#### 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