$C^{1,\beta}$ regularity of viscosity solutions via a continuous-dependence result

Abstract

In this article, we are interested in the existence, uniqueness, and regularity of solutions of fully nonlinear parabolic equations, with initial data $u_0$, in the whole space $\mathbb R^n$. Our main result is the existence of a strictly subquadratic solution with a local $C^{1,\beta}$ regularity with respect to the space variable, assuming $C^{1,\alpha}$ regularity on $u_0$ and local uniform ellipticity of the equation. Our proof relies on a result of N. Zhu which shows the local $C^{1,\beta}$ regularity of the solution provided it is Lipschitz continuous and H\"older continuous in t, with an exponent $ {\gamma>\frac{1}{2}}$; we obtain this last property through a continuous-dependence result. Then we investigate further regularity for the solution using results of L. Wang.