A generalization of Savin's small perturbation theorem for fully nonlinear elliptic equations and applications

Zhenyu Fan

REVIEW 3 major objections 7 minor 38 references

A generalization of Savin's small perturbation theorem for fully nonlinear elliptic equations and applications

T0 review · 3 major / 7 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read For locally uniformly elliptic fully nonlinear equations, uniformly flat viscosity solutions with Hölder-small forcing and coefficient oscillation are automatically C^{2,α} inside the ball.

desk verdict Serious, referee-worthy paper with a plausible main theorem and a genuinely different proof strategy, but the proof as written has a real gap in the Hölder estimate and the sigma_k application leans on an unsupported measure-theoretic lemma. read the letter →

arxiv 2509.01138 v1 pith:GJ2KGD3W submitted 2025-09-01 math.AP

Zhenyu Fan This is my paper · ORCID

classification math.AP MSC 35B6535D4035J60

keywords SavinsmallperturbationtheoremfullynonlinearellipticequationsviscositysolutionsC^{2α}regularitysigma-kHessianequationpartialweakHarnackinequalityslidingparaboloids

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to prove a robust regularity principle: if a viscosity solution of a fully nonlinear elliptic equation is uniformly tiny, and both the right-hand side and the equation's dependence on x are Hölder-small at the same scale, then the solution is automatically C^{2,α} in the interior. Savin's small perturbation theorem already did this for homogeneous problems without lower-order terms and with no x-dependence; the paper removes both restrictions under a Lipschitz structure condition in (p,z) and uniform continuity of the Hessian coefficient. The proof's engine is a compactness argument: badly behaved flat solutions are rescaled until they converge to a solution of the linearized constant-coefficient equation, which must be smooth. The advertised reward is a partial regularity theorem for σ_k(D^2u)=f: any k-convex viscosity solution with positive Lipschitz f is twice differentiable almost everywhere and C^{2,α} off a Lebesgue-null singular set.

What carries the argument

The machinery is the method of sliding paraboloids: for a supersolution one slides concave paraboloids from below, estimates the measure of the contact set through Pucci operators and the area formula, and iterates by enlarging the paraboloid opening rather than rescaling the solution, which would change the ellipticity constants in locally uniformly elliptic problems. This yields the weak Harnack inequality and Hölder estimates. The second engine is Lemma 3.1 (improvement of flatness), which upgrades a quadratic approximation of a flat solution at scale r to a better quadratic approximation at scale ηr; repeated application gives the pointwise C^{2,α} estimate that is Theorem 1.7.

What would settle it

A 2-convex viscosity solution of σ_2(D^2u)=f in R^n with n≥5, with positive Lipschitz f, whose singular set has positive Lebesgue measure would falsify Proposition 1.10. Equally decisive for the main theorem: a sequence of flat solutions (u_j, f_j) satisfying Theorem 1.7's hypotheses with δ_j→0 but no uniform C^{2,α} bound on B_{1/2} would disprove Theorem 1.7.

Watch

Extended reading notes

Core claim

Theorem 1.7 states that if F is elliptic and ρ-locally uniformly elliptic, vanishes at the origin, satisfies the structure inequality |F(M,p,z,x)−F(M,q,s,x)| ≤ b0|p−q| + c0|z−s|, and has DM F uniformly continuous, then any viscosity solution of F(D^2u,Du,u,x)=f with ||u||_{L∞} ≤ δ, ||f||_{C^{0,α}} ≤ δ, and |F(M,p,z,x)−F(M,p,z,x')| ≤ δ|x−x'|^α is automatically C^{2,α}(B_{1/2}) with a universal estimate. The constants depend only on n, α, ρ, λ, Λ, b0, c0, and the modulus of continuity ωF. The proof obtains weak Harnack and Hölder estimates for locally uniformly elliptic equations by sliding paraboloids, avoiding the rescaling that would destroy local uniform ellipticity, then uses a compactnes

Load-bearing premise

For the advertised σ_k application, the load-bearing premise is that every 2-convex function has a distributional Hessian representable as a matrix of Radon measures; the paper cites this to a theorem stated for k>n/2 and does not derive it for k=2 in dimensions n≥5, so Proposition 1.10 collapses if that representation fails.

Editorial extensions

If this is right

If Theorem 1.7 is correct, the classical small-perturbation regularity phenomenon extends to nonhomogeneous fully nonlinear equations with Hölder data, requiring no convexity or concavity of F.
The regularity principle implies partial regularity for σ_k Hessian equations: every k-convex viscosity solution of σ_k(D^2u)=f with positive Lipschitz f is twice differentiable a.e. and its singular set has Lebesgue measure zero.
The sliding-paraboloid weak Harnack and Hölder estimates in Section 2 apply to locally uniformly elliptic equations without rescaling, giving a reusable toolbox beyond the main theorem.
For uniformly elliptic F=F(M,x), Theorem 1.7 recovers and extends the earlier nonhomogeneous small-perturbation result, and for general locally uniformly elliptic F it supplies a proof different from a previous claimed one that the paper says contained a mistake.

Reading between the lines

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

If the uniform continuity of DM F were weakened to a Dini or VMO modulus, the same compactness-limiting scheme might yield C^{1,β} or C^{2,Dini} variants of the conclusion; the paper does not pursue this.
The σ_k application would likely extend to 1<k≤n/2 in any dimension once the Radon-measure representation of the Hessian is available for k-convex functions in that range; the paper's proof currently inherits that representation from a theorem stated for k>n/2.
One could test the sharp scaling of the thresholds by checking whether the δ in Theorem 1.7 must depend on α through a power law for model equations such as σ_k(D^2u)=f, where explicit examples might show the optimal relation.
Combining Theorem 1.7 with higher W^{3,ε}-type estimates and the blow-up analysis used for uniformly elliptic equations may upgrade the Lebesgue-null singular set in Proposition 1.10 to a Hausdorff-dimension estimate; the paper leaves that step implicit.

Share X Bluesky LinkedIn Reddit HN

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

Short version: this paper deserves a serious referee, but it is not ready as written. The main theorem—Savin's small perturbation result in the nonhomogeneous case with Hölder-small f and x-dependence—is the expected statement, and the proof via sliding paraboloids is coherent and genuinely different from the flawed Lian-Zhang attempt. The compactness argument in Lemma 3.1 is substantially worked out, and the paper is honest about the provenance of the theorem. No fitted parameters, no self-citation games.

The soft spots are real. