Partial Regularity For Nonlinear Elliptic Systems And Calculus Of Variations With Orlicz-Growth

In this thesis,we mainly studies the partial regularity for weak solutions of elliptic systems and the minimal problem of vector-valued nonlinear functionals these two prob-lems.The first one is to study the partial Holder continuity of the weak solutions of the nonlinear elliptic equations under the condition of Orlicz growth.The second one is to show the partial regularity of the vector function minimizers of functional with Orlicz growth under the VMO coefficient and Sobolev space Wloc 1,s(?)coefficient,respectively.The specific content is as follows.In Chapter 1,we summarize the background of the paper and the latest developments in the relevant literature,introduce the concepts of the related functions,spaces and so on.We also recall the related lemmas,and then summarizes the main results of this paper.In Chapter 2,we consider the following nonlinear elliptic equation systems of diver-gence under the condition of ?-growth:-div(A(x,u,Du))=b(x,u,Du),x ? ?,where A:?× RN × RNn?RNn,b:? × RN × RNn? RN(n?2,N?1)satisfying?-growth condition,? is an Orlicz function satisfying both the A2 and V2 conditions,andt?'(t)??(t)andt??(t)??'(t).Then when A satisfys modulus of continuity condition,we can obtain the partial Holder continuity for the weak solutions u ? W1,?(?,RN)of the elliptic systems.In addition,under the function A(x,u,Du)about x,u is Holder continuous,it can be further proved that Du is still partially Holder continuous.In Chapter 3,we consider the variation problem as follows:Let ? be the bounded domain in Rn,n? 2,N? 1,if u:? Rn? RN is the vector-valued minimizers of integral functionals(?) where ? is an Orlicz function satisfying both the ?2 and ?2 conditions,andt?'(t)??(t)andt??(t)??'(t).If a(x)? VMO(Vanishing Mean Oscillations,x?a(x)can be discontinuous),then the variational minimal weak solution u is Holder continuous with arbitrary Holder exponent??(0,1).Further,if the coefficient ? ?(?)? L?(?),where the integrable exponent s>1 of a(x)is arbitrary,then Du is partially Holder continuous.