Page 478 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 478
GEOMETRIC-FUNCTIONAL INEQUALITIES AND RELATED TOPICS (MS-23)
An eigenvalue problem in anisotropic Orlicz-Sobolev spaces
Angela Alberico, a.alberico@iac.cnr.it
Institute for Applied Mathematics “M. Picone” (IAC),
Italian National Research Council (CNR), Napoli, Italy
Coauthors: Giuseppina di Blasio, Filomena Feo
The existence of eigenfunctions for a class of fully anisotropic elliptic equations is established.
The relevant equations are associated with constrained minimization problems for integral func-
tionals depending on the gradient of competing functions through general anisotropic Young
functions. In particular, the latter need neither be radial, nor have a polynomial growth, and are
not even assumed to satisfy the so called ∆2−condition. In particular, our analysis requires the
development of some new aspects of the theory of anisotropic Orlicz-Sobolev spaces.
Fractional Orlicz-Sobolev spaces
Andrea Cianchi, andrea.cianchi@unifi.it
Università di Firenze, Italy
Coauthors: Angela Alberico, Lubos Pick, Lenka Slavíková
Optimal embeddings for fractional-order Orlicz–Sobolev spaces are presented. Related Hardy
type inequalities are proposed as well. Versions for fractional Orlicz-Sobolev seminorms of
the Bourgain-Brezis-Mironescu theorem on the limit as the order of smoothness tends to 1 and
of the Maz’ya-Shaposhnikova theorem on the limit as the order of smoothness tends to 0 are
established.
A liquid-solid phase transition in a simple model for swarming
Rupert Frank, rlfrank@caltech.edu
Caltech, United States, and LMU Munich, Germany
We consider a non-local shape optimization problem, which is motivated by a simple model
for swarming and other self-assembly/aggregation models, and prove the existence of different
phases. In particular, we show that in the large mass regime the ground state density profile
is the characteristic function of a round ball. An essential ingredient in our proof is a strict
rearrangement inequality with a quantitative error estimate. The talk is based on joint work
with E. Lieb.
Approximation and nuclear embeddings in weighted function spaces
Dorothee Haroske, dorothee.haroske@uni-jena.de
Friedrich Schiller University Jena, Germany
Coauthor: Leszek Skrzypczak
We study nuclear embeddings for weighted spaces of Besov and Triebel-Lizorkin type where
the weight belongs to some Muckenhoupt class and is essentially of polynomial type. Here
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An eigenvalue problem in anisotropic Orlicz-Sobolev spaces
Angela Alberico, a.alberico@iac.cnr.it
Institute for Applied Mathematics “M. Picone” (IAC),
Italian National Research Council (CNR), Napoli, Italy
Coauthors: Giuseppina di Blasio, Filomena Feo
The existence of eigenfunctions for a class of fully anisotropic elliptic equations is established.
The relevant equations are associated with constrained minimization problems for integral func-
tionals depending on the gradient of competing functions through general anisotropic Young
functions. In particular, the latter need neither be radial, nor have a polynomial growth, and are
not even assumed to satisfy the so called ∆2−condition. In particular, our analysis requires the
development of some new aspects of the theory of anisotropic Orlicz-Sobolev spaces.
Fractional Orlicz-Sobolev spaces
Andrea Cianchi, andrea.cianchi@unifi.it
Università di Firenze, Italy
Coauthors: Angela Alberico, Lubos Pick, Lenka Slavíková
Optimal embeddings for fractional-order Orlicz–Sobolev spaces are presented. Related Hardy
type inequalities are proposed as well. Versions for fractional Orlicz-Sobolev seminorms of
the Bourgain-Brezis-Mironescu theorem on the limit as the order of smoothness tends to 1 and
of the Maz’ya-Shaposhnikova theorem on the limit as the order of smoothness tends to 0 are
established.
A liquid-solid phase transition in a simple model for swarming
Rupert Frank, rlfrank@caltech.edu
Caltech, United States, and LMU Munich, Germany
We consider a non-local shape optimization problem, which is motivated by a simple model
for swarming and other self-assembly/aggregation models, and prove the existence of different
phases. In particular, we show that in the large mass regime the ground state density profile
is the characteristic function of a round ball. An essential ingredient in our proof is a strict
rearrangement inequality with a quantitative error estimate. The talk is based on joint work
with E. Lieb.
Approximation and nuclear embeddings in weighted function spaces
Dorothee Haroske, dorothee.haroske@uni-jena.de
Friedrich Schiller University Jena, Germany
Coauthor: Leszek Skrzypczak
We study nuclear embeddings for weighted spaces of Besov and Triebel-Lizorkin type where
the weight belongs to some Muckenhoupt class and is essentially of polynomial type. Here
476
