Regularity For Weak Solutions Of Obstacle Problems To Degenerate Elliptic Equations And Systems

The degenerate elliptic obstacle problems arise in various branches of the applied sciences,such as mechanical engineering,mathematical finance,image re-construction.It is an important topic in the study of partial differential equations and their applications.This thesis is mainly concerned with the regularity for weak solutions to the degenerate elliptic obstacle problems constructed by non-commutative vector fields.These results extend and improve the corresponding results in the Euclidean setting.This work is composed of the following three partsThe first part is concerned with CX1,α regularity for weak solutions to a class of quasilinear p-Laplacian type degenerate elliptic obstacle problems with VMO coefficients.We first establish the local boundedness of horizontal gradients of weak solutions to a degenerate elliptic equation with nondiagonal constant coef-ficients by using the Lp estimates for uniformly hypoelliptic operators and the Calderon-Zygmund theory.Then the interior Holder regularity of the weak so-lution and its gradient is obtained with different assumptions on the coefficients With the similar approach,we also study the interior regularity of weak solution-s to obstacle problems for nondiagonal quasilinear degenerate elliptic systems of smooth Hormander vector fieldsIn the second part,we treat the interior Holder regularity for weak solutions to degenerate elliptic equations with drift on homogeneous group.We can not establish the Poincare inequality with drift in this case.To overcome this difficulty,we first prove a Sobolev type inequality for weak solutions and then obtain the higher Lp estimates for the gradients by establishing the representation formula of weak solutions with the help of the fundamental solution of the Hormander’s operators.The interior Holder continuity for the weak solutions is derived by using the representation formula of the gradients and estimates of singular integralsThe third part is devoted to the study of higher integrability for very weak solutions to a class of nonlinear degenerate elliptic obstacle problems structured on smooth Hormander vector fields.We first establish the local higher integra-bility by using the Hardy-Littlewood maximal functions and obstacle functions to construct suitable test functions and by using the reverse Holder inequality in Carnot-Caratheodory space.Furthermore,we also derive the global higher inte-grability for very weak solutions with a capacitary condition on Ω by giving a Sobolev type inequality with a capacity term.