People

Faculty

• Claudia Anedda (Assistant professor)
• Lucio Cadeddu (Assistant professor)
• Fabrizio Cuccu (Assistant professor)
• Silvia Frassu (Assistant professor)
• Antonio Greco (Associate professor)
• Antonio Iannizzotto (Associate professor)
• Monica Marras (Associate professor)
• Giuseppe Viglialoro (Associate professor)

Research Interests

1.1 KELLER-SEGEL MODELS

1.1.1 Attraction-repulsion chemotaxis models. In Alzheimer's 
disease, the aggregation of microglia is idealized by introducing in 
the classical Keller-Segel system two different chemical 
distributions: the chemoattractant v, attracting the cells and the 
chemorepellent w, repulsing. When they linearly evolve, some results 
are available; but what happens if the chemorepellent and the 
chemoattractant signals are produced/consumed with other laws? Which 
of these impacts might, for instance, prevent any gathering of the 
cell density u, or at least expand its lifespan?

1.1.2 Indirect consumption models. When the attractive signal v is 
directly consumed by the cells, the stimulus tends to vanish and 
smoothness is more conceivable. This is not so clear in models with 
singular chemosensitivities of the form 1/v, which are undefined at 
v=0. We study the behavior in time of solutions (especially globality 
and boundedness) to Keller-Segel systems with singular cross-terms, 
when the signal is indirectly absorbed by a further chemical.

1.2 POPULATION DYNAMICS

The topic of our investigation is the optimization of the first 
(principal) eigenvalue of the Laplace operator with an indefinite 
weight and different boundary conditions, the weight varying in a 
class of rearrangements of bounded functions. We are interested in 
proving existence of optimizers, their qualitative properties and 
related problems of symmetry and symmetry breaking.

The methods and tools needed to accomplish this kind of investigations 
are the theory of Sobolev spaces, maximum principles for elliptic 
equations, classical variational techniques, and the theory of 
rearrangements of measurable functions.

This kind of optimization problems is connected with the study of 
reaction-diffusion equations in Mathematical Ecology, in particular in 
population dynamics, in order to understand the behaviour or the 
survival of living organisms in a given (often heterogeneous) 
environment.

If the modelled population inhabits a bounded region, it is 
interesting to study how the spatial arrangements of favourable and 
unfavourable habitats in the environment affect the survival of the 
population.

The homogeneous Dirichlet boundary conditions indicate that the 
exterior environment is completely hostile and any individual reaching 
the boundary dies.

The Neumann boundary conditions mean that the boundary acts as a 
barrier, i.e. any individual reaching the boundary returns to the 
interior. The Robin boundary conditions are intermediate between the 
previous conditions and they can approach them as borderline cases.

Our main aim is the investigation of symmetry properties, already 
known under the Dirichlet boundary conditions, in the case of Neumann 
and Robin boundary conditions. Biologically, this issue is related to 
the study of the effects of a fragmented environment on a population.

Finally, this kind of mathematical problem also arises in Physics, 
when one models a vibrating membrane. In this context, the first 
eigenvalue describes the principal natural frequency of the membrane 
and its minimization models the physical problem of building a 
composite membrane with the least principal frequency, given a fixed 
amount of different materials.

2 THIN FILMS

In the literature the great part of the results concerns the case of 
only one equation, because fourth order parabolic equations describe a 
variety of physical processes. In particular, in the thin film theory, 
such problems describe the evolution of the epitaxial growth of a thin 
film where the solution u(x; t) is the height from the surface of the 
film.

In 2015, Xu /et al./ investigated a fourth order parabolic problem in 
a bounded domain and, by using the potential well method, showed that 
the solutions exist globally or blow up in finite time, depending on 
whether or not the initial data are in the potential well.

By different methods Philippin proved that the solution of a fourth 
order parabolic problem cannot exist for all time, i.e. it blows up in 
L²-norm in a finite time T and an upper bound for T is derived; in 
addition, he constructed (under certain conditions on the data) a 
lower bound for T by using a first order differential inequality method.

In this context, our aim is to investigate the behavior in time of the 
solution using the method introduced by G. Philippin: in particular we 
are interested in studying the blow-up phenomenon of the solution in 
finite time t* and determining a time interval [0,T], with T a lower 
bound of t*, so we get a safe interval of existence of the solution.

