For the Cauchy problem of a class of fully nonlinear degenerate parabolic equations, this paper studies the existence, uniqueness and regularity of viscosity solutions; these results apply to Hamilton-Jacobi-Bellman (HJB for short) equation, Leland equation and equations of p-Laplacian type, which find a lot of applications in fluid mechanics, stochastic control theory and optimal portfolio selection and transaction cost problems in finance.; Further studies are done on the properties of viscosity solutions of the above models: (1) Bernstein estimates (especially C 1,α estimates) and convexity of viscosity solutions of the HJB equation; (2) monotonicity in time and in Leland constant of the viscosity solutions to the Leland equation and the relationship between Leland solutions and Black-Scholes solutions; (3) the existence and Lipschitz continuity of the free boundaries of viscosity solutions for fully nonlinear equations ut + F(Du, D2u) = 0, with p-Laplacian equation as model. Our study extends the application of viscosity solution theory and aids in the qualitative analysis and numerical computation of the above models.; To construct continuous viscosity solutions, we make use of Perron Method and various estimates by virtue of viscosity solution theory; we generalize Bernstein estimates and Kruzhkov's regularization theorem in time from smooth solutions to viscosity solutions; our method applies to initial boundary value problem, though the estimates of uniformly continuous moduli near the boundary need to be obtained and suitable viscosity sub- and super-solutions need to be constructed; to study the Leland equation, we transform it into standard form by Euler transformation and linear translation, then study the property of the viscosity solutions by virtue of comparison principle; to study the properties of the free boundary of equations of p-Laplacian type, we employ comparison principle, reflection principle, moving plane method and the construction of sub and super solutions.
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机译：针对一类完全非线性退化的抛物方程的柯西问题，研究了粘性解的存在性，唯一性和正则性。这些结果适用于汉密尔顿-雅各比-贝尔曼（HJB）方程，Leland方程和p-Laplacian型方程，它们在流体力学，随机控制理论以及最优投资组合选择和金融交易成本问题中有许多应用。 ;对以上模型的粘度溶液的性质进行了进一步的研究：（1）H.B.Bernstein估计（尤其是 C 1，α估计）和HJB粘度溶液的凸性方程; （2）Leland方程的粘度解的时间和Leland常数的单调性以及Leland解和Black-Scholes解之间的关系; （3）完全非线性方程 u t + F （ Du <的粘性解的自由边界的存在和Lipschitz连续性/ italic>， D 2 u ）= 0，以p-Laplacian方程为模型。我们的研究扩展了粘度溶液理论的应用，并有助于上述模型的定性分析和数值计算。为了构造连续的粘度溶液，我们利用Perron方法和各种基于粘度溶液理论的估计。我们及时地将Bernstein估计和Kruzhkov的正则定理从光滑解转换为粘度解。我们的方法适用于初始边界值问题，尽管需要获得边界附近均匀连续模量的估计值，并且需要构造合适的粘度子解和超解。为了研究Leland方程，我们通过欧拉变换和线性平移将其转换成标准形式，然后利用比较原理研究粘度溶液的性质。为了研究p-Laplacian型方程自由边界的性质，我们采用比较原理，反射原理，移动平面方法以及子解和超解的构造。
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