On the study of the theory of viscosity solutions and its new developments

在A.Swiech的两篇研究论文中，在适当的假设下，建立了完全非线性二阶一致椭圆型/抛物型偏微分方程解的Lp粘性解的Aleksandrov-Bakelman-Pucci极大值原理，其中一阶导数、无界系数、无界非齐次项可能是超线性增长项。Perron方法通过引入半连续Lp-粘性解，首次应用于完全非线性椭圆型偏微分方程解的Lp-粘性解.对于数学金融学中出现的几个非线性变分不等式，在H.Morimoto，H.Morimoto和S.Sakaguchi的研究论文中，通过证明相关值函数具有足够的正则性，构造了最优控制.

The Aleksandrov-Bakelman-Pucci maximum principle for Lp-viscosity solutions of fully nonlinear second order uniformly elliptic/parabolic partial differential equations with possibly superllinear growth terms of the first derivatives, unbounded coefficients, unbounded inhomogeneous terms has been established under appropriate hypotheses in two research papers with A. Swiech. Some counter-examples have been presented when there are no hypotheses.Perron's method has been first applied to Lp-viscosity solutions of fully nonlinear elliptic partial differential equations by introducing semicontinuous Lp-visosity solutions.For several nonlinear variational inequalities arising in Mathematical Finance, optimal controls have been constructed by showing that associated value functions admit enough regularity in research papers with H. Morimoto, and H. Morimoto and S. Sakaguchi.