Sharp form of the dependence of interior $C^{2,α}$ Monge–Ampère estimates on $α$ and on the $C^α$ norm of the right-hand side

Determine the sharp form of the function $ω(α,‖f‖_{C^α})$ in the interior Schauder-type estimate $‖D^2u‖_{C^α(B_{1/2})}≤ω(α,‖f‖_{C^α(B_1)})$ for strictly convex solutions of the Monge–Ampère equation $det D^2u=f$ in $B_1⊂R^n$ with $0<λ≤f≤Λ$ and a fixed modulus of strict convexity. Figalli, Jhaveri and Mooney (J. Funct. Anal. 270 (2016)) proved the upper bounds $ω≲‖f‖_{C^α}^{ρ}$ ($ρ>1$ depending on the data; the proof yields $ρ=2σ/α$ in terms of the strict-convexity exponent $σ>1$) and, for $‖f‖_{C^α}=1$, $ω≤exp((C/α)log(1/α))$, and constructed two-dimensional families with a uniform modulus of strict convexity showing that the dependence on $‖f‖_{C^α}$ is at least superlinear and that $ω≥e^{c/α}$. The surviving open core is: (i) whether the upper bound can be improved to the pure exponential scale $e^{C/α}$ matching the lower examples, or whether examples force growth between $e^{c/α}$ and $exp((C/α)log(1/α))$ or beyond; (ii) the optimal power exponent in the $‖f‖_{C^α}$-dependence and its dependence on dimension and the ellipticity data; and (iii) sharpness constructions in dimensions $n≥3$, where no analogue of the two-dimensional examples is known. Later work has consolidated these bounds in neighboring boundary-normalized settings but has not determined the sharp law in the original class.

Background

For the Poisson equation $\Delta v=g$ in $B_1⊂R^n$, classical Schauder estimates give $‖v‖_{C^{2,α}(B_{1/2})}≤C(‖v‖_{L^∞(B_1)}+‖g‖_{C^α(B_1)})$ for every $α∈(0,1)$, with $C∼1/α$ as $α→0$. Here $C^α$ denotes the Hölder space of functions whose increments are $O(|x-y|^α)$, $C^{2,α}$ the space of twice continuously differentiable functions whose Hessian $D^2v$ lies in $C^α$, and $‖·‖_{C^{2,α}}$ the corresponding norm. The same quantitative behavior — linear in $1/α$ and in $‖g‖_{C^α}$ — holds for all uniformly elliptic equations, linear or fully nonlinear. The Monge–Ampère equation

$$det D^2u=f$$

prescribes the Hessian determinant of a convex potential $u$ (geometrically, the Gaussian curvature of the graph of $u$ up to a factor). It is the model degenerate elliptic equation and is invariant under arbitrary affine changes of variables, so no uniformly elliptic structure is available globally. The natural solution concept is the Alexandrov solution — a convex function whose Monge–Ampère measure, the image of Lebesgue measure under the subdifferential, has density $f$ — and positive regularity requires strict convexity of the solution. Caffarelli proved interior $C^{1,α}$, $W^{2,p}$ and $C^{2,α}$ estimates for solutions with $0<λ≤f≤Λ$ that are strictly convex, together with a localization property: the sublevel sets (sections) $Z_h(x)={u<u(x)+⟨∇u(x),·-x⟩+h}$ are equivalent to ellipsoids by John's lemma, so after an affine normalization that makes a section round the equation becomes uniformly elliptic and regularity follows. A modulus of strict convexity is a fixed lower bound $cr^σ$ (with $σ>1$ depending only on $n,λ,Λ$ for normalized data) on the gap between $u$ and its tangent planes; fixing it is what makes the estimates in this problem uniform. These are regularity results, however, without explicit dependence of the constants on the data. The quantitative question studied by Figalli–Jhaveri–Mooney is: among strictly convex solutions of $det D^2u=f$ in $B_1$ with $0<λ≤f≤Λ$, $f∈C^α(B_1)$, and a fixed modulus of strict convexity, how does the best function $ω:R_+×R_+→R_+$ in

$$‖D^2u‖_{C^α(B_{1/2})}≤ω(α,‖f‖_{C^α(B_1)})$$

(whose existence follows from Caffarelli's $C^{2,α}$ estimates by compactness) depend on its arguments? Earlier quantitative attempts include the continuity estimates of Jian–Wang (2007), based on iterating a sequence of sections, whose Hölder estimate appeared to depend linearly on $‖f‖_{C^α}$, and the $C^{2,α}$ a priori estimate of Chen–Wang (2014) in the real and complex settings. Figalli–Jhaveri–Mooney showed the truth is genuinely nonlinear. Theorem 1.1: $‖D^2u‖_{C^α(B_{1/2})}≤C‖f‖_{C^α(B_1)}^{ρ}$ with $C,ρ>1$ depending on $n,λ,Λ,α$ (and $C$ also on the modulus of strict convexity); the proof yields the explicit exponent $ρ=2σ/α$, where $σ=σ(n,λ,Λ)>1$ is the strict-convexity exponent. Theorem 1.2: for $‖f‖_{C^α(B_1)}=1$ and $α∈(0,1/2]$, $‖D^2u‖_{C^α(B_{1/2})}≤exp((C/α)log(1/α))$. Theorem 1.3: two-dimensional families with a uniform modulus of strict convexity and $0<λ≤f≤Λ$ for which the dependence on $‖f‖_{C^α}$ is at least superlinear (any fixed superlinear exponent is achievable by increasing the anisotropy parameter of the construction, with the family's ellipticity constants then depending on that parameter) and, with $‖f_α‖_{C^α}≤C$, $‖D^2u_α‖_{C^α(B_{1/2})}≥e^{c/α}$. The examples are built from solutions invariant under anisotropic rescalings $u(x_1,x_2)=κ^{-(γ+1)}u(κx_1,κ^γx_2)$, whose sections become very eccentric before the equation linearizes around a paraboloid — the mechanism, identified in the paper's appendix, that distinguishes the degenerate affine-invariant Monge–Ampère equation from uniformly elliptic equations, where the dependence is linear in both $1/α$ and the data norm. The gap between these upper and lower bounds is the subject of this problem.

Problem Statement

Among strictly convex solutions $u$ of the Monge–Ampère equation $det D^2u=f$ in $B_1⊂R^n$ with $0<λ≤f≤Λ$, $f∈C^α(B_1)$, and a fixed modulus of strict convexity (with constants allowed to depend on $n,λ,Λ$ and that modulus, exactly as in Theorems 1.1–1.3 of Figalli–Jhaveri–Mooney), determine the sharp form of the function $ω$ in the interior estimate

$$‖D^2u‖_{C^α(B_{1/2})}≤ω(α,‖f‖_{C^α(B_1)}).$$

The sharp form of $ω$ is determined by matching upper bounds and explicit sharpness constructions in the same class; the currently open components, which jointly constitute this single objective, are:

1. Sharp $α$-dependence. The known bounds are $ω≥e^{c/α}$ (two-dimensional examples with $‖f‖_{C^α}≤C$ and a uniform modulus of strict convexity) and $ω≤exp((C/α)log(1/α))$ (for $‖f‖_{C^α}=1$, $α≤1/2$). Determine whether the upper bound can be improved to the pure exponential scale $ω≲e^{C/α}$, matching the examples, or whether there exist solutions in the stated class forcing growth strictly between $e^{c/α}$ and $exp((C/α)log(1/α))$, or faster than $exp((C/α)log(1/α))$.

2. Sharp dependence on $‖f‖_{C^α}$. The known bounds are the power upper bound $‖D^2u‖_{C^α}≲‖f‖_{C^α}^{2σ/α}$ (the exponent obtained in the proof, $σ=σ(n,λ,Λ)>1$ the strict-convexity exponent) and two-dimensional examples forcing superlinear growth (any fixed superlinear exponent is achievable in two dimensions by increasing the anisotropy parameter of the construction, with the family's ellipticity constants then depending on that parameter). Determine the optimal exponent $ρ$ as a function of $n,λ,Λ$ and $α$ — in particular whether $ρ$ must diverge like $1/α$ — and what the minimal exponent is in dimensions $n≥3$.

3. Sharpness constructions in $n≥3$. All known amplifying examples live in two dimensions, where the determinant of the anisotropically homogeneous ansatz reduces to a one-dimensional problem. Construct analogous families of strictly convex solutions with a uniform modulus of strict convexity and bounded data in dimensions $n≥3$, or show that the optimal form of $ω$ in higher dimensions differs from the two-dimensional picture.

The three components are linked by the same estimate and constrain one another; a resolution of any component must be compatible with the others through the common bound on $ω$.

References

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