# Fully non-linear PDE

A nice method of obtaining existence of solutions of many geometrically defined (and hence highly degenerate) parabolic systems (such as mean curvature flow) involves the reduction of the system to a single scalar equation whose solutions provide solutions of the original geometric problem by providing a suitable graph height function.

Once the scalar equation is obtained, the author simply appeals to the "standard theory of scalar parabolic PDE"; however, the standard theorem' required is certainly not standard, as it needs to provide solutions of a fully non-linear, second order, strictly (not uniformly) parabolic PDE over a closed Riemannian manifold (not simply $\mathbb{R}^n$). So my questions are as follows:

1) Where can I find the above theorem' (if it exists)?

2) I have seen the above theorem but with the uniform parabolicity assumption. Does the case I need (strictly, but not necessarily uniformly, parabolic) follow easily? Seems reasonable since the spatial manifold is compact.

Thanks for any help, ML

## 1 Answer

I think the standard reference is probably Olʹga Aleksandrovna Ladyzhenskai͡a, Vsevolod Alekseevich Solonnikov, Nina N. Ural'tseva Linear and Quasi-linear Equations of Parabolic Type, but you probably need to work to get your equations to fit their theorem. There certainly is no easy proof of existence of parabolic equations general enough for use in Riemannian geometry.

• Thanks Ben. As its name suggests, that reference only deals with quasi-linear equations. I require local existence in the fully non-linear (strictly rather than uniformly) parabolic setting. The mean curvature flow reduces to a quasi-linear equation, but other flows, such as the harmonic mean curvature flow, do not, and the fully non-linear theory is necessary. In some settings, the equations I'm considering can be reduced to scalar equations over the sphere, or a cylinder, or some other explicit Riemannian manifold, so references to results for equations over those spaces would also be nice. – User0.9999999..... Sep 15 '13 at 6:23
• Also, I'm not after a result with an easy proof (clearly a result as general as the one I'm after won't be easy, at least not quantitatively). But a `black box' reference or two would suffice. – User0.9999999..... Sep 15 '13 at 6:25