具有非标准增长条件的微分形式的Caccioppoli不等式及其高阶可积性

CACCIOPPOLI INEQUALITY AND HIGHER INTEGRABILITY FOR DIFFERENTIAL FORMS WITH NON-STANDARD GROWTH CONDITIONS

本文主要研究了微分形式中的相关不等式.利用A-调和方程的性质及与该方程相关的弱逆Holder不等式和一类满足非标准增长条件的Young函数的性质,获得了一类特殊的微分形式(即非齐次A-调和张量)在该类Young函数作用下的Caccoppoli不等式及其高阶可积性.该结论将微分形式中Caccoppoli不等式由Lp空间推广到了由该类Young函数构成的Orlicz空间,同时验证了该Caccoppoli不等式可以用于微分形式的定量估计和定性分析.

In this paper,we mainly study the related inequalities for differential forms.By using the properties of A-harmonic equation,the weak inverse Holder inequality associated with the equation and the properties of a class of Young functions satisfying non-standard growth conditions,we obtain the Caccopoli in-equality and its high-order integrability for a special differential form(i.e.,the non-homogeneous A-harmonic tensor)under the action of this kind of Young functions.This conclusion extends the Caccopoli inequality for differential form from Lp space to Orlicz space composed of young functions of this kind,and verifies that the Caccopoli inequality can be used for quantitative estimation and qualitative analysis of differential forms.