A higher integrability theorem from a reverse weighted inequality

2019 ◽  
Vol 51 (6) ◽  
pp. 967-977 ◽  
Author(s):  
Samir Saker ◽  
Donal O'Regan ◽  
Ravi Agarwal
Keyword(s):  
Weighted Inequality ◽  
Integrability Theorem ◽  
Higher Integrability

Autonomous Geometrical Mechanics

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Phase Space ◽  
Contact Structure ◽  
Invariant Tori ◽  
Time Dependent ◽  
Contact Structures ◽  
Natural Setting ◽  
Adiabatic Invariance ◽  
Jet Bundle ◽  
Integrability Theorem ◽  
Extended Phase

This chapter examines the structure of the phase space of an integrable system as being constructed from invariant tori using the Arnold–Liouville integrability theorem, and periodic flow and ergodic flow are investigated using action-angle theory. Time-dependent mechanics is formulated by extending the symplectic structure to a contact structure in an extended phase space before it is shown that mechanics has a natural setting on a jet bundle. The chapter then describes phase space of integrable systems and how tori behave when time-dependent dynamics occurs. Adiabatic invariance is discussed, as well as slow and fast Hamiltonian systems, the Hannay angle and counter adiabatic terms. In addition, the chapter discusses foliation, resonant tori, non-resonant tori, contact structures, Pfaffian forms, jet manifolds and Stokes’s theorem.

scholarly journals On a weighted inequality of Hardy type in spaces Lp(⋅)

2009 ◽  
Vol 353 (2) ◽  
pp. 521-530 ◽  
Author(s):  
Farman I. Mamedov ◽  
Aziz Harman
Keyword(s):  
Weighted Inequality ◽  
Hardy Type

scholarly journals Higher Integrability of Weak Solutions to a Class of Double Obstacle Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Zhenhua Hu ◽  
Shuqing Zhou
Keyword(s):  
Differential Equation ◽  
Weak Solutions ◽  
Second Order ◽  
Obstacle Problems ◽  
Elliptic Differential Equation ◽  
Higher Integrability ◽  
Global Higher Integrability ◽  
Quasilinear Elliptic

We first introduce double obstacle systems associated with the second-order quasilinear elliptic differential equationdiv(A(x,∇u))=div f(x,u), whereA(x,∇u),f(x,u)are twon×Nmatrices satisfying certain conditions presented in the context, then investigate the local and global higher integrability of weak solutions to the double obstacle systems, and finally generalize the results of the double obstacle problems to the double obstacle systems.

scholarly journals On some reverse integral inequalities

1990 ◽  
Vol 49 (2) ◽  
pp. 319-326 ◽  
Author(s):  
Luciana Nania
Keyword(s):  
Integral Inequalities ◽  
Reverse Inequality ◽  
Higher Integrability ◽  
Decreasing Functions

AbstractWe prove the higher integrability of nonnegative decreasing functions, verifying a reverse inequality, and we calculate the optimal integrability exponent for these functions.

scholarly journals Higher Integrability for Minimizers of the Mumford–Shah Functional

2014 ◽  
Vol 213 (2) ◽  
pp. 491-502 ◽  
Author(s):  
Guido De Philippis ◽  
Alessio Figalli
Keyword(s):  
Higher Integrability
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