Page 598 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 598
OURNEY FROM PURE TO APPLIED MATHEMATICS (MS-53)
Linear non-degeneracy and uniqueness of the bubble solution for the
critical fractional Hénon equation in RN
Begoña Barrios, bbarrios@ull.edu.es
Universidad de La Laguna, Spain
In this talk we show a linear non-degeneracy result of positive radially symmetric solutions of
(−∆)su = |x| uα N +2s+2α in RN ,
N −2s
where (−∆)s is the fractional Laplacian operator , 0 < s < 1, α > −2s and N > 2s.
Moreover, as a consequence, a uniqueness result of those solutions with Morse index equal to
one is obtained. In particular, we get that the ground state solution is unique. Our approach
follows some ideas developed in the deep, and celebrated, papers done by R. Frank and E.
Lenzmann (Acta Math. 2013) and R. Frank, E. Lenzmann, L. Silvestre (Comm. Pure Appl.
Math. 2016) but, of course, our proofs are not based on ODE arguments as occurs in the local
case. Our non-degeneracy result extends, in the radial setting, some known theorems proved by
J. Dávila, M. del Pino and Y. Sire (Proc. Amer. Math. Soc. 2013) and by F. Gladiali, M. Grossi
and S.L.N. Neves (Adv. Math. 2013). However, due to the nature of the fractional operator and
the weight in nonlinearity, we also argue in a different way than these authors do.
The results presented in this talk have been obtained in collaboration with S. Alarcón and
A. Quaas.
On the p-adic geometry of Shimura varieties
Ana Caraiani, a.caraiani@imperial.ac.uk
Imperial College London, United Kingdom
The Langlands program is a vast network of conjectures that connect many areas of pure mathe-
matics, such as number theory, representation theory, and harmonic analysis. Shimura varieties
are certain algebraic-geometric objects that play an important role in the Langlands program.
They have a nice moduli interpretation and they provide, in many cases, a geometric realisation
of the global Langlands correspondence. I will illustrate the beautiful geometry of Shimura
varieties using the simplest example, that of the modular curve. I will then mention some recent
results that use p-adic geometry, specifically the theory of perfectoid spaces.
Very weak solutions to PDEs in inhomogeneous and anisotropic spaces
Iwona Chlebicka, i.chlebicka@mimuw.edu.pl
University of Warsaw, Poland
I will discuss well-posedness and regularity to nonlinear PDEs of simple divergent form
−divA(x, ∇u) = f or ∂tu − divA(t, x, ∇u) = f,
where the datum f is merely integrable or even is a measure, whereas the growth of A is
governed by an inhomogeneous and fully anisotropic N -function M (x, ∇u). Inhomogeneity
596
Linear non-degeneracy and uniqueness of the bubble solution for the
critical fractional Hénon equation in RN
Begoña Barrios, bbarrios@ull.edu.es
Universidad de La Laguna, Spain
In this talk we show a linear non-degeneracy result of positive radially symmetric solutions of
(−∆)su = |x| uα N +2s+2α in RN ,
N −2s
where (−∆)s is the fractional Laplacian operator , 0 < s < 1, α > −2s and N > 2s.
Moreover, as a consequence, a uniqueness result of those solutions with Morse index equal to
one is obtained. In particular, we get that the ground state solution is unique. Our approach
follows some ideas developed in the deep, and celebrated, papers done by R. Frank and E.
Lenzmann (Acta Math. 2013) and R. Frank, E. Lenzmann, L. Silvestre (Comm. Pure Appl.
Math. 2016) but, of course, our proofs are not based on ODE arguments as occurs in the local
case. Our non-degeneracy result extends, in the radial setting, some known theorems proved by
J. Dávila, M. del Pino and Y. Sire (Proc. Amer. Math. Soc. 2013) and by F. Gladiali, M. Grossi
and S.L.N. Neves (Adv. Math. 2013). However, due to the nature of the fractional operator and
the weight in nonlinearity, we also argue in a different way than these authors do.
The results presented in this talk have been obtained in collaboration with S. Alarcón and
A. Quaas.
On the p-adic geometry of Shimura varieties
Ana Caraiani, a.caraiani@imperial.ac.uk
Imperial College London, United Kingdom
The Langlands program is a vast network of conjectures that connect many areas of pure mathe-
matics, such as number theory, representation theory, and harmonic analysis. Shimura varieties
are certain algebraic-geometric objects that play an important role in the Langlands program.
They have a nice moduli interpretation and they provide, in many cases, a geometric realisation
of the global Langlands correspondence. I will illustrate the beautiful geometry of Shimura
varieties using the simplest example, that of the modular curve. I will then mention some recent
results that use p-adic geometry, specifically the theory of perfectoid spaces.
Very weak solutions to PDEs in inhomogeneous and anisotropic spaces
Iwona Chlebicka, i.chlebicka@mimuw.edu.pl
University of Warsaw, Poland
I will discuss well-posedness and regularity to nonlinear PDEs of simple divergent form
−divA(x, ∇u) = f or ∂tu − divA(t, x, ∇u) = f,
where the datum f is merely integrable or even is a measure, whereas the growth of A is
governed by an inhomogeneous and fully anisotropic N -function M (x, ∇u). Inhomogeneity
596
