|
 Scientific motivation
The area of nonlinear evolution equations is a
research topic that has seen an enormous development in the last years, and will witness something
similar also in the years to come.
Research collaborations already exist between several of the invited speakers, but the common interests
and the very similar research fields (like the parabolic Harnack estimates for degenerate/singular
operators) make it quite natural to forecast new interactions. Since we plan to leave space for the talks
of several young researchers, which will most likely include doctoral and post-doctoral students already
working with some of the main speakers, we expect that their role in fostering new projects will be
quite relevant.
The topics on which the conference will focus are manifold, and we are going to sketch some of them
hereafter. Recent results on some of them are available, and will be exposed in the conference, and we
expect the conference to boost further result in at least some of the arguments listed below.
- The asymptotic behaviour of solutions to singular nonlinear differential equations depends strongly
on the class of initial data considered. It has been studied in detail in the case initial data are
close to special explicit fundamental solutions in recent papers of Blanchet, Bonforte, Dolbeault,
Grillo, Vazquez, but one has no clue about rates of convergence for, e.g., compactly supported
data. Entropy methods are not available so far, but the analysis of some special cases points
towards a connection between rates of convergence to equilibrium and spectral gap of suitable
linear operators, which have often a geometric significance.
- The propagation of positivity and the Harnack inequality for singular evolution equations is a
field in which recent and crucial developments are due to DiBenedetto, Gianazza and Vespri and,
separately, to Bonforte and Vazquez. Extension to logarithmic diffusions or to the very fast case
are still open.
- The study of the role of curvature in the nonlinear evolution equations on manifolds is open. For
example it is not known whether fundamental solution describing the asymptotics of solutions for
general data exist, nor it is known whether such solutions, if any, have compact support for all
times as in the Euclidean case, or are instead full support. No geometrical condition distinguishing
the two cases is known. On a Riemannian manifolds, the fact that the Ricci curvature is bounded
from below is known to imply the validity of a Harnack inequality in the linear case, but there is
no clue about its role in the nonlinear case.
- The role of the Wasserstein metric could be important in defining fundamental solutions: it is
indeed known that they are, in the Euclidean case, fixed points of a suitable map w.r.t. the such
metric.
- In a recent important paper Acerbi and Mingione prove new results in the Calderon-Zygmund
theory for nonlinear evolution equation, with no use at all of harmonic analysis tools. However, a
lot remains to be studied as concerns optimal regularity of the solutions of such problems.
- The regularity of the extremal solution of semilinear elliptic problems of Gelfand type (of second
order) is an open problem in several cases. Brezis and Vazquez have determined the spatial dimentions
in which regularity holds true. Recently, Cabré and Capella have continued such investigation.
Berchio, Cassani and Gazzola (among others) have studied similar problems for fourth order problems.
One shall try to describe the behaviour of the extremal solution, to give estimates for the
maximal parameter and to determine the critical dimension for different boundary conditions.
There will be a
session of contributed talks: it will be mainly dedicated to young
participants, and hence we strongly encourage all interested PhD and
PostDocs students to submit a title and a short abstract to the
Scientific Committee. This has to be done done using the Registration Form (see below).
|
