| The purpose of the present thesis is to investigate a special class of degenerate elliptic equations with boundary value conditions arising from geometry.The study of degenerate elliptic equations can be traced back to 1910's, first appeared in Picone's thesis. After half century's development in this field, O.A.Oleinik and E.V.Radkeic, published their classic monograph Second order equations with nonnegative characteristic form in 1971. Their monograph summarized the theory developed before 1970's, and established a general framework of the theory of linear degenerate elliptic equations. They stated the existence of weak solution in If space and some Hilbert spaces for the general boundary value problem of linear degenerate elliptic equations, and made a certain further contribution to the regularity theory. During the past three decades, several progresses have been made in the research of second order degenerate elliptic equations. For example, works of L.CafTarelli, L.Nirenberg, J.Spruck, H.Brezis, P.L.Lions, E.B.Fabes, C.E.Kenig, D.Jerison,and Fanghua Lin.The degenerate elliptic problems we shall study is very closely related to rigidity problems arising from infinitesimal isometric deformation, as well as other geometry problem, such as minimal surface in hyperbolic space. In particular, the existence of solution with high order regularity is very important to investigate geometry problems. One would like to know under what conditions the solution of such equations are as smooth as the given data. The theory on well-posedness and regularity of solutions to such class of degenerate problems, plays a crucial role in the above fields. Anyway, such equations are deserved to be investigated vastly. So far such problems might not be able to be treated by some standard methods. Maybe they will stimulate a general study of linear, semilinear, quasilinear, and nonlinear degenerate elliptic equations.The present thesis is divided into four chapters.Chapetr I is the introduction. We introduce briefly the above mentioned history of degenerate elliptic equations and the geometric background of this class problem we are concerned with. In the last part of this chapter,we will summarize the main results of existence, uniqueness and regularity for solution of such kind of problems.In chapter II, we consider a class of boundary value problem for second order degenerate elliptic equations on a bounded periodic domain ft, which is homeomorphic to the cylindrical surface. The well-posedness and regularity problems of such equations will be dealed with elliptic regularization method. Firstly, we construct a family of auxiliary problems with a small parameter e and a large parameter A. The energy estimates will be derivied after some lengthy calculation. Secondly, the above priori estimates and the Banach-Saks theorem provide the existence of H1 weak solution for the auxiliary problems without small parameter e. Based on this fact we get the uniqueness of H1 weak solution by the principle that strong solution of degenerate equation is identical with its weak solution. Thirdly, we can view the degenerate PDE as an ODE with respect to degenerate direction. So an integral representation of the weak solution's 1-order derivative can be obtained to establish two fundmental lemma under two different cases. Lastly, we will carry out an iteration method, with the help of the interior regularity theorem of elliptic equation, to obtain high order regularity and the corresponding estimates for solutions of the auxiliary problems without small parameter e. Eventually, one may remove the large parameter A from the auxiliary problems by Fredholm-Riesz-Schauder theory and extremal principle. Thus the well-posedness and the regularity of the original problem will be obtained in a standard way.For the sake of convenience, we introduce some symbols. ConsiderLu = y(uxx + uyy) + aux + buy + cu, in C R2+,where ; and set a0 = a(x, 0),b0 = b(x, 0).Now, the main results in chapter I are as follows.Theorem 1.3.1. Su...
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