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

Courses by:

- 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.

- Patrizia Pucci, Università di Perugia, Perugia (Italy)

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.

- Susanna Terracini, Università di Torino, Torino (Italy)

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.

Registration:

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.

How to reach us:

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.

Accommodation:

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.