Regularity For Minimizers Of Functionals And Solutions To Elliptic Systems

There are five chapters in this paper.Chapters 1 and 5 are Introduction and Conclusions,respectively.The second chapter deals with anisotropic integral functionals and nonlinear elliptic systems defined on vector valued mapping u =(u1,…,uN):?(?)Rn?RN.We present monotonic-ity inequalities on the density f:?×RN×n?R and the matrix a=(ai?):?×RN×n?RN×n,which guarantee global bounds of u.In Chapter 3,we deal with regularity for minimizers to anisotropic integral functionals and solutions to some anisotropic problems with nonstandard growth.In Section 3.1,we deal with the problem with f(x,z)satisfy some coercivity condition.We consider a minimizer u among all functions that agree on the boundary(?)? with some fixed boundary value u*.And we assume that the function ? =max{u*,(?)} makes the density f(x,Du)more integrable under the obstacle prob-lem and we prove that the minimizer u enjoy higher integrability.In Section 3.2,we deal with the problem and the Caratheodory functions ai:?×Rn?R satisfy some coercivity condition.We assume that the function ? =max{u*,(?)} makes ai(x,D?)more integrable than L'i(?),i = 1,…,n require,and then we prove that the solution u enjoys higher integrability.In Chapter 4,we consider regularity properties for weak solutions u of the nonhomoge-neous quasilinear elliptic systemThe off-diagonal coefficients are small when(?)u(?)is large,the faster off-diagonal coefficients aij??(x,y)and fi? decay,the higher integrability of u becomes.