Jueves 6 de mayo, 15 - 17 horas.

Alessandro Audrito (University of Zurich).

Título: Boundary regularity of solutions to some nonlocal elliptic Neumann

problems.

Abstract: I will present some recent results concerning the regularity up to the

boundary of solutions to the Neumann problem for the fractional

Laplacian: if u is a weak solution to (−∆)^su = f in Ω, N_su = 0 in Ω^c,

then u is C^α up to the boundary for some α ∈ (0, 1). Moreover, if s >

1/2, then u ∈ C^2s−1+α(Ω).

This is a joint work with J.-C. Felipe-Navarro (UPC) and X. Ros-Oton (UB).

Azahara de la Torre (Universidad de Granada).

Título: From Conformal Geometry to the study of elliptic PDEs.

Abstract: The so called Yamabe problem in Conformal Geometry is finding a metric

conformal to a given one and which has constant scalar curvature. From

the analytic point of view, this problem becomes a semilinear elliptic

PDE with critical (for the Sobolev embedding) power non-linearity. If we

study the problem in the Euclidean space, allowing the presence of

nonzero-dimensional singularities can be transformed into reducing the

non-linearity to a Sobolev-subcritical power. A quite recent notion of

non-local curvature gives rise to a parallel study which weakens the

geometric assumptions giving rise to a non-local semilinear elliptic

(Sobolev-critical) PDE.

In this talk, we will focus on the Euclidean space in the presence of

singularities of maximal possible dimension. In both cases, local and

non-local, we will construct singular solutions of a local/non-local

semilinear elliptic equation with superlinear nonlinearity which is

subritical for Sobolev embeddings, but it is critical for the existence

of singular solutions (below such threshold the singularity is removable).

This is a joint work with H. Chan.

Zoom link: https://uniroma1.zoom.us/j/87961090612?pwd=clZHOUhNbFAvVDE2eWRuM3MxZE02dz09

Webpage: https://sites.google.com/view/pdesespanaitalia.