scholarly journals Interpolation inequalities in \begin{document}$ \mathrm W^{1,p}( {\mathbb S}^1) $\end{document} and carré du champ methods

2020 ◽  
Vol 40 (1) ◽  
pp. 375-394
Author(s):  
Jean Dolbeault ◽  
◽  
Marta García-Huidobro ◽  
Rául Manásevich ◽  
◽  
...  
Keyword(s):  
Interpolation Inequalities ◽  
Carré Du Champ

scholarly journals Improved Interpolation Inequalities and Stability

2020 ◽  
Vol 20 (2) ◽  
pp. 277-291
Author(s):  
Jean Dolbeault ◽  
Maria J. Esteban
Keyword(s):  
Euclidean Space ◽  
Symmetry Breaking ◽  
Stereographic Projection ◽  
Optimal Interpolation ◽  
Stability Estimates ◽  
Interpolation Inequalities ◽  
Carré Du Champ ◽  
Optimal Constants ◽  
Optimal Functions

AbstractFor exponents in the subcritical range, we revisit some optimal interpolation inequalities on the sphere with carré du champ methods and use the remainder terms to produce improved inequalities. The method provides us with lower estimates of the optimal constants in the symmetry breaking range and stability estimates for the optimal functions. Some of these results can be reformulated in the Euclidean space using the stereographic projection.

scholarly journals Solutions of Navier-Stokes Equations with Coriolis Force in the Rotational Framework

2021 ◽  
Vol 31 (5) ◽  
pp. 54-57
Keyword(s):  
Coriolis Force ◽  
Existence And Uniqueness ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Mixed Norm ◽  
Uniqueness Of Solutions ◽  
Navier Stokes Equations ◽  
Interpolation Inequalities ◽  
Stokes Problems ◽  
Point Argument

In this article, for 0 ≤m<∞ and the index vectors q=(q_1,q_2 ,q_3 ),r=(r_1,r_2,r_3) where 1≤q_i≤∞,1<r_i<∞ and 1≤i≤3, we study new results of Navier-Stokes equations with Coriolis force in the rotational framework in mixed-norm Sobolev-Lorentz spaces H ̇^(m,r,q) (R^3), which are more general than the classical Sobolev spaces. We prove the existence and uniqueness of solutions to the Navier-Stokes equations (NSE) under Coriolis force in the spaces L^∞([0, T]; H ̇^(m,r,q) ) by using topological arguments, the fixed point argument and interpolation inequalities. We have achieved new results compared to previous research in the Navier-Stokes problems.

scholarly journals Inequalities involving Aharonov–Bohm magnetic potentials in dimensions 2 and 3

2020 ◽  
pp. 2150006
Author(s):  
Denis Bonheure ◽  
Jean Dolbeault ◽  
Maria J. Esteban ◽  
Ari Laptev ◽  
Michael Loss
Keyword(s):  
Magnetic Field ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Euclidean Spaces ◽  
Hardy Inequalities ◽  
Interpolation Inequalities ◽  
Nonlinear Interpolation ◽  
Magnetic Potentials ◽  
Aharonov Bohm

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schrödinger operators involving Aharonov–Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions 2 and 3. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in the presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions [Formula: see text] and [Formula: see text].

Extension Theorems on Weighted Sobolev Spaces and Some Applications

2006 ◽  
Vol 58 (3) ◽  
pp. 492-528 ◽  
Author(s):  
Seng-Kee Chua
Keyword(s):  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Poincaré Inequality ◽  
Poincare Inequality ◽  
Extension Theorems ◽  
Doubling Weight ◽  
Interpolation Inequalities

AbstractWe extend the extension theorems to weighted Sobolev spaces on (ε, δ) domains with doubling weight w that satisfies a Poincaré inequality and such that w–1/p is locally Lp′. We also make use of the main theorem to improve weighted Sobolev interpolation inequalities.

Sobolev Interpolation Inequalities with Weights

1991 ◽  
Vol 323 (1) ◽  
pp. 263 ◽  
Author(s):  
Cristian E. Gutierrez ◽  
Richard L. Wheeden
Keyword(s):  
Interpolation Inequalities
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