Surveys in Differential Geometry

Volume 19 (2014)

The regularity of solutions in degenerate geometric problems

Pages: 83 – 110

DOI: https://dx.doi.org/10.4310/SDG.2014.v19.n1.a4

Author

Panagiota Daskalopoulos (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

Abstract

We discuss the optimal regularity of solutions to degenerate elliptic and parabolic fully nonlinear partial differential equations, in particular the evolution of a hypersurface $M^n_t$ in $\mathbb{R}^{n+1}$ by powers of its Gaussian curvature and other nonlinear functions of its principal curvatures. We will also discuss the regularity question related to the Weyl problem with nonnegative curvature, which involves a fully-nonlinear degenerate elliptic equation of Monge-Ampère type.

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Published 6 March 2015