimal partition energies in different classes of partitions; while the lower bounds are reminiscent of the classic isoperimetric inequalities for metri...

We study evolution equations on networks that can be modeled by means of hyperbolic systems. We extend our previous findings in Kramar et al. (Linear hyperbolic systems on networks. arXiv:2003.08281, 2020) by discussing well-posedness under rather general transmission conditions that might be either of stationary or dynamic type—or a combination of...

The aim of the paper is to study the problem $u_{tt}-c^2\Delta u=0$ in $\mathbb{R}\times\Omega$, $\mu v_{tt}- \text{div}_\Gamma (\sigma \nabla_\Gamma v)+\delta v_t+\kappa v+\rho u_t =0$ on $\mathbb{R}\times \Gamma_1$, $v_t =\partial_\nu u$ on $\mathbb{R}\times \Gamma_1$,$\partial_\nu u=0$ on $\mathbb{R}\times \Gamma_0$, $u(0,x)=u_0(x)$ and $u_t(0,x...

This paper contains two contributions in the study of optimal transport on metric graphs. Firstly, we prove a Benamou-Brenier formula for the Wasserstein distance, which establishes the equivalence of static and dynamical optimal transport. Secondly, in the spirit of Jordan-Kinderlehrer-Otto, we show that McKean-Vlasov equations can be formulated a...

We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partition...

Discrete-state stochastic models are a popular approach to describe the inherent stochasticity of gene expression in single cells. The analysis of such models is hindered by the fact that the underlying discrete state space is extremely large. Therefore hybrid models, in which protein counts are replaced by average protein concentrations, have beco...

We define and study two new kinds of ``effective resistances'' based on hubs-biased --hubs-repelling and hubs-attracting -- models of navigating a graph/network. We prove that these effective resistances are squared Euclidean distances between the vertices of a graph. They can be expressed in terms of the Moore-Penrose pseudoinverse of the hubs-bia...

We study hyperbolic systems of one - dimensional partial differential equations under general , possibly non-local boundary conditions. A large class of evolution equations, either on individual 1- dimensional intervals or on general networks , can be reformulated in our rather flexible formalism , which generalizes the classical technique of first...

We establish metric graph counterparts of Pleijel's theorem on the asymptotics of the number of nodal domains $\nu_n$ of the $n$-th eigenfunction(s) of a broad class of operators on compact metric graphs, including Schr\"odinger operators with $L^1$-potentials and a variety of vertex conditions as well as the $p$-Laplacian with natural vertex condi...

We study higher-order elliptic operators on one-dimensional ramified structures (networks). We introduce a general variational framework for fourth-order operators that allows us to study features of both hyperbolic and parabolic equations driven by this class of operators. We observe that they extend to the higher-order case and discuss well-posed...

In the original version of this article, the $ symbols were misinterpreted as part of the article text and placed in the article as it is.

We analyze properties of semigroups generated by Schrödinger operators Δ−V or polyharmonic operators −(−Δ)m, on metric graphs both on Lp-spaces and spaces of continuous functions. In the case of spatially constant potentials, we provide a semi-explicit formula for their kernel. Under an additional sub-exponential growth condition on the graph, we p...

We study evolution equations on networks that can be modeled by means of hyperbolic systems. We extend our previous findings in \cite{KraMugNic20} by discussing well-posedness under rather general transmission conditions that might be either of stationary or dynamic type - or a combination of both. Our results rely upon semigroup theory and element...

We study properties of spectral minimal partitions of metric graphs within the framework recently introduced in [Kennedy et al. (2020), arXiv:2005.01126]. We provide sharp lower and upper estimates for minimal partition energies in different classes of partitions; while the lower bounds are reminiscent of the classic isoperimetric inequalities for...

We study the one-dimensional Laplace operator with point interactions on the real line identified with two copies of the half-line \([0,\infty )\). All possible boundary conditions that define generators of \(C_0\)-semigroups on \(L^2\big ([0,\infty )\big )\oplus L^2\big ([0,\infty )\big )\) are characterized. Here, the Cayley transform of the matr...

