Titles and abstracts
(ICREA & Universitat Politècnica de Catalunya)
Stable solutions to semilinear elliptic equations are smooth up to dimension 9
The regularity of stable solutions to semilinear elliptic PDEs has been studied since the 1970's.
In dimensions 10 and higher, there exist singular stable energy solutions.
In this talk I will describe a recent work in collaboration with
Figalli, Ros-Oton, and Serra, where we prove that stable solutions are
smooth up to the optimal dimension 9. This answers to an open problem
posed by Brezis in the mid-nineties concerning the regularity of
extremal solutions to Gelfand-type problems.
Maso (SISSA, Trieste)
New results on the
jerky crack growth in elasto-plastic materials
In the framework of a model for the
quasistatic crack growth in elasto-plastic homogeneous materials in the
planar case, we study the properties of the length of the crack as a
function of time.
We prove that, under suitable technical assumptions on the crack path,
this monotone function is a pure jump function. Under stronger
assumptions we prove also that the number of jumps is finite.
Irene Fonseca (Carnegie Mellon University)
Phase separation in heterogeneous
A variational model in the context of
the gradient theory for fluid-fluid phase transitions with small scale
heterogeneities is studied. In the case where the scale of the small
heterogeneities is of the same order of the scale governing the phase
transition, the interaction between homogenization and the phase
transitions process leads to an anisotropic interfacial energy.
Bounds on the homogenized surface tension are established. In addition,
a characterization of the large-scale limiting behavior of viscosity
solutions to non-degenerate and periodic Eikonal equations in
half-spaces is given.
This is joint work with Riccardo Cristoferi (Radboud University, The
Netherlands), Adrian Hagerty (USA), Cristina Popovici (USA), Rustum
Choksi (McGill, Canada), Jessica Lin (McGill, Canada), and Raghavendra
Venkatraman (NYU, USA).