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2004 $C^{1,\beta}$ regularity of viscosity solutions via a continuous-dependence result
Mariane Bourgoing
Adv. Differential Equations 9(3-4): 447-480 (2004).


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.


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Mariane Bourgoing. "$C^{1,\beta}$ regularity of viscosity solutions via a continuous-dependence result." Adv. Differential Equations 9 (3-4) 447 - 480, 2004.


Published: 2004
First available in Project Euclid: 18 December 2012

zbMATH: 1107.35032
MathSciNet: MR2100635

Primary: 35K55
Secondary: 35B50, 35B65

Rights: Copyright © 2004 Khayyam Publishing, Inc.


Vol.9 • No. 3-4 • 2004
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