We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partition...

We study diffusion-type equations supported on structures that are randomly varying in time. After settling the issue of well-posedness, we focus on the asymptotic behavior of solutions: our main result gives sufficient conditions for pathwise convergence in norm of the (random) propagator towards a (deterministic) steady state. We apply our findin...

We analyze properties of semigroups generated by Schr\"odinger operators $-\Delta+V$ or polyharmonic operators $-(-\Delta)^m$, on metric graphs both on $L^p$-spaces and spaces of continuous functions. In the case of spatially constant potentials, we provide a semi-explicit formula for their kernel. Under an additional sub-exponential growth conditi...

We study hyperbolic systems of one-dimensional partial differential equations under general, possibly non-local boundary conditions. A large class of evolution equations, either on individual 1-dimensional intervals or on general networks, can be reformulated in our rather flexible formalism, which generalizes the classical technique of first-order...

We study the differential operator $A=\frac{d^4}{dx^4}$ acting on a connected network $\mathcal{G}$ along with $\mathcal L^2$, the square of the discrete Laplacian acting on a connected combinatorial graph $\mathsf{G}$. For both operators we discuss self-adjointness issues, well-posedness of the associated parabolic problems \[ \frac{df}{dt}=-\math...

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...

Linear evolution equations are considered usually for the time variable being defined on an interval where typically initial conditions or time-periodicity of solutions are required to single out certain solutions. Here we would like to make a point of allowing time to be defined on a metric graph or network where on the branching points coupling c...

This book presents novel results by participants of the conference “Control theory of infinite-dimensional systems” that took place in January 2018 at the FernUniversität in Hagen. Topics include well-posedness, controllability, optimal control problems as well as stability of linear and nonlinear systems, and are covered by world-leading experts i...

This book contains contributions from the participants of the research group hosted by the ZiF - Center for Interdisciplinary Research at the University of Bielefeld during the period 2013-2017 as well as from the conclusive conference organized at Bielefeld in December 2017. The contributions consist of original research papers: they mirror the sc...

We consider two $C_0$-semigroups on functions spaces or, more generally, Banach lattices and give necessary and sufficient conditions for the orbits of the first semigroup to dominate the orbits of the second semigroup for large times. As an important special case we consider an $L^2$-space and self-adjoint operators $A$ and $B$ which generate $C_0...

Metric graphs are often introduced based on combinatorics, upon "associating" each edge of a graph with an interval; or else, casually "gluing" a collection of intervals at their endpoints in a network-like fashion. Here we propose an abstract, self-contained definition of metric graph. Being mostly topological, it doesn't require any knowledge fro...

Metric graphs are often introduced based on combinatorics, upon "associating" each edge of a graph with an interval; or else, casually "gluing" a collection of intervals at their endpoints in a network-like fashion. Here we propose an abstract, self-contained definition of metric graph. Being mostly topological, it doesn't require any knowledge fro...

We study higher order elliptic operators on one-dimensional ramified structures (networks). We first introduce a general variational framework for fourth order operators that allows us to study features of both hyperbolic and parabolic equations driven by this class of operators that easily extend to the higher order case. In particular, we discuss...

We investigate self-adjoint extensions of the minimal Kirchhoff Laplacian on an infinite metric graph. More specifically, the main focus is on the relationship between graph ends and the space of self-adjoint extensions of the corresponding minimal Kirchhoff Laplacian $\mathbf{H}_0$. First, we introduce the notion of finite and infinite volume for...

A community structure is an important non-trivial topological feature of a complex networks. Indeed community structures are a typical feature of social networks, tightly connected groups of nodes in the World Wide Web usually correspond to pages on common topics, communities in cellular and genetic networks are related to functional modules [46].

The aim of this paper is to study the wellposedness and $L^2$-regularity, firstly for a linear heat equation with dynamic boundary conditions by using the approach of sesquilinear forms, and secondly for its backward adjoint equation using the Galerkin approximation and the extension semigroup to a negative Sobolev space.

A method for estimating the spectral gap along with higher eigenvalues of nonequilateral quantum graphs has been introduced by Amini and Cohen-Steiner recently: it is based on a new transference principle between discrete and continuous models of a graph. We elaborate on it by developing a more general transference principle and by proposing altern...

We study the one-dimensional Laplace operator with point interactions on the real line identified with two copies of the half-line $[0,\infty)$. All possible boundary conditions that define generators of $C_0$-semigroups on $L^2\big([0,\infty)\big)\oplus L^2\big([0,\infty)\big)$ are characterized. Here, the Cayley transform of the boundary conditio...

In analogy to a characterisation of operator matrices generating $C_0$-semigroups due to R. Nagel (\cite{[Na89]}), we give conditions on its entries in order that a $2\times 2$ operator matrix generates a cosine operator function. We apply this to systems of wave equations, to second order initial-boundary value problems, and to overdamped wave equ...

Discrete-state stochastic models are a popular approach to describe the inherent stochasticity of gene expression in single cells. The analysis of such models is hindered by the fact that the underlying discrete state space is extremely large. Therefore hybrid models, in which protein counts are replaced by average protein concentrations, have beco...

We develop a variational approach in order to study qualitative properties of non-autonomous 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...

A widely used approach to describe the dynamics of gene regulatory networks is based on the chemical master equation, which considers probability distributions over all possible combinations of molecular counts. The analysis of such models is extremely challenging due to their large discrete state space. We therefore propose a hybrid approximation...

We present a systematic collection of spectral surgery principles for the Laplacian on a metric graph with any of the usual vertex conditions (natural, Dirichlet or $\delta$-type), which show how various types of changes of a local or localised nature to a graph impact the spectrum of the Laplacian. Many of these principles are entirely new, these...

We study Schroedinger operators with Robin boundary conditions on exterior domains in $\R^d$. We prove sharp point-wise estimates for the associated semi-groups which show, in particular, how the boundary conditions affect the time decay of the heat kernel in dimensions one and two. Applications to spectral estimates are discussed as well.

We derive a number of upper and lower bounds for the first nontrivial eigenvalue of a finite quantum graph in terms of the edge connectivity of the graph, i.e., the minimal number of edges which need to be removed to make the graph disconnected. On combinatorial graphs, one of the bounds is the well-known inequality of Fiedler, of which we give a n...

We develop the theory of linear evolution equations associated with the adjacency matrix of a graph, focusing in particular on infinite graphs of two kinds: uniformly locally finite graphs as well as locally finite line graphs. We discuss in detail qualitative properties of solutions to these problems by quadratic form methods. We distinguish betwe...

In the present paper the Airy operator on star graphs is defined and studied. The Airy operator is a third order differential operator arising in different contexts, but our main concern is related to its role as the linear part of the Korteweg-de Vries equation, usually studied on a line or a half-line. The first problem treated and solved is its...

We review the theory of Cheeger constants for graphs and quantum graphs and their present and envisaged applications.

We study the diffusion of epidemics on networks that are partitioned into local communities. The gross structure of hierarchical networks of this kind can be described by a quotient
graph. The rationale of this approach is that individuals infect those belonging to the same community with higher probability than individuals in other communities. In...

We prove Cheeger inequalities for p-Laplacians on finite and infinite
weighted graphs. Unlike in previous works, we do not impose boundedness of the
vertex degree, nor do we restrict ourselves to the normalized Laplacian and,
more generally, we do not impose any boundedness assumption on the geometry.
This is achieved by a novel definition of the m...

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 invest...

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 ea...

The p-Laplacian operators have a rich analytical theory and in the last few years they have also offered efficient tools to tackle several tasks in machine learning. During the workshop mathematicians and theoretical computer scientists working on models based on p-Laplacians on graphs and manifolds have presented the latest theoretical development...

