The Holder Derivative Of The Koch Cuver And The Holder Laplace Operators On The Sierpinski Gasket

Fractals are very unsmooth sets.Defining a derivative on a fractal has always been a.challenge.Sc:holars spent.long time studying on the differential equations of fractals was troubled by this problem.R.Strichartz tried to extend the Gauss-Green integral formula on the Sierpinski gasket(SG)using of different measures still caused con-fusion".He commented:"The La.placia.n on SG is not a differen-tial operator in Usual sense." J.Kigami,whose comment was more straightforward,saying "Since fractal like the Sierpinski gasket and Koch curve do not have any structures,to define differential opera-tors like the Laplacian is not possible from the classical viewpoint of analysis.Such a difficulty is a.new challenge in mathematics."We.consider two Holder conditions for the von Koch curve and the har-monic functions of Sierpinski gasket,with indexes of log 3/log 4 andlog3/5-/log2 respectively,and introcduced the Holcder interval(cdyadic)derivatives.The log 3/log 4 Holder derivative of the Koch curve is actuated obtained.It is also proven that the derivatives of harmonicfunctions in the Laplacian of SG are log3/5 log2-Holder derivatives.The results have a certain significance on the establishment of fractal analysis.