Page 480 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 480
GEOMETRIC-FUNCTIONAL INEQUALITIES AND RELATED TOPICS (MS-23)
[4] I. Skrzypczak. Hardy-type inequalities derived from p-harmonic problems, Nonlinear
Anal., 93 (2013), pp. 30–50.
Compactness of Sobolev embeddings with upper Ahlfors regular measures
Zdeneˇk Mihula, mihulaz@karlin.mff.cuni.cz
Charles University, Faculty of Mathematics and Physics, Czech Republic
A lot of different Sobolev-type embeddings (e.g., classical Sobolev embeddings in the Eu-
clidean setting, boundary trace embeddings, trace embeddings on manifolds, some weighted
Sobolev embeddings), which are often treated separately, can be viewed as special instances
of Sobolev embeddings with respect to upper Ahlfors regular measures (i.e., Borel measures
whose decay on balls is bounded from above by a power of their radii). The aim of this talk is
to present in some sense sharp compactness results for such embeddings in the general setting
of rearrangement-invariant spaces.
Measure of noncompactness of Sobolev embeddings
Vít Musil, vit.musil@unifi.it
Università degli Studi di Firenze, Italy
Coauthors: Luboš Pick, Jan Lang, Miroslav Olšák
A bounded set can be covered by a single ball of some radius. Sometimes several balls of a
smaller radius can also cover the set. A compact set can be covered by finitely many balls
of arbitrary small radius. The smallest radius that allows to cover the set with finitely many
balls therefore describes sets laying in between boundedness and compactness. Such quantity
is called a measure of non-compactness.
Based on the property of images of the unit balls, linear mappings between Banach spaces
are also classified as bounded or compact and to those staying in between, we can assign the
measure of non-compactness as well.
An important instance of an operator is a Sobolev embedding. Compactness of a Sobolev
embedding can constitute a crucial step in many applications in partial differential equations,
probability theory, calculus of variations, mathematical physics and other disciplines. In the
non-compact case, more subtle techniques have to be developed and the measure of non-
compactness plays an indispensable role here.
We give a survey of some recent new results on measure of non-compactness of Sobolev
embeddings and related mappings.
478
[4] I. Skrzypczak. Hardy-type inequalities derived from p-harmonic problems, Nonlinear
Anal., 93 (2013), pp. 30–50.
Compactness of Sobolev embeddings with upper Ahlfors regular measures
Zdeneˇk Mihula, mihulaz@karlin.mff.cuni.cz
Charles University, Faculty of Mathematics and Physics, Czech Republic
A lot of different Sobolev-type embeddings (e.g., classical Sobolev embeddings in the Eu-
clidean setting, boundary trace embeddings, trace embeddings on manifolds, some weighted
Sobolev embeddings), which are often treated separately, can be viewed as special instances
of Sobolev embeddings with respect to upper Ahlfors regular measures (i.e., Borel measures
whose decay on balls is bounded from above by a power of their radii). The aim of this talk is
to present in some sense sharp compactness results for such embeddings in the general setting
of rearrangement-invariant spaces.
Measure of noncompactness of Sobolev embeddings
Vít Musil, vit.musil@unifi.it
Università degli Studi di Firenze, Italy
Coauthors: Luboš Pick, Jan Lang, Miroslav Olšák
A bounded set can be covered by a single ball of some radius. Sometimes several balls of a
smaller radius can also cover the set. A compact set can be covered by finitely many balls
of arbitrary small radius. The smallest radius that allows to cover the set with finitely many
balls therefore describes sets laying in between boundedness and compactness. Such quantity
is called a measure of non-compactness.
Based on the property of images of the unit balls, linear mappings between Banach spaces
are also classified as bounded or compact and to those staying in between, we can assign the
measure of non-compactness as well.
An important instance of an operator is a Sobolev embedding. Compactness of a Sobolev
embedding can constitute a crucial step in many applications in partial differential equations,
probability theory, calculus of variations, mathematical physics and other disciplines. In the
non-compact case, more subtle techniques have to be developed and the measure of non-
compactness plays an indispensable role here.
We give a survey of some recent new results on measure of non-compactness of Sobolev
embeddings and related mappings.
478