3 SYMMETRY IN OVERDETERMINED PROBLEMS

We investigate both a fully overdetermined problem of Serrin type and 
a partially overdetermined problem in a conical domain. We plan to 
establish symmetry results for some particularly meaningful elliptic 
operators. A fundamental nonlinear (unless p=2) elliptic operator is 
the p-Laplacian $\Delta_p u = div(|\nabla u|^{p-2} \nabla u)$. Using a 
viscosity interpretation, the parameter p is allowed to take any value 
between 1 and infinity, including the extrema. The novelty of our 
research lies in the fact that we deal with a non-autonomous Neumann 
condition. In the case when p>1, by means of a comparison with 
Euclidean balls, we expect to prove radial symmetry of the domain and 
the solution. A similar result is also expected in the special case 
when p=1: however, since the comparison principle fails for such an 
operator, we intend to exploit its geometric interpretation as the 
mean curvature of the level sets of the solution.

4 SOLVING NONLOCAL, NONLINEAR PROBLEMS

The natural extension of the well-known fractional Laplacian is the 
fractional p-Laplacian, namely a nonlocal nonlinear operator in 
divergence form, which represents the case study for a wide class of 
operators of the same type, arising from game theory (the ‘tug of war’ 
with drifts) and image restoring. More generally, this family of 
operators can be used to model diffusive processes and equilibria 
involving nonlocal interactions, as is the case in phase transitions, 
population dynamics, and continuum mechanics.

Such operators require ad hoc techniques, since (contrary to the 
linear fractional Laplacian) they do not allow for a definition 
through Fourier transforms, or a simple Caffarelli-Silvestre type 
extension approach to retrieve local problems. Neither they can be 
dealt with like the classical p-Laplacian, since the useful method of 
reflections through boundary will not work here. Typically, one must 
construct barriers and sub-supersolutions in order to locally control 
the solutions, in a sense, giving up equations for inequalities.

At present, the theory for standard elliptic and parabolic equations 
driven by the fractional p-Laplacian is getting established. So, 
researchers are in a good position to tackle some special problems 
involving further difficulties:

1. in the elliptic (stationary) framework, we plan to extend to the 
quasilinear nonlocal equations with sublinear reactions the optimal 
solvability result of Brézis-Oswald; to investigate a Serrin-type 
overdetermined problem aiming at a nonexistence result for all 
bounded, smooth domains except the ball; and to establish a functional 
framework for nonlinear equations with singular terms at the origin, 
proving existence of a solution through a fixed-point theoretical 
approach;

2. in the parabolic (evolutive) framework, we plan to study 
quantitative Harnack-type estimates for the solutions of equations 
with classical time derivative and fractional p-Laplacian space 
diffusion, focusing on an effective control on the oscillation of 
solutions rather than with regularity targets.

5 APPLICATIONS OF MATHEMATICAL ANALYSIS AND NUMBER THEORY

We study applications of ODE's to problems inspired by biology and 
nutrition, e.g. to build a mathematical model for an optimal diet 
plan. Moreover, applications of number theory and symmetry schemes are 
used to investigate hidden patterns in music theory, composition and 
art in general.

Events

• IV Workshop on Trends in Nonlinear Analysis (Cagliari, September 13-14, 2022)
• Partial Differential Equations in Analysis and Mathematical Physics (Santa Margherita di Pula, May 30 - June 1, 2019)
• III Workshop on Trends in Nonlinear Analysis in honor of Stella Piro Vernier (Cagliari, September 7-9, 2017)
• Research Meeting on Nonlocal Operators (Cagliari, October 6-8, 2016)
• Geometric properties of solutions to elliptic and parabolic problems (Santa Margherita di Pula, September 19-21, 2016)
• II Workshop on Trends in Nonlinear Analysis (Cagliari, September 24-26, 2015)
• Trends in Nonlinear Analysis (Cagliari, March 21-22, 2014)
• Nonlinear elliptic and parabolic problems (Cagliari, June 28-29, 2012, in honour of prof. Giovanni Porru)

Research Collaborations

• Yutaro Chiyo (Tokyo University of Science)
• Rafael Rodríguez Galván (Universidad de Cádiz)
• Francesca Gladiali (Università di Sassari)
• Johannes Lankeit (Leibniz Universität Hannover)
• Tongxing Li (Shandong University)
• Marcello Lucia (The City University of New York)
• Benyam Mebrate (Wollo University, Dessie, Ethiopia)
• Sunra Mosconi (Università di Catania)
• Dimitri Mugnai (Università della Tuscia)
• Nikolas S. Papageorgiou (National Technical University of Athens)
• Pieralberto Sicbaldi (Universidad de Granada)
• Vasile Staicu (Universidade de Aveiro)
• Yuya Tanaka (Tokyo University of Science)
• Cornelis van der Mee (Università di Cagliari)
• Tobias Weth (Goethe-Universität Frankfurt am Main)
• Thomas Woolley (Cardiff University)
• Tomomi Yokota (Tokyo University of Science)

Selected Publications

Alumni

Former Members

• Todor Gramchev (R.I.P.)
• Anna Piro Grimaldi
• Stella Piro Vernier
• Giovanni Porru
• Francesco Ragnedda