## Advances in Differential Equations

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

### $C^{1,\beta}$ 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867951

**Mathematical Reviews number (MathSciNet)**

MR2100635

**Zentralblatt MATH identifier**

1107.35032

**Subjects**

Primary: 35K55: Nonlinear parabolic equations

Secondary: 35B50: Maximum principles 35B65: Smoothness and regularity of solutions

#### Citation

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