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I would like to know whether viscosity solutions to $u_{t} - F( D^{2} (u) ) = 0$ are $C^{1, \alpha}$ analogous to the elliptic case as in the book by Caffarelli and Cabre .

Here F is assumed to be uniformly elliptic .

$D^{2}(u)$ is the spatial Hessian of $u$.

An answer would be appreciated.

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The result holds true in the parabolic case for F uniformly elliptic.

For a proof, you can take a look at the series of papers by Lihe Wang where boundary regularity is also investigated: http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160450202/full (SECTION 4) http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160450103/abstract

See also https://projecteuclid.org/euclid.bams/1183555461 and the very interesting book of Krylov http://www.springer.com/us/book/9781402003349.

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  • $\begingroup$ Is there some hope to have that for all $t\in[0,T]$, the function $x\mapsto u(t,x)$ is analytic ? $\endgroup$
    – NancyBoy
    Commented Jan 16 at 17:11

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