In this paper, regularities of a certain functional minimizer are studied in the Newton space. Newton space is a generalization of Sobolev space in a metric measure space with some extra conditions. The functional isrnof the type F( u , gu ) = ∫(u , gu) d/u on Newton spaces, with gpu - c | u |p≦ f(u , gu) ≦ gpu +c | u |p, forrnsome c > 0. The solutions of the variational problem with the functional are proved to be locally Holderrncontinuous. The main approach is De Giorgi's iterations.%主要研究了牛顿空间中泛函F(u,gμ)=∫f(u,gu)dμ的极小问题,其中对某常数c＞0,泛函满足gpu-c｜u｜p≤f(u,gu)≤gpu+c｜u｜p.牛顿空间是Sobolev空间在度是空间中的推广,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迭代的方法研究了该泛函极小的正则性,证明了该泛函极小的H(o)lder连续性.这使得我们在不求精确解的情况下,利用方程本身的条件可以对方程解进行数值估计.
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