Viscosity solution theory for fully nonlinear equations and its applications

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The basic theory for viscosity solutions of fully nonlinear second order elliptic partial differential equations is studied. In case when uniformly elliptic equations contain unbounded coefficients to the first derivatives, it is proved that the weak Harnack inequality holds for Lp-viscosity solutions. As applications, it turns out that qualitative properties such as the strong maximum principle, the maximum principle for unbounded domains, the Phragmen-Lindelov theorem etc. are shown.In case when degenerate elliptic equations contain the first derivative terms with supearlinear growth, by setting appropriate function spaces, to which viscosity solutions belong, the comparison principle for them is proved.