Azahara DeLaTorre Pedraza
Sapienza Università di Roma.
Date(s) : 24/05/2022 iCal
11 h 00 min - 12 h 00 min
The so called Yamabe problem in Conformal Geometry consists in 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 PDE.
In this talk, we will focus on the study of the solutions of these non-local (or fractional) semilinear elliptic PDEs which represent metrics which are singular along nonzero-dimensional singularities. In collaboration with Ao, Chan, Fontelos, González and Wei, we covered the construction of solutions which are singular along (zero and positive dimensional) smooth submanifolds. This was done through the development of new methods coming from conformal geometry and Scattering theory for the study of non-local ODEs. Due to the limitations of the techniques we used, the particular case of “maximal’’ dimension for the singularity was not covered. In a recent work, in collaboration with H. Chan, we covered this specific dimension constructing and studying singular solutions of critical dimension. This was done through the study of the Lane-Emden-Serrin equation.