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

Mariane Bourgoing

#### 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.

#### Article information

Source
Adv. Differential Equations, Volume 9, Number 3-4 (2004), 447-480.

Dates
First available in Project Euclid: 18 December 2012

Bourgoing, Mariane. $C^{1,\beta}$ regularity of viscosity solutions via a continuous-dependence result. Adv. Differential Equations 9 (2004), no. 3-4, 447--480. https://projecteuclid.org/euclid.ade/1355867951