A priori estimates for solutions of fully nonlinear equations with convex level set

Abstract

We derive an a priori C 2,α estimate for solutions of the fully non-linear elliptic equation F(D 2 u) = 0, provided the level set Σ = {M | F(M) = 0} satisfies: (a) Σ ∩ {M | Tr M = t} is strictly convex for all constants t; (b) the angle between the identity matrix I and the normal F ij to Σ is strictly positive on the non-convex part of Σ. Moreover, we do not need any convexity assumption on F in the course of the proof for the two dimensional case, as the classical result indicates.. II ≥ ω(Λ) > 0, (2) (Non-convex part, strict transversality) The angle between I and the normal (F ij) to Σ, (I, F ij) ≥ Θ(Λ) > 0, provided that M ≤ Λ.

Convergence of the $J$-flow on toric manifolds

“…We now show the C 2,α -estimates for equation (25). The difficulty is that the operator is neither concave, nor convex, and the result of Caffarelli-Yuan [5] also does not apply directly.…”

“…The difficulty is that the operator M → T r(M ) + d det(M ) is neither concave, nor convex for d > 0, and so the standard Evans-Krylov theory does not apply. In addition the level sets {M : det(M ) = 1 − t} are not uniformly convex as t → 1, and so the results of Caffarelli-Yuan [5] also do not apply directly. Instead we obtain the interior C 2,α estimates by showing that det(D 2 h) 1/n is a supersolution for the linearized equation, to which the techniques of [5] can be applied.…”

$$C^{2,\alpha }$$ C 2 , α estimates for nonlinear elliptic equations of twisted type

“…The conclusion of Theorem 1.3 follows immediately from Proposition 2.2, either by appealing directly to Savin's small perturbations theorem [13, Theorem 1.3], or by following the argument in Caffarelli-Yuan [4].…”

“…Caffarelli-Cabré [3] proved an Evans-Krylov type estimate for functions F which are the minimum of convex and concave functions-in particular, their result applies to equations of Issac's type. Caffarelli-Yuan [4] proved C 2,α a priori estimates under the assumption that level set {F = 0} has uniformly convex intersection with a family of planes. In essence, this allows one of the principle curvatures of {F = 0} to be negative.…”

A priori estimates for classical solutions of fully nonlinear elliptic equations

“…Without the convex (or concave) hypothesis on F , Caffarelli and Yuan [4] proved that the viscosity solutions of (1) are C 2,α if the level set = {M : F (M ) = 0} satisfies: (i) ∩{M : tr(M ) = t} is strictly convex for all constants t; (ii) the angle between the identity matrix I and the normal F ij to is strictly positive on the non-convex part of .…”