Regularity For The Solutions To Double Obstacle Problems

This thesis is composed of four chapters.The first chapter is an introduction and mainly discusses the background of our work and narrates our main work.The second chapter is preliminaries and gives some basic knowledge and preliminary lemmas we needed in the thesis.In the third chapter, the regularity of weak solutions to double obstacle problems to quasilinear elliptic partial differential equations div(A(x,â–½u))=divf(x) (1) is discussed. Under certain conditions on the equations, the local and global higher integrability of the weak solutions to (1) is obtained by constructing special test functions and using the Holder's inequality, the Poincare's inequality, the Young's inequality and reverse Holder inequality. These results generalize the results in corresponding literatures.In the fourth chapter, the regularity of very weak solutions to double obstacle problems to non-homogeneous A-harmonic equations-Div(A(x,Du(x)))=f(x,u(x)) (2) Under certain conditions on the equations, the local higher integrability of the very weak solutions to (2) is obtained by constructing suitable test functions via Hodge decomposition and using the Holder's inequality, the Poincare's inequality, the Young's inequality, reverse Holder inequality and some basic inequalities. The qualities of the very weak solutions to double obstacle problems has not been studied so far.