$$C^{2,\alpha }$$ C 2 , α regularities and estimates for nonlinear elliptic and parabolic equations in geometry

Abstract

We give sharp C 2,α estimates for solutions of some fully nonlinear elliptic and parabolic equations in complex geometry and almost complex geometry, assuming a bound on the Laplacian of the solution. We also prove the analogous results to complex Monge-Ampère equations with conical singularities. As an application, we obtain a local estimate for Calabi-Yau equation in almost complex geometry. We also improve the C 2,α regularities and estimates for viscosity solutions to some uniformly elliptic and parabolic equations. All our results are optimal regarding the Hölder exponent.

A parabolic approach to the Calabi-Yau problem in HKT geometry

“…Then E is a compact convex subset of Sym(4n, R). We observe that Φ and p satisfy the assumptions in [7,Theorem 5.1]. Indeed, This together with n r=1 u rr ≤ C gives a uniform lower bound for each u rr .…”

“…Step 4. A general result in [7] implies a uniform Hölder estimate on the second derivatives of ϕ, thus a classical bootstrapping argument using Schauder estimates implies T = ∞ and a uniform bound on |∇ k ϕ| for k ≥ 1 (Lemmas 8 and 9).…”

The Monge–Ampère equation for non-integrable almost complex structures

“…Proof of Theorem 1.2. Thanks to Propositions 3.1 and 5.1, we can obtain a priori C 2,α estimates from the main result of [44], for some 0 < α < 1 (and in fact for all 0 < α < 1 by [10]). Higher order estimates follow after differentiating the equation and applying the usual bootstrapping method.…”

Transverse fully nonlinear equations on Sasakian manifolds and applications

“…In the foliated local coordinate patch (U ; x 1 , · · · , x r , z 1 , · · · , z n ), we work with the qualities of (z 1 , · · · , z n ). Therefore, given (7.8), (7.18) and (7.19), the C 2,α estimate (7.20) follows from the Evans-Krylov theory (see [64,15]) and high order estimate (7.21) follows from the bootstrapping argument.…”