度量空间中某类泛函极小的局部有界性

The Local Boundedness of an Certain Functional Minimizer on Metric Spaces

主要研究了牛顿空间中泛函F(u,u)=∫f(u,gu)dμ的极小问题,其中对某常数c>0,泛函满足条件gup-c|u|p≤f(u,gu)≤gpu+c|u|p.牛顿空间是Soolev空间在度量空间中的推广,具有更一般的性质.在该空间中研究偏微分方程,对建立更一般的偏微分方程理论具有指导和开拓意义.本文利用De Giorgi迭代的方法验证了该泛函极小的局部有界性,而这一性质的成立为我们今后进一步研究该泛函极小的正则性奠定了基础.

In our paper,the boundedness of an certain functional minimizer is considered on the so called Newton space,which is a generalization of Sobolev space in a metric measure space with some extra conditions.This kind of problem is important and instutive for us to develop theory of partial differential equations on more general metric spaces than Sobolev spaces.The functional is of the type F（u,gu）=∫f（u,gu）dμ on Newton spaces,with gpu-c｜u｜p≤f（u,gu）≤gpu＋c｜u｜p,for some c0.In the main theorem the minimizer is proved to be locally bounded with the aids of the De Giorgi iteration method.