Weak reverse Hölder inequality of weakly A-harmonic sensors and Hölder continuity of A-harmonic sensors

Wang Tingting, Bao, Bao Gejun

Description

In this paper, we obtain the weak reverse Hölder inequality of weakly A-harmonic sensors and establish the Hölder continuity of A-harmonic sensors. Mathematics Subject Classification 2010: 58A10 · 35J60 In this paper, we obtain the weak reverse Hölder inequality of weakly A-harmonic sensors and establish the Hölder continuity of A-harmonic sensors. Mathematics Subject Classification 2010: 58A10 · 35J60

Sujets

• Hölder condition

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 6
• Langue: English

Extrait

Wang and BaoJournal of Inequalities and Applications2011,2011:99 http://www.journalofinequalitiesandapplications.com/content/2011/1/99

R E S E A R C H

Weak reverse Hölder inequality Aharmonic sensors and Hölder Aharmonic sensors * Tingting Wang and Gejun Bao

* Correspondence: [e-mail]. cn Department of Mathematics, Harbin Institute of Technology, Harbin 150001, PR China

Open Access

of weakly continuity

of

Abstract In this paper, we obtain the weak reverse Hölder inequality of weakly Aharmonic sensors and establish the Hölder continuity of Aharmonic sensors. Mathematics Subject Classification 2010:58A10 ∙ 35J60 Keywords:Weak reverse Hölder inequality, Weakly Aharmonic sensors, Hölder continuity

1 Introduction In this paper, we consider the Aharmonic equation ∗ d A x,du= 0

(1:1)

l n l n where mappingA:Ω×Λ(ℝ)®Λ(ℝ) satisfies the following assumptions for fixed 0 <a≤b<∞: (1)Asatisfies the Carathéodory measurability condition; l n (2) for a.e.xÎΩand allξÎΛ(ℝ)

p A x,ξ,ξ ≥α|ξ|,

p− |A x,ξ| ≤β|ξ|

l n (3) for a.e.xÎΩand allξÎΛ(ℝ),lÎℝ

A x,λξ

p−2 =λ|λ|A x,ξ

(1:2)

Here, 1 <p<∞is a fixed exponent associated with (1.1). Remark:The notions and basic theory of exterior calculus used in this paper can be found in [1] and [2], we do not mention them here. Definition 1.1[2]A solution u to(1.1), called Aharmonic tensor, is an element of the 1,p l−1 Sobolev space W(, such that  A(x,du),dφdx=

1,p l 1 for alljÎW(Ω,Λ)with compact support. In particular, we impose the growth condition

A x,ξ

∙ξ≈ |ξ|

© 2011 Wang and Bao; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.