School on Nonlinear Elliptic Problems
within the GNAMPA Project 2013
Problemi nonlocali di tipo laplaciano frazionario
Dipartimento di Matematica e Applicazioni
Università di Milano 'Bicocca'
January 20-24, 2014
Nonlinear PDEs can be used to describe a wide variety of phenomena arising in different contexts such as geometry, physics, mechanics, engineering and, more recently, life sciences, just to name a few.
The aim of the school is to present some recent results and future trends on Nonlinear Elliptic Problems and their applications, by leading together experts in this field.
The courses organized within the school are addressed to Ph.D. students as well as Post-Doctoral and active researchers interested mostly in Nonlinear Analysis, Partial Differential Equations and their many applications.
Organizers: Giovanni Molica Bisci, Simone Secchi and Raffaella Servadei
Supported by:
Dipartimento di Matematica e Applicazioni, Università di Milano 'Bicocca'
Università 'Mediterranea' di Reggio Calabria
GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni)
ERC Project EPSILON (Elliptic Pde's and Symmetry of Interfaces and Layers for Odd Nonlinearities)
Courses by:
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Patrizia Pucci, Università di Perugia, Perugia (Italy)
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Susanna Terracini, Università di Torino, Torino (Italy)
Registration:
How to reach us:
Accommodation:
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Xavier Cabré, ICREA and Universitat Politècnica de Catalunya, Barcelona (Spain)
Title of the course: The influence of fractional diffusion in Allen-Cahn and KPP type equations
Abstract: In this mini-course I will start explaining basic ideas concerning fractional
Laplacians as well as the essential tools to treat nonlinear equations involving
these operators. The main part of the course will present results on stationary
solutions of fractional equations (mainly of bistable or Allen-Cahn type)
and on front propagation with fractional diffusion for Fisher-KPP, combustion,
and bistable reactions.
The course will focus in the following papers (see arXiv):
- In several works in collaboration with Y. Sire and E. Cinti, we study the
existence, uniqueness, and qualitative properties of layer or heteroclinic
solutions to fractional elliptic equations involving a bistable nonlinearity. A
Hamiltonian quantity and sharp energy estimates play here a central role,
and we will also present some of their more recent applications given by
other authors;
- I will explain the results in several articles with J.-M. Roquejoffre and A.-C.
Coulon where we establish the exponential in time propagation of fronts for
the Fisher-KPP equation with fractional diffusion, both in homogeneous and
in periodic media;
- Finally I will describe very recent results with N. Consul and J.V. Mandè on
traveling fronts for the following fractional-diffusion type problem: the
classical homogeneous heat equation in a half-plane with a boundary
Neumann condition of bistable or combustion type.
Here you can find the slides of the course.
Title of the course: On higher order p-Kirchhoff problems
Abstract: In the course we present some recent existence theorems for
nontrivial stationary solutions of problems involving
involving the \(p\)-polyharmonic Kirchhoff operator in bounded domains.
The \(p\)-polyharmonic operators \(\Delta^L_{p}\) were recently introduced in
[F. Colasuonno and P. Pucci, Multiplicity of solutions for
\(p(x)\)-polyharmonic elliptic Kirchhoff equations,
Nonlinear Anal. 74 (2011) 5962-5974] for all orders L and
independently, in the same volume of the journal, in [V.F. Lubyshev,
Multiple solutions of an even-order nonlinear problem with convex-concave
nonlinearity, Nonlinear Anal. 74 (2011) 1345-1354] only for \(L\) even.
The results are then extended to non-degenerate \(p(x)\)-polyharmonic
Kirchhoff operators.
Several useful properties of the underlying functional
solution space \([W^{L,p}_0(\Omega)]^d\), endowed with the natural norm
arising from the variational structure of the problem, are also proved both
in the homogeneous case \(p\equiv \)Const
and in the non-homogeneous case \(p=p(x)\).
In the latter some sufficient
conditions on the variable exponent \(p\) are given to prove the positivity of
the infimum \(\lambda_1\) of the Rayleigh quotient for the \(p(x)\)-polyharmonic
operator \(\Delta^L_{p(x)}\).
Other related problems will be also presented, as well as open problems.
Here you can find the slides (Part 1) and slides (Part 2) of the course.
Title of the course: Geometric aspects in competition-diffusion problems
Abstract: In this mini-course we will present some recent results and various open
problems related to the entire solutions and to the De Giorgi
conjecture for competition-diffusion systems.
In order to understand the interactions of the species in
competition, we will consider the problem of the classification
of the entire solutions of systems with polynomial growth (also
in the case of non-local diffusion).
The problem is related to the structure of the multiple cluster
points of the limiting profile of the segregated species, to
their regularity and to the rate convergence of the competition-diffusion
systems.
Here you can find the slides of the course.
If you are willing to participate, please let us know by filling in
the registration form. Participants
are encouraged to propose a talk: the organizing committee will select
a few proposals. The deadline for the submission of the proposal is January 8, 2014.
The workshop will be held in room 3014 (third floor) of
the Dipartimento di
Matematica e Applicazioni of the Università di Milano
Bicocca. Please
follow these
instructions to reach us. If you need to take a bus or a tram,
please consider that traffic jams happen on a regular basis in Milan,
and tram number 7 is particularly slow. Buses and the subway are
better choices.
Milan is a big city, and it would be impossible to collect a list of
hotels, B&B's, residences and other structures where you can book a
room. Moreover, some structures may be crowded by people attending
fairs and other big events. Our suggestion is to use a search engine
on the Internet to find the best offer for your budget. We suggest
that you reserve your room as soon as possible. Finally, always
check how long it will take to reach the workshop from your hotel:
this can be done on
the website of
ATM. If your hotel is close to the railway station Milano Porta
Garibaldi, you can also board any train that stops at Milano
Greco Pirelli, just in front of the Dipartimento di Matematica e Applicazioni. You
can use a standard ATM ticket, and trains are frequent.
Here you can find the poster, the schedule of the school and
the abstracts of the talks.
Here you can find some pictures of the school.
For further information, please contact
nep2014@unimib.it.