Estimation Of Capacity And Schramm-Loewner Equation For Multiple Slits

Stochastic Loewner evolution or Schramm Loewner evolution(SLE),introduced by O.Schramm,is a one-parameter family of random planar growth processes described by solving the classical Loewner differential equation with a one-dimensional Brownian motion as the driving term.In this thesis,our main work is as follows.First,the capacities of hulls in a strip domain are estimated.The capacity of hull in a strip domain is given by expectation expression;some inequalities concerning capacities of hulls in a strip domain are established;the geometric explanation of capacity of hull in a strip domain is given.Secondly,the existence and uniqueness theorem for dipolar Schramm-Loewner equation of multiple slits is established.Using Arzel `a-Ascoli Theorem it is first proved that the set of driving functions associated with ? is a compact subset of the Banach space C[0,1].Next,based on the 1-slit dipolar Schramm-Loewner equation a family of Loewner equations corresponding to ? is constructed by the method of step function.Utilizing the continuous dependence of Loewner chains and the pre-compactness of driving functions it is derived that ? can be generated from a dipolar Schramm-Loewner differential equation.Moreover,it is showed that the dipolar Schramm-Loewner equation is unique via employing the estimations of capacities of hulls in the strip region.