Some Problems Of Viscosity Solutions For Second Order Parabolic Equations

This paper is devoted to study of existence and comparison theorems of viscosity solutions of Dirichlet problem of second order parabolic equations.In Chapter One, the author briefly reviews history of viscosity solutions and then sets forth the problems discussed in this paper .In Chapter Two, the following Dirichlet problem is considered Using Perron's method, the theorem is proved.Theorem: Suppose that there is a subsolution (_|u) and a supersolution (u|-) of (Ⅰ) that satisfy Then w( t , x ) = sup {u (t , x ) :(_|u)≤u≤(u|-) and u is a subsolution of (Ⅰ) } is a solution of (Ⅰ).Remark: For definitions of (_|u) and (u|-) ,see Theorem 2.3.1 of P14.Moreover , examples of existence of viscosity solutions are presented.In Chapter Three, the author considers the following Dirichlet problem:The author proves the comparison theorem for (Ⅱ).Theorem: Assume that H satisfies some proper conditions(See Theorem 3.2.1 of P21). If u is a bounded usc subsolution of (Ⅱ) and if v is a bounded lsc supersolution of (Ⅱ), then u≤v (t , x )∈Q_T Moreover if (u|-) and (v|-) on (Q_T|—) are defined by setting then (u|-) and (v|-) are still an usc subsolution and a lsc supersolution of (Ⅱ) respectively , and (u|-)≤(v|-) (t , x )∈(Q_T|—)Moreover, using the comparison theorem, the existence of viscosity solutions of (Ⅱ) is obtained.