The Øresund Seminar on Analysis, PDE & Related Topics

Abstract: 

Abstract: 

The talk will present some recent results about the free boundary problem for an incompressible, irrotational liquid drop of nearly spherical shape with capillarity. The first part of the talk will deal with the extension of some classical results from the flat case to the spherical geometry, including the reduction to a problem on the boundary, the Hamiltonian structure, the analyticity and tame estimates for the Dirichlet-Neumann operator in Sobolev class, and a linearization formula for it, both by a differential geometry approach and with the method of the 'good unknown of Alinhac'. The second part of the talk will deal with the bifurcation of travelling waves, which are nontrivial fixed profiles rotating with constant angular velocity. Bifurcation from both simple and multiple eigenvalues is considered, where the Hamiltonian and variational structure of the problem, together with its conserved quantities, play a key role.

From works in collaboration with Vesa Julin, Domenico Angelo La Manna and Giuseppe La Scala.

Abstract: 

Abstract: 

For a dynamical system, modelled by the iterations of a mapping T:X→X, an invariant measure is a measure μ on X such that the measure of a set is the same as the measure of the pre-image under T of the set. For such measures, the distribution of a typical orbit is exactly the measure μ.

Given a smooth function φ, and a parametrised family of dynamical systems (X, T_t, μ_t), the response funtion R(t) = ∫φ dμ_t describes how typical orbits change with the parameter t. In some cases, R is smooth, which is called linear response, and in other cases R is only Hölder continuous, which is called fractional response.

I am going to talk about work together with Viviane Baladi about fractional response in the quadratic family, T_t(x) = t - x^2. From previous work, it is known that the response function R is essentially ½-Hölder continuous, and we aim at proving that this is essentially sharp.