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Evans-Krylov C^{2,alpha} estimate

Gives a uniform C2,αC^{2,\alpha} bound by Evans–Krylov
compactness for the closedness step
Claim·§A The Dirichlet problem on a periodic fiber·inside thmDirichlet problem for quaternionic Monge-Ampère

Statement (paper)

Along the continuity path, the solutions ψ\psi satisfy a uniform C2,αC^{2,\alpha} bound for some α>0\alpha>0. The real Hessians D2ψD^2\psi lie in the compact convex set K={M:∣M∣≤C1, cI≤H(M)≤CI}\mathcal K=\{M:|M|\le C_1,\ cI\le\mathcal H(M)\le CI\}, on which F(M)=log⁡det⁡H(M)F(M)=\log\det\mathcal H(M) is concave and uniformly elliptic, so the equation F(D2ψ)−log⁡gt=0F(D^2\psi)-\log g_t=0 with boundary data B0B_0 satisfies the hypotheses of the boundary and interior Evans--Krylov estimates.

In plain English

Along the continuity path, the solutions ψ\psi obey a uniform C2,αC^{2,\alpha} bound for some α>0\alpha>0. This holds because D2ψD^2\psi stays in a compact convex set where F=log⁡det⁡HF=\log\det\mathcal H is concave and uniformly elliptic, so the boundary and interior Evans–Krylov estimates apply.

Why it's here

The previous claim's full C2C^2 bound and positivity of H(D2ψ)\mathcal H(D^2\psi) confine D2ψD^2\psi to the set K\mathcal K, which makes the equation concave and uniformly elliptic. Evans–Krylov then upgrades C2C^2 control to a uniform Hölder bound on second derivatives. This gives the compactness that the closedness and smoothness step needs to pass to limits and bootstrap regularity up to the boundary.
Classified routine:
Direct application of the interior and boundary Evans–Krylov C^{2,α} estimates (Evans 1982, Krylov 1983) for concave uniformly elliptic equations, once the preceding C^2 bound confines D^2ψ to a compact set on which F is concave and uniformly elliptic.

Uses

claimDouble normal derivative bound and full C2

Used by

claimClosedness of solvable parameters and boundary smoothness

Sources

PaperOpen
Section
§A The Dirichlet problem on a periodic fiber
Page
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sections/dirichlet.tex:171
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Friction

  • info
    inferred_dep
    outline: "Evans-Krylov C^{2,alpha} estimate" rests on claim-thm-dirichlet-4