scholarly journals How to Recognize Polynomials in Higher Order Sobolev Spaces

2013 ◽  
Vol 112 (2) ◽  
pp. 161 ◽  
Author(s):  
Bogdan Bojarski ◽  
Lizaveta Ihnatsyeva ◽  
JUHA KINNUNEN JUHA KINNUNEN
Keyword(s):  
Sobolev Spaces ◽  
Remainder Term ◽  
Integral Condition ◽  
Higher Order ◽  
Order Case

This paper extends characterizations of Sobolev spaces by Bourgain, Brézis, and Mironescu to the higher order case. As a byproduct, we obtain an integral condition for the Taylor remainder term, which implies that the function is a polynomial. Similar questions are also considered in the context of Whitney jets.

Higher order Adams’ inequality with the exact growth condition

2018 ◽  
Vol 20 (06) ◽  
pp. 1750072 ◽  
Author(s):  
Nader Masmoudi ◽  
Federica Sani
Keyword(s):  
Growth Condition ◽  
Sobolev Spaces ◽  
Higher Order ◽  
Sharp Inequality ◽  
Second Order ◽  
Original Form ◽  
Order Case ◽  
Whole Space ◽  
Exact Growth Condition

Adams’ inequality is the complete generalization of the Trudinger–Moser inequality to the case of Sobolev spaces involving higher order derivatives. The failure of the original form of the sharp inequality when the problem is considered on the whole space [Formula: see text] served as a motivation to investigate in the direction of a refined sharp inequality, the so-called Adams’ inequality with the exact growth condition. Due to the difficulties arising in the higher order case from the lack of direct symmetrization techniques, this refined result is known to hold on first- and second-order Sobolev spaces only. We extend the validity of Adams’ inequality with the exact growth to higher order Sobolev spaces.

Weighted higher order exponential type inequalities in metric spaces and applications

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huiju Wang ◽  
Pengcheng Niu
Keyword(s):  
Homogeneous Space ◽  
Lie Groups ◽  
Sobolev Spaces ◽  
Exponential Type ◽  
Metric Spaces ◽  
Type Inequality ◽  
Higher Order ◽  
Geodesic Space ◽  
Stratified Lie Groups ◽  
Weighted Case

AbstractIn this paper, we establish weighted higher order exponential type inequalities in the geodesic space {({X,d,\mu})} by proposing an abstract higher order Poincaré inequality. These are also new in the non-weighted case. As applications, we obtain a weighted Trudinger’s theorem in the geodesic setting and weighted higher order exponential type estimates for functions in Folland–Stein type Sobolev spaces defined on stratified Lie groups. A higher order exponential type inequality in a connected homogeneous space is also given.

scholarly journals Approximation in higher-order Sobolev spaces and Hodge systems

2019 ◽  
Vol 276 (5) ◽  
pp. 1430-1478 ◽  
Author(s):  
Pierre Bousquet ◽  
Emmanuel Russ ◽  
Yi Wang ◽  
Po-Lam Yung
Keyword(s):  
Sobolev Spaces ◽  
Higher Order

scholarly journals A duality-type method for the obstacle problem

2013 ◽  
Vol 21 (3) ◽  
pp. 181-196 ◽  
Author(s):  
Diana Rodica Merlusca
Keyword(s):  
Sobolev Spaces ◽  
Obstacle Problem ◽  
Optimization Problem ◽  
Quadratic Optimization ◽  
Higher Order ◽  
Type Method ◽  
Quadratic Optimization Problem ◽  
Approximate Problem ◽  
Duality Property ◽  
New Type

Abstract Based on a duality property, we solve the obstacle problem on Sobolev spaces of higher order. We have considered a new type of approximate problem and with the help of the duality we reduce it to a quadratic optimization problem, which can be solved much easier.

scholarly journals The multivariate Faà di Bruno formula and multivariate Taylor expansions with explicit integral remainder term

2007 ◽  
Vol 48 (3) ◽  
pp. 327-341 ◽  
Author(s):  
Roy B. Leipnik ◽  
Charles E. M. Pearce
Keyword(s):  
Remainder Term ◽  
Natural Generalization ◽  
Higher Order ◽  
Composite Function ◽  
Multivariate Case ◽  
Taylor Expansions ◽  
Univariate Case ◽  
Faà Di Bruno Formula ◽  
Derivatives Of

AbstractThe Faà di Bruno formulæ for higher-order derivatives of a composite function are important in analysis for a variety of applications. There is a substantial literature on the univariate case, but despite significant applications the multivariate case has until recently received limited study. We present a succinct result which is a natural generalization of the univariate version. The derivation makes use of an explicit integralform of the remainder term for multivariate Taylor expansions.

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