InthistalkbasedonjointworkwithKienzlerandVazquezIwill explainhowCaffarelli’sstrategyofprovingregularityforfree boundaryproblemscanbeimplementedfortheporousmediumequation. ElementsoftheproofisaglobalexistenceresultforLipschitz…
Categoria evento: Analysis Seminar
Nonlocal phase transitions and minimal surfaces – Enrico Valdinoci
We present some recent results on a nonlocal version of the Allen-Cahn equation, which is also related, at a large scale, to surfaces which minimise a nonlocal perimeter functional. We discuss some features, such as symmetry property, regularity and…
A direct approach to an epiperimetric inequality for Free-Boundary problems – Luca Spolaor
Using a direct approach, we prove a 2-dimensional epiperimetric inequality for the one-phase problem in the scalar and vectorial cases and for the double-phase problem. From this we deduce the $C^{1,α}$ regularity of the free-boundary in the scalar…
Cohesive behaviour arising in homogenization of Mumford-Shah type functionals – Marco Barchiesi (Universita’ di Napoli Federico II)
I will discuss the effective properties of a composite material whose micro-structure is constituted by a brittle material with periodically distributed soft inclusions. In particular, I will show that the presence of the soft inclusions gives rise,…
A PDE approach to a 2-dimensional matching problem – Federico Stra (Scuola Normale Superiore Pisa)
We prove asymptotic results for 2-dimensional random matching problems. In particular, we obtain the leading term in the asymptotic expansion of the expected quadratic transportation cost for empirical measures of two samples of independent uniform…
A variational model for chiral liquid crystals – Marcello Ponsiglione (Universita’ di Roma La Sapienza)
I will introduce a discrete variational model for chiral liquid crystals. The order parameter in this model is an angle measuring the orientation of the chiral molecules. The corresponding energy is a slight variant of the XY energy, which also…
Ground states of a two phase model with cross and self attractive interactions – Lucia De Luca (TU Muenchen )
We consider a variational model for two interacting species (or phases), subject to cross and self attractive interactions. We show existence and several qualitative properties of minimizers. Depending on the strengths of the attractive forces,…
Dynamic evolutions for a peeling test in dimension one. – Giuliano Lazzaroni (SISSA)
We present a simplified model of dynamic crack propagation, where the equation of elastodynamics is coupled with Griffith’s principle. In recent years there has been an increasing interest in studying systems where second-order equations for…
Obstacle-Type Problems and Generalizations to the Fully Non-Linear Setting – Andreas Minne (Scuola Normale Superiore)
The obstacle problem is more than half a century old, but remains a relevant topic in mathematics due to the many variants and applications that occur in, for instance, physics, finance and potential theory. In the seminar we will briefly go over…
From coarea formula to level set differential equations – Valentino Magnani (Universita’ di Pisa)
We present how the problem of coarea formula in the Heisenberg group leads us rather unexpectedly to a system of “level set differential equations”, generated by a defining mapping of very low regularity. In this system we cannot write classical…