Page 520 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 520
NONLOCAL OPERATORS AND RELATED TOPICS (MS-55)
positive dimension of the junction set and some role played by the geometry of the domain.
Such difficulties are overcome by a double approximation procedure: by approximating the po-
tential with functions vanishing near the boundary and the Dirichlet N -dimensional region with
smooth (N + 1)-dimensional sets with straight vertical boundary, it is possible to construct a
sequence of approximating solutions which enjoy enough regularity to derive Pohozaev type
identities, needed to obtain Almgren type monotonicity formulas and consequently to perform
blow-up analysis.
Three balls inequalities for discrete Schrödinger operators
Aingeru Fernández Bertolin, aingeru.fernandez@ehu.eus
University of the Basque Country, Spain
Coauthors: Luz Roncal, Angkana Rüland, Diana Stan
In this talk we will study the so-called three balls inequalities in the discrete setting, which in
the continuous setting would be stated as inequalities of the form
u L2(B1) ≤ C u α u ,1−α
L2 (B1/2 )
L2(B2)
for some α ∈ (0, 1) and functions u such that P u = 0 for a given operator P . Such an inequality
is used then to study propagation of smallness and unique continuation.
In the discrete setting it is known that these properties cannot be true in general, even in the
simplest case of discrete harmonic functions. However, one can prove a similar inequality in
the lattice with a correction term that tends to zero as the step size of the lattice tends to zero.
In this talk we will see how we can extend this discrete inequality for the discrete Laplacian
to a large variety of discrete magnetic Schrödinger operators, by means of a discrete Carleman
estimate for the discrete Laplace operator.
Non-local ODEs in conformal geometry
María Del Mar González, mariamar.gonzalezn@uam.es
Universidad Autónoma de Madrid, Spain
When one looks for radial solutions of an equation with fractional Laplacian, it is not generally
possible to use usual ODE methods. If such equation has some conformal invariances, then
one may rewrite it in Emden-Fowler (cylindrical) coordinates and to use the properties of the
conformal fractional Laplacian on the cylinder. After giving the necessary background, we
will briefly consider two particular applications of this technique: 1. Symmetry breaking, non-
degeneracy and uniqueness for the fractional Caffarelli-Kohn-Nirenberg inequality (joint work
with W. Ao and A. DelaTorre). 2. Existence and regularity for fractional Laplacian equations
with drift and a critical Hardy potential (joint with H. Chan, M. Fontelos and J. Wei).
518
positive dimension of the junction set and some role played by the geometry of the domain.
Such difficulties are overcome by a double approximation procedure: by approximating the po-
tential with functions vanishing near the boundary and the Dirichlet N -dimensional region with
smooth (N + 1)-dimensional sets with straight vertical boundary, it is possible to construct a
sequence of approximating solutions which enjoy enough regularity to derive Pohozaev type
identities, needed to obtain Almgren type monotonicity formulas and consequently to perform
blow-up analysis.
Three balls inequalities for discrete Schrödinger operators
Aingeru Fernández Bertolin, aingeru.fernandez@ehu.eus
University of the Basque Country, Spain
Coauthors: Luz Roncal, Angkana Rüland, Diana Stan
In this talk we will study the so-called three balls inequalities in the discrete setting, which in
the continuous setting would be stated as inequalities of the form
u L2(B1) ≤ C u α u ,1−α
L2 (B1/2 )
L2(B2)
for some α ∈ (0, 1) and functions u such that P u = 0 for a given operator P . Such an inequality
is used then to study propagation of smallness and unique continuation.
In the discrete setting it is known that these properties cannot be true in general, even in the
simplest case of discrete harmonic functions. However, one can prove a similar inequality in
the lattice with a correction term that tends to zero as the step size of the lattice tends to zero.
In this talk we will see how we can extend this discrete inequality for the discrete Laplacian
to a large variety of discrete magnetic Schrödinger operators, by means of a discrete Carleman
estimate for the discrete Laplace operator.
Non-local ODEs in conformal geometry
María Del Mar González, mariamar.gonzalezn@uam.es
Universidad Autónoma de Madrid, Spain
When one looks for radial solutions of an equation with fractional Laplacian, it is not generally
possible to use usual ODE methods. If such equation has some conformal invariances, then
one may rewrite it in Emden-Fowler (cylindrical) coordinates and to use the properties of the
conformal fractional Laplacian on the cylinder. After giving the necessary background, we
will briefly consider two particular applications of this technique: 1. Symmetry breaking, non-
degeneracy and uniqueness for the fractional Caffarelli-Kohn-Nirenberg inequality (joint work
with W. Ao and A. DelaTorre). 2. Existence and regularity for fractional Laplacian equations
with drift and a critical Hardy potential (joint with H. Chan, M. Fontelos and J. Wei).
518
