Page 479 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 479
GEOMETRIC-FUNCTIONAL INEQUALITIES AND RELATED TOPICS (MS-23)
we can extend our previous results on the compactness of corresponding embeddings. The
concept of nuclearity goes back to Grothendieck (1955) and was the basis for many fundamental
developments in functional analysis. Recently we noticed a refreshed interest to study such
questions in special situations. This led us to the investigation in the weighted setting. We
obtain complete characterisations for the nuclearity of the corresponding embedding. Essential
tools are a discretisation in terms of wavelet bases, operator ideal techniques, as well as a very
useful result of Tong (1969) about the nuclearity of diagonal operators acting in p spaces.
Attainability of the best Sobolev constant in a ball
Noriusuke Ioku, ioku@tohoku.ac.jp
Tohoku University, Japan
The best constant in the Sobolev inequality in the whole space is attained by the Aubin–Talenti
function; however, this does not happen in bounded domains because of the break down of the
dilation invariance. In this talk, we investigate a nonlinear scale invariant form of the Sobolev
inequality in a ball and show that its best constant is attained by functions of the Aubin–Talenti
type.
Caccioppoli–type estimates and Hardy–type inequalities derived from
weighted p–harmonic problems
Agnieszka Kałamajska, kalamajs@mimuw.edu.pl
University of Warsaw, Poland
Let u : Rn ⊇ Ω → R be the nontrivial and nonnegative solution to the following anticoercive
partial differential inequality of elliptic type involving weighted p–Laplacian:
−∆p,au := −div(a(x)|∇u|p−2∇u) ≥ b(x)Φ(u)χ{u>0},
where Φ : (0, ∞) → R is the given continuous function with certain properties. After ob-
taining Caccioppoli–type estimates for u, we derive from them several variants of Hardy–type
inequalities in weighted Sobolev setting, some of them holding with best constants.
The talk will be based on joint work with Iwona Chlebicka and Pavel Drabek [1]. It ex-
tends earier result from [4] to the weighted setting. Both approaches are besed on the modified
techniques due to Pohozhaev and Mitidieri [3] from [2].
References
[1] I. Chlebicka, P. Drabek, A. Kałamajska, Caccioppoli–type estimates and Hardy–type in-
equalities derived from weighted p–harmonic problems, Rev. Mat. Complut. 32 (2019)(3),
pp. 601–630.
[2] A. Kałamajska, K. Pietruska-Pałuba, and I. Skrzypczak. Nonexistence results for differen-
tial inequalities involving A-Laplacian, Adv. Differential Equations, 17(3-4) (2012), pp.
307–336.
[3] È. Mitidieri and S. I. Pokhozhaev. Absence of positive solutions for quasilinear ellip-
tic problems in RN , Tr. Mat. Inst. Steklova 227 (1999) (Issled. po Teor. Differ. Funkts.
Mnogikh Perem. i ee Prilozh. 18), pp. 192–222.
477
we can extend our previous results on the compactness of corresponding embeddings. The
concept of nuclearity goes back to Grothendieck (1955) and was the basis for many fundamental
developments in functional analysis. Recently we noticed a refreshed interest to study such
questions in special situations. This led us to the investigation in the weighted setting. We
obtain complete characterisations for the nuclearity of the corresponding embedding. Essential
tools are a discretisation in terms of wavelet bases, operator ideal techniques, as well as a very
useful result of Tong (1969) about the nuclearity of diagonal operators acting in p spaces.
Attainability of the best Sobolev constant in a ball
Noriusuke Ioku, ioku@tohoku.ac.jp
Tohoku University, Japan
The best constant in the Sobolev inequality in the whole space is attained by the Aubin–Talenti
function; however, this does not happen in bounded domains because of the break down of the
dilation invariance. In this talk, we investigate a nonlinear scale invariant form of the Sobolev
inequality in a ball and show that its best constant is attained by functions of the Aubin–Talenti
type.
Caccioppoli–type estimates and Hardy–type inequalities derived from
weighted p–harmonic problems
Agnieszka Kałamajska, kalamajs@mimuw.edu.pl
University of Warsaw, Poland
Let u : Rn ⊇ Ω → R be the nontrivial and nonnegative solution to the following anticoercive
partial differential inequality of elliptic type involving weighted p–Laplacian:
−∆p,au := −div(a(x)|∇u|p−2∇u) ≥ b(x)Φ(u)χ{u>0},
where Φ : (0, ∞) → R is the given continuous function with certain properties. After ob-
taining Caccioppoli–type estimates for u, we derive from them several variants of Hardy–type
inequalities in weighted Sobolev setting, some of them holding with best constants.
The talk will be based on joint work with Iwona Chlebicka and Pavel Drabek [1]. It ex-
tends earier result from [4] to the weighted setting. Both approaches are besed on the modified
techniques due to Pohozhaev and Mitidieri [3] from [2].
References
[1] I. Chlebicka, P. Drabek, A. Kałamajska, Caccioppoli–type estimates and Hardy–type in-
equalities derived from weighted p–harmonic problems, Rev. Mat. Complut. 32 (2019)(3),
pp. 601–630.
[2] A. Kałamajska, K. Pietruska-Pałuba, and I. Skrzypczak. Nonexistence results for differen-
tial inequalities involving A-Laplacian, Adv. Differential Equations, 17(3-4) (2012), pp.
307–336.
[3] È. Mitidieri and S. I. Pokhozhaev. Absence of positive solutions for quasilinear ellip-
tic problems in RN , Tr. Mat. Inst. Steklova 227 (1999) (Issled. po Teor. Differ. Funkts.
Mnogikh Perem. i ee Prilozh. 18), pp. 192–222.
477
