Page 491 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 491
HARMONIC ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS (MS-28)
Fractional Degenerate Poincaré-Sobolev inequalities
Carlos Perez Moreno, cperez@bcamath.org
University of Basque Country and BCAM, Spain
In this lecture we will discuss some recent results concerning fractional Poincaré and Poincaré-
Sobolev inequalities with weights. These results improve some celebrated results by Bourgain-
Brezis-Minorescu, Maz’ya-Shaponiskova unified by M. Milman. Our approach is based on
methods from Harmonic Analysis. We will first consider the usual context of cubes but also
we will discuss some new results in the multiparameter setting improving some results by Shi-
Torchinsky and Lu-Wheeden from the 90’s.
Regularity of solutions of complex coefficient elliptic systems: the
p-ellipticity condition
Jill Pipher, Jill_Pipher@Brown.edu
Brown University, United States
Solving boundary value problems for divergence form real elliptic equations has been an active
and productive area of research ever since the foundational work of De Giorgi - Nash - Moser es-
tablished Hölder continuity of solutions when the operator coefficients are merely bounded and
measurable. The solutions to such real-valued equations share some important properties with
harmonic functions: maximum principles, Harnack principles, and estimates up to the boundary
that enable one to solve Dirichlet problems in the classical sense of nontangential convergence.
Weak solutions of complex elliptic equations and elliptic systems do not necessarily share these
good properties of continuity or maximum principles.
In joint work with M. Dindoš, we introduced in 2017 a structural condition (p-ellipticity) on
divergence form complex elliptic equations that was inspired by a condition related to Lp con-
tractivity due to Cialdea and Maz’ya. The p-ellipticity condition was simultaneously discovered
by Carbonaro and Dragicˇevic´ to prove a bilinear embedding result. Subsequently, the condition
has proven useful in the study of well-posedness of a degenerate elliptic operator associated
with domains with lower-dimensional boundary.
In this talk we discuss p-ellipticity for complex divergence form equations, and then describe
recent work, joint with J. Li and M. Dindoš, extending this condition to elliptic systems. In
particular, we discuss applications to the Dirichlet problem for the Lamé systems.
Lp estimates for wave equations with specific Lipschitz coefficients
Pierre Portal, pierre.portal@anu.edu.au
Australian National University, Australia
Coauthor: Dorothee Frey
For the standard linear wave equation ∂t2u = ∆u, the solution at time t belongs to Lp(Rd) for
1 1
p − 2 |,p
initial data u(0, .) ∈ W ,(d−1)| ∂tu(0, .) = 0. This is a classical result of Peral/Myachi
from the 1980’s, which motivated the development of Fourier Integral Operator theory. It is
optimal in terms of the order of the Sobolev space of initial data. In this talk, we discuss an
489
Fractional Degenerate Poincaré-Sobolev inequalities
Carlos Perez Moreno, cperez@bcamath.org
University of Basque Country and BCAM, Spain
In this lecture we will discuss some recent results concerning fractional Poincaré and Poincaré-
Sobolev inequalities with weights. These results improve some celebrated results by Bourgain-
Brezis-Minorescu, Maz’ya-Shaponiskova unified by M. Milman. Our approach is based on
methods from Harmonic Analysis. We will first consider the usual context of cubes but also
we will discuss some new results in the multiparameter setting improving some results by Shi-
Torchinsky and Lu-Wheeden from the 90’s.
Regularity of solutions of complex coefficient elliptic systems: the
p-ellipticity condition
Jill Pipher, Jill_Pipher@Brown.edu
Brown University, United States
Solving boundary value problems for divergence form real elliptic equations has been an active
and productive area of research ever since the foundational work of De Giorgi - Nash - Moser es-
tablished Hölder continuity of solutions when the operator coefficients are merely bounded and
measurable. The solutions to such real-valued equations share some important properties with
harmonic functions: maximum principles, Harnack principles, and estimates up to the boundary
that enable one to solve Dirichlet problems in the classical sense of nontangential convergence.
Weak solutions of complex elliptic equations and elliptic systems do not necessarily share these
good properties of continuity or maximum principles.
In joint work with M. Dindoš, we introduced in 2017 a structural condition (p-ellipticity) on
divergence form complex elliptic equations that was inspired by a condition related to Lp con-
tractivity due to Cialdea and Maz’ya. The p-ellipticity condition was simultaneously discovered
by Carbonaro and Dragicˇevic´ to prove a bilinear embedding result. Subsequently, the condition
has proven useful in the study of well-posedness of a degenerate elliptic operator associated
with domains with lower-dimensional boundary.
In this talk we discuss p-ellipticity for complex divergence form equations, and then describe
recent work, joint with J. Li and M. Dindoš, extending this condition to elliptic systems. In
particular, we discuss applications to the Dirichlet problem for the Lamé systems.
Lp estimates for wave equations with specific Lipschitz coefficients
Pierre Portal, pierre.portal@anu.edu.au
Australian National University, Australia
Coauthor: Dorothee Frey
For the standard linear wave equation ∂t2u = ∆u, the solution at time t belongs to Lp(Rd) for
1 1
p − 2 |,p
initial data u(0, .) ∈ W ,(d−1)| ∂tu(0, .) = 0. This is a classical result of Peral/Myachi
from the 1980’s, which motivated the development of Fourier Integral Operator theory. It is
optimal in terms of the order of the Sobolev space of initial data. In this talk, we discuss an
489