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.

We consider the so-called \emph{discrete $p$-Laplacian}, a nonlinear
difference operator that acts on functions defined on the nodes of a possibly
infinite graph. We study the associated nonlinear Cauchy problem and identify
the generator of the associated nonlinear semigroups. We prove higher order
time regularity of the solutions. We investigate...

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 o...

The first step in the study of evolution equations is the choice of suitable functions spaces that capture the structure of the problem. In this chapter we introduce some relevant spaces of functions on networks. We assume the theory of Lebesgue spaces to be known to the reader, whilst the fundamental aspects of the theory of Sobolev spaces are sum...

It is well-known from elementary linear algebra that the solution of the linear Cauchy problem $$\displaystyle{ \left \{\begin{array}{rcll} \frac{\mathit{dx}} {\mathit{dt}} (t)& =&Ax(t), &\qquad t \in \mathbb{R}, \\ x(0)& =&x_{0} \in {\mathbb{C}}^{\ltimes }, \end{array} \right. }$$ associated with an n × n matrix A is given by x(t): = e
tA
x
0, whe...

The aim of this final chapter is to discuss how possible symmetries in an (oriented) weighted graph influence the behavior of evolution equations—either on or on the metric graph over it. We will use the word “symmetry” in a rather broad sense, to mean different notions of structural regularity of graphs of the Laplacian on.
The common thread in ou...

Applying the theorems of Hille–Yosida or Lumer–Phillips is sometimes unsatisfactory, as they make no claim about possible regularity gain (either in space or time) of solutions. In this chapter we specialize our previous investigations to parabolic equations. These are evolution equations, typically associated with diffusive processes, whose foremo...

In the previous chapters we have introduced a convenient functional analytical framework for studying evolution equations on network-like structures, along with some first examples. In the next chapters we will devote much attention to the investigation of properties of partial differential equations that are motivated by applications, and we will...

The theory of forms presented in Chap. 6 was originally developed in order to extend the study of parabolic problems beyond the setting of the Spectral Theorem, in much the same way the Lax–Milgram Lemma extended the applicability of the Riesz–Fréchet Theorem. This program was successful: Nowadays many relevant results on linear parabolic problems...

In this chapter we are going to review a manifold of operators defined on networks. We will see later on that most of these operators arise in connection with some relevant evolution equation on networks. However, here we are not yet going to discuss any dynamical system or partial differential equation. Rather, our aim is to explain the interplay...

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.

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 semi...

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...

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, th...

Introduction.- Operators on Networks.- Function Spaces on Networks.- Operator Semigroups.- And Now Something Completely Different: A Crash Course in Cortical Modeling.- Sequilinear Froms and Analytic Semigroups.- Evolution Equations Associated With Self-adjoint Operators.- Symmetry Properties.- Index.

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 symmetr...

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 abs...

We study the discrete version of the p-Laplace operator. Based on its variational properties we discuss some features of the associated parabolic problem. We prove well-posedness of the problem and obtain information about positivity and comparison principles as well as compatibility with the symmetries of the underlying graph. Our methods consist...

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-posed...

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...

We characterize the domain of a Fleming-Viot type operator of the form Lφ(x):= ∑Ni=1 xi(1 - xi)Diiφ(x) + ∑Ni=1(αi(1 - xi) - αi+1xi)Diφ(x) on Lp([0,1]N, μ) for 1 < p < ∞, where μ is the corresponding invariant measure. Our approach relies on the characterization of the domain of the one-dimensional Fleming-Viot operator and the Dore-Venni operator s...

We study the existence of strong solutions for a class of stochastic differential equations in an infinite dimensional space. Our investigation is specially motivated by the stochastic version of a common model of potential spread in a dendritic tree. We do not assume the noise in the junction points to be Markovian. In fact, we allow for long-rang...