$C^{2,\alpha}$ estimates for solutions to almost linear elliptic equations

Arunima Bhattacharya; Micah Warren

REVIEW 4 major objections 3 minor 22 references

$C^{2,\alpha}$ estimates for solutions to almost linear elliptic equations

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Pith's one-line read This paper proves C^{2,\alpha} interior estimates for viscosity solutions of uniformly elliptic equations that are C^1-close to linear, with an explicit closeness bound.

desk verdict A novel and credible C^{2,alpha} estimate for almost-linear fully nonlinear equations, with a genuine but fixable domain-transfer error in the main theorem. read the letter →

arxiv 1908.03654 v1 pith:2BZRMGTE submitted 2019-08-09 math.AP

Arunima Bhattacharya , Micah Warren This is my paper · ORCID

classification math.AP MSC 35J6035B6535D40

keywords fullynonlinearellipticequationsviscositysolutionsC^{2\alpha}estimatesalmostlinearoperatorsuniformellipticityEvans–Krylovregularitynonconvexexplicitclosenessconstant

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The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves interior $C^{{2,\alpha}}$ estimates for viscosity solutions of fully nonlinear uniformly elliptic equations F($D^{2}$u)=f that are $C^{1}$-close to a linear operator. The closeness is measured by requiring the derivative map DF to vary by at most \varepsilon across all symmetric matrices, and the paper computes an explicit \varepsilon_0(n,\$\lambda$,\Lambda,\$\alpha$) that guarantees the estimate. For the homogeneous equation the conclusion is a $C^{{2,\alpha}}$ bound on B_{1/(4\Lambda)} controlled by the sup norm of u, and for the inhomogeneous equation with f\in C^\$\alpha$ the bound is controlled by the sup norm of u plus the C^\$\alpha$ norm of f. The interest is that no concavity or convexity of F is assumed, so the result extends the classical Evans\,--\,Krylov regularity boost to a class of nonconvex equations.

What carries the argument

The load-bearing object is the almost-linearity condition \|DF(M)-DF(N)\|\le\varepsilon for all symmetric matrices M,N, which measures how far F's derivative is from being constant. Under this condition the paper reduces F to an operator whose derivative at zero is the identity by an affine change of coordinates, then uses a two-scale iteration: at each scale it approximates u by a quadratic polynomial P with F($D^{2}$P)=0, comparing u to a mollified harmonic function via the Krylov\,--\,Safanov H\"older estimate and harmonic derivative estimates. The quadratic polynomial lemma supplies a uniform pointwise approximation |u-P|\le C|x|^{2+\$\alpha$}, and a pointwise-to-H\"older lemma converts that approximation into a $C^{{2,\alpha}}$ bound. For the inhomogeneous equation, the same polynomial iteration is run with a smallness condition on the L^n average of f, using a $C^{{1,1}}$ approximation lemma from Caffarelli\,--\,Cabr\'e.

What would settle it

Test the unstated inclusion used around (2.44): take n=2, \$\lambda$=1/2, \Lambda=1, and W=DF(0)=\operatorname{diag}(1/2,1). Then A=$W^{{-1/2}}$=\operatorname{diag}(\sqrt{2},1), and the point x=(1/4,0) lies in B_{1/(4\Lambda)}=B_{1/4}, but \sqrt{\Lambda}A^T x=(\sqrt{2}/4,0) has norm about 0.354, outside B_{1/4}. This explicit failure shows the proof's estimate on the claimed ball is not established in the regime \$\lambda$\Lambda<1; a counterexample or a corrected transfer step would settle the matter.

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Extended reading notes

Core claim

The central claim is Theorem 1.3: for any ellipticity constants \$\lambda$,\Lambda and any \$\alpha$\in(0,1), there is an explicit \varepsilon_0(n,\$\lambda$,\Lambda,\$\alpha$)>0 such that if F is almost linear with constant \varepsilon_0 and u is a viscosity solution of F($D^{2}$u)=0 in B_1, then u is $C^{{2,\alpha}}$ on B_{1/(4\Lambda)} with \|$D^{2}$u\|_{C^\$\alpha$(B_{1/(4\Lambda)})} \le C_1\|u\|_{L^\infty(B_1)}. The paper also proves the inhomogeneous version, Theorem 1.4: if f\in C^\$\alpha$, then solutions of F($D^{2}$u)=f are $C^{{2,\alpha}}$ in B_{1/2} with norm controlled by \|u\|_{L^\infty}+\|f\|_{C^\$\alpha$}. The proof does not require F to be concave or convex; the only structural assumption is the almost-linear closeness condition.

Load-bearing premise

The proof transfers the estimate back to the original solution through an affine change of coordinates, and the claimed final ball is only contained in the ball where the rescaled estimate applies when the product of the ellipticity constants is at least one; the paper does not handle the other case.

Editorial extensions

If this is right

Any viscosity solution of F(D^2u)=0 with F almost linear at the explicit threshold is automatically C^{2,\alpha} in the interior, so the equation has classical solutions with second derivatives H\"older continuous.
The estimate is scale-invariant in the spirit of classical Schauder theory: the C^{2,\alpha} norm on a smaller ball is controlled linearly by the sup norm, so the result can be used as an a priori estimate in existence proofs.
For equations with H\"older right-hand side, the same regularity holds and the estimate depends additively on the C^\alpha norm of f, making the result suitable for perturbation and fixed-point arguments.
Because no concavity or convexity is assumed, the result adds a genuinely nonconvex class of uniformly elliptic equations to the known examples of fully nonlinear equations that enjoy Evans\,--\,Krylov-type regularity.
The closeness constant \varepsilon_0 is explicit, so the theorem gives a concrete quantitative threshold that can be checked for a given operator.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The authors do not pursue it, but the explicit \varepsilon_0 is likely far from optimal: the proof's constants are chosen by crude estimates, so a sharper computation of the admissible closeness window would make the result more directly applicable to perturbations of specific linear operators.
The affine normalization to the Laplacian suggests a natural extension to operators close to any constant-coefficient elliptic operator L rather than only the Laplacian; such an extension would also address the \lambda\Lambda\ge 1 restriction currently implicit in the proof's final ball transfer.
A testable refinement would be to perturb the Pucci extremal operators by a small Lipschitz term and compute the maximal \varepsilon for which C^{2,\alpha} estimates still hold; the theorem guarantees existence of such an \varepsilon, but numerical counterexamples near the threshold could locate the true constant.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Jim,

This paper proves interior C^{2,alpha} estimates for viscosity solutions of F(D^2u)=0 when F is uniformly elliptic and its derivative is globally epsilon0-close to a constant. The result is new: Cordes-Nirenberg gives C^{1,alpha} for linear equations, Savin requires the solution to be close to a quadratic, and the Nadirashvili-Vladut examples show general uniform ellipticity doesn't boost regularity. The proof uses a polynomial-approximation iteration rather than integral estimates, and the inhomogeneous version in Theorem 1.4 is a useful extension. The rescaling that reduces DF(0) to the identity is clean, and the Cordes-Nirenberg appendix is a helpful historical note.

The soft spots are real but not fatal.

First, the final transfer in Theorem 1.3 has a genuine domain gap. The step needs sqrt(Lambda) A^T B_{1/(4Lambda)} subset B_{1/4}. Since ||A|| = 1/sqrt(lambda), the image radius is 1/(4 sqrt(lambda Lambda)), which exceeds 1/4 when lambda Lambda < 1. So the estimate as written only reaches B_{sqrt(lambda)/(4 sqrt(Lambda))} in that regime, not the claimed B_{1/(4Lambda)}. That's a fixable problem—rescaling the domain or restating the theorem with a smaller ball—but it's not just cosmetic.

Second, the explicit constants have a bookkeeping error. In (2.40), the bound should give a factor 1/r0^{2+alpha} rather than 1/r0^{1+alpha}, and this propagates to (2.29) and (1.7). The moral is unchanged, but the printed constants aren't correct.

Third, Lemma 3.1 is delegated with 'the obvious approximation argument.' The boundary regularity needed to get h in C^2 with the stated estimates isn't shown. This is minor and standard, but the proof is not self-contained there.

Also, the word 'explicit' is doing some work: epsilon0 depends on the Krylov-Safanov constants, which aren't explicit. That's an overstatement, not a flaw in the strategy.

The central argument is coherent and the theorem believable once the domain is fixed. The paper deserves a serious referee; the gaps are the kind a good referee can identify and the authors can patch. I'd send it out.

For your own work, don't quote the B_{1/(4Lambda)} domain without checking the lambda Lambda >= 1 condition. The estimate on the smaller ball seems solid, and the method is worth knowing.

Referee Report

4 major / 3 minor

Summary. This paper proves interior C^{2,alpha} estimates for viscosity solutions of uniformly elliptic fully nonlinear equations F(D^2u)=0 and F(D^2u)=f, assuming F is uniformly differentiable and its derivative has oscillation bounded by an explicit epsilon0. The main results are Theorem 1.3 (homogeneous case) and Theorem 1.4 (inhomogeneous case), with explicit constants. The proof uses an approximation-by-polynomials iteration: Lemma 2.1 constructs a quadratic polynomial approximating the solution at small scale when the operator is close to the Laplacian; Proposition 2.4 iterates this to get a global estimate; Theorem 1.3 transfers it to operators close to a general linear operator via an affine change of variables; Theorem 1.4 derives the inhomogeneous estimate using approximation by homogeneous solutions and scaling.

Significance. If the proof were complete, the result would be a valuable extension of Evans-Krylov type C^{2,alpha} regularity to non-concave, non-convex operators that are C^1-close to linear, with explicit bounds on the closeness. The paper also contains a useful review of the Cordes-Nirenberg theory and a clean appendix lemma converting pointwise Holder estimates into C^{2,alpha} estimates. However, several gaps in the proof affect the main theorems as stated.

major comments (4)

[Proof of Theorem 1.3, transfer step near (2.44)] The proof concludes the C^{2,alpha} estimate on B_{1/(4Lambda)} from the estimate for tilde v on B_{1/4}, using that sqrt(Lambda) A^T x is in B_{1/4} for x in B_{1/(4Lambda)}. Since ||A^T|| = lambda^{-1/2}, the image of B_{1/(4Lambda)} under sqrt(Lambda) A^T has radius 1/(4 sqrt(lambda Lambda)), which exceeds 1/4 when lambda Lambda < 1. The theorem is stated for all positive lambda and Lambda without this restriction, so the stated domain is not reached by the proof in that regime. The statement should either assume lambda Lambda >= 1 or replace B_{1/(4Lambda)} by B_{1/(4 sqrt(lambda Lambda))} throughout, with the constant C_1 adjusted accordingly.
[Proof of Proposition 2.4, transfer to arbitrary x0 after (2.41)] The argument defining v(x')=4u(x'/2+x0) gives ||v||_{L∞(B1)} <= 4M, so applying (2.40) to v and rescaling back yields |u(x)-P_{x0}(x)| <= M C'_0 2^{2+alpha}|x-x0|^{2+alpha}, not M C'_0 2^{alpha}|x-x0|^{2+alpha}. The displayed constant is therefore missing a factor 4; this propagates into the formula for C_1 in (1.7).
[Proof of Theorem 1.4, reduction before (3.28)] The proof defines tilde f = f - f(0) and then asserts F_T(D^2 tilde u) = tilde f / T. However, the equation is F(D^2u)=f=tilde f+f(0), and for a fully nonlinear F one cannot subtract the constant f(0) from the equation without changing the operator. Moreover, the hypothesis (3.2) of Lemma 3.2 forces the L^n averages of f over B_r to be O(r^alpha), which excludes an arbitrary constant part f(0). A correct reduction would need to subtract a quadratic polynomial satisfying F(tI)=f(0), possible by ellipticity and the intermediate value theorem, and then track the extra quadratic term in the estimates; this step is absent.
[Claim 2.5 in the proof of Proposition 2.4] Lemma 2.1 is applied to F_i(N)=F(N+D^2P_i), but its hypothesis (2.2) is not verified. The text checks only that F_i inherits almost-linearity, namely |DF_i(M)-DF_i(N)| <= tilde epsilon0, and ellipticity. From (2.28) for F one obtains |F_i(N)-tr(N)| <= tilde epsilon0(||N||+2||D^2P_i||), and ||D^2P_i|| is not small compared with the decreasing norms ||v_i||_{L∞}; hence the application of Lemma 2.1 as stated is not justified. The induction in Proposition 2.4 needs either a strengthened form of Lemma 2.1 or a different argument.

minor comments (3)

[Lemma 2.1, statement vs. proof] The expression for C0 in part (ii) of Lemma 2.1 does not match the derivation in (2.16)-(2.17); please reconcile the displayed constants.
[Throughout] There are several typographical errors and formatting inconsistencies, including 'niether' in the introduction, 'secord order' in Corollary 4.2, and irregular spacing around displayed formulas; a careful proofreading pass is needed.
[Appendix 1] In Corollary 4.2, the hypothesis should state that the estimate (4.9) holds for every y in B_{1/2} and all x for which the left side is defined; the current wording is slightly ambiguous about the domain of the estimate.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the estimates are derived from standard external regularity theorems and self-contained approximation arguments; the only self-citation is historical and non-load-bearing.

full rationale

The derivation chain is self-contained. Theorem 1.3 is built from Lemma 2.1 and Proposition 2.4, which use the Krylov–Safanov theorem, harmonic boundary-value approximation, comparison principles, and harmonic Taylor remainder bounds to construct approximating quadratic polynomials; the iterative argument then uses only the already-established approximation at each scale, without assuming the conclusion. Theorem 1.4 applies Theorem 1.3 to a rescaled equation and uses the Caffarelli–Cabre approximation lemma, with the rescaled right-hand side satisfying the required smallness by construction from the C^alpha assumption. The only self-citation, to Streets–Warren [SW16], appears in the historical introduction and carries no argumentative weight. The apparent difficulty in the final affine transfer when lambda*Lambda < 1 is a geometric domain-estimate correctness issue, not a circularity: it is not a reduction of the conclusion to an assumption, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof depends on standard elliptic PDE background theorems and on the explicit construction of auxiliary constants (r0, gamma, mu, delta, epsilon0). No numbers are fitted to data, and no new entities are postulated. The hand-chosen constants are all functions of n, lambda, Lambda, alpha and of constants K1, K2, C3 imported from cited theorems.

assumptions (5)

standard math Krylov-Safanov theorem: viscosity solutions of uniformly elliptic equations are C^{alpha0} and satisfy the estimate (2.4).
Invoked in Lemma 2.1 to control mollification error via ||u_gamma - u||_infty <= K1 gamma^{alpha0} M; the constants K1 and alpha0 are imported from [KS81, Theorem 1].
standard math Harmonic boundary regularity: a harmonic function with C^3 boundary data is C^2 in the interior with a universal constant (Theorem 2.3).
Used in Lemma 2.1 to bound ||D^2 h||_infty by ||u_gamma||_{C^3}, producing the constant K2 in (2.21).
standard math Comparison principle for viscosity solutions of uniformly elliptic equations.
Used in Lemma 2.1 to bound |u - h| via sub- and supersolutions built from F(D^2h) and the closeness condition.
standard math Caffarelli-Cabre approximation lemma (Lemma 3.1): solutions of F(D^2u)=f are approximated by C^{1,1} solutions of the homogeneous equation with error controlled by ||f||_{L^n}.
Load-bearing for Theorem 1.4; cited to [CC95, Lemma 7.9] with an 'obvious approximation argument' rather than a full proof.
standard math Existence of viscosity solutions to the Dirichlet problem for uniformly elliptic F(D^2w)=0 with continuous boundary data.
Implicit in Lemma 3.1 and the definition of C^{1,1} interior estimates; standard Perron method.

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Pith. "Pith review of $C^{2,\alpha}$ estimates for solutions to almost linear elliptic equations." pith.science (2026). https://pith.science/paper/2BZRMGTE

@misc{pith2026190803654,
  author       = {Pith},
  title        = {Pith review of: $C^2,\alpha$ estimates for solutions to almost linear elliptic equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2BZRMGTE}},
  note         = {Machine review of arXiv:1908.03654}
}

abstract

In this paper, we show $C^{2,\alpha}$ interior estimates for viscosity solutions of fully non-linear, uniformly elliptic equations, which are close to linear equations and we also compute an explicit bound for the closeness.

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Reference graph

Works this paper leans on

22 extracted references · 20 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Armstrong, Luis E

Scott N. Armstrong, Luis E. Silvestre, and Charles K. Smart, Partial regularity of solutions of fully nonlinear, uniformly elliptic equations, Communications on Pure and Applied Mathematics 65, no. 8, 1169--1184

work page

[2] [2]

J \" o rgens , Michigan Math

Eugenio Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K . J \" o rgens , Michigan Math. J. 5 (1958), 105--126. 0106487

work page 1958

[3] [3]

43, American Mathematical Soc., 1995

Luis A Caffarelli and Xavier Cabr \'e , Fully nonlinear elliptic equations, vol. 43, American Mathematical Soc., 1995

work page 1995

[4] [4]

Caffarelli, Interior C^ 2, regularity theory for a class of nonconvex fully nonlinear elliptic equations , J

Xavier Cabr\' e and Luis A. Caffarelli, Interior C^ 2, regularity theory for a class of nonconvex fully nonlinear elliptic equations , J. Math. Pures Appl. (9) 82 (2003), no. 5, 573--612. 1995493

work page 2003

[5] [5]

China Math

Yi Cao, DongSheng Li, and LiHe Wang, A priori estimates for classical solutions of fully nonlinear elliptic equations, Sci. China Math. 54 (2011), no. 3, 457--462. 2775423

work page 2011

[6] [6]

Collins, C^ 2, estimates for nonlinear elliptic equations of twisted type , Calc

Tristan C. Collins, C^ 2, estimates for nonlinear elliptic equations of twisted type , Calc. Var. Partial Differential Equations 55 (2016), no. 1, Art. 6, 11. 3441283

2016

[7] [7]

3, 278--312

Heinz Otto Cordes, \"U ber die erste randwertaufgabe bei quasilinearen differentialgleichungen zweiter ordnung in mehr als zwei variablen , Mathematische Annalen 131 (1956), no. 3, 278--312

work page 1956

[8] [8]

H. O. Cordes, Zero order a priori estimates for solutions of elliptic differential equations, Proc. S ympos. P ure M ath., V ol. IV , American Mathematical Society, Providence, R.I., 1961, pp. 157--166. 0146511

work page 1961