## Analysis & PDE

• Anal. PDE
• Volume 8, Number 5 (2015), 1145-1164.

### On estimates for fully nonlinear parabolic equations on Riemannian manifolds

#### Abstract

We present some new ideas to derive a priori second-order estimates for a wide class of fully nonlinear parabolic equations. Our methods, which produce new existence results for the initial-boundary value problems in $ℝn$, are powerful enough to work in general Riemannian manifolds.

#### Article information

Source
Anal. PDE, Volume 8, Number 5 (2015), 1145-1164.

Dates
Revised: 13 February 2015
Accepted: 30 April 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.apde/1510843132

Digital Object Identifier
doi:10.2140/apde.2015.8.1145

Mathematical Reviews number (MathSciNet)
MR3393676

Zentralblatt MATH identifier
1323.35075

#### Citation

Guan, Bo; Shi, Shujun; Sui, Zhenan. On estimates for fully nonlinear parabolic equations on Riemannian manifolds. Anal. PDE 8 (2015), no. 5, 1145--1164. doi:10.2140/apde.2015.8.1145. https://projecteuclid.org/euclid.apde/1510843132

#### References

• L. Caffarelli, L. Nirenberg, and J. Spruck, “The Dirichlet problem for nonlinear second-order elliptic equations, III: Functions of the eigenvalues of the Hessian”, Acta Math. 155:3–4 (1985), 261–301.
• B. Guan, “The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds”, preprint, 2014.
• B. Guan, “Second-order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds”, Duke Math. J. 163:8 (2014), 1491–1524.
• N. Ivochkina and O. Ladyzhenskaya, “Flows generated by symmetric functions of the eigenvalues of the Hessian”, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. $($POMI$)$ 221:26 (1995), 127–144, 258. Reprinted in J. Math. Sci. 87:2 (1997), 3353–3365.
• H. Jiao and Z. Sui, “The first initial-boundary value problem for a class of fully nonlinear parabolic equations on Riemannian manifolds”, Int. Math. Res. Not. 2015:9 (2015), 2576–2595.
• Y. Y. Li, “Some existence results for fully nonlinear elliptic equations of Monge–Ampère type”, Comm. Pure Appl. Math. 43:2 (1990), 233–271.
• G. M. Lieberman, Second order parabolic differential equations, World Scientific, River Edge, NJ, 1996.
• N. S. Trudinger, “On the Dirichlet problem for Hessian equations”, Acta Math. 175:2 (1995), 151–164.
• J. Urbas, “Hessian equations on compact Riemannian manifolds”, pp. 367–377 in Nonlinear problems in mathematical physics and related topics, II, edited by M. S. Birman et al., Int. Math. Ser. (N. Y.) 2, Kluwer/Plenum, New York, 2002.
• G. Wei and W. Wylie, “Comparison geometry for the Bakry–Emery Ricci tensor”, J. Differential Geom. 83:2 (2009), 377–405.