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# Surveys in Differential Geometry

## Volume 19 (2014)

### The regularity of solutions in degenerate geometric problems

Pages: 83 – 110

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

#### Author

#### 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.