The Regularity Of Nonlinear Elliptic Problems In Musielak-orlicz-sobolev Spaces

In this paper,we discuss the regularity of nonlinear elliptic problems in Musielak-Orlicz-Sobolev space,under some appropriate assumptions on the N(?)-fucntions.Starting from the boundary value problems of differential equations,the Sobolev embed?ding theorems and boundary trace embedding theorems are very important for the study of the Neumann boundary value problems of differential equations.In chapter 3,we use the properties of N(?)function in Musielak-Orlicz-Sobolev space,and give some trace em-bedding properties in more general function space(Musielak-Sobolev space)on bounded domains by means of some classical embedding theorems and trace theorems in hyperplanes.The trace embedding theorems in the Musielak-Sobolev space include the trace on the inner lower dimensional planes and the trace on the boundary.Furthermore,we get the com-pact trace embedding theorems.These conclusions generalize the relevant conclusions in the Orlicz-Sobolev space and the variable exponent Sobolev space.In fact,there are some important classical regularity results for the minimizers of integral functionals within the classical Sobolev,variable exponent,Orlicz and Musielak-Orlicz-Sobolev framework settings in the literature.In Chapter 4,under the framework of Musielak-Orlicz-Sobolev space,we study the regularity of minimizer of the more general integral in Musielak-Orlicz-Sobolev space from the point of partial differential equation-s.We re-establish De Giorgi iterations with the help of recently emerging tools in the Musielak-Orlicz-Sobolev space,and prove the regularity of local minimizers of a class of energy functional E(v)=E(v,?)??_?f(x,v(x),?v(x))dx,When f satisfies some growth conditions,we prove that the local minimizers of the energy functional are Holder continuous.And we can also demonstrate the regularity of weak solutions for a class of fully nonlinear elliptic equation When L,F satisfy some increasing conditions,we demonstrate that the weak solutions of the equation are local Holder continuous.In Chapter 5,we use the properties of(?)function and the Poincare type inequalities in Musielak-Orlicz-Sobolev space to generate a series of integral inequalities.Then using the Calderon-Zygmund decomposition,we give an estimate for the gradient of the local minimizers of a class of energy functional in Musielak-Orlicz-Sobolev space.This conclusion is a generalization of the classical Calderon-Zygmund theory in linear case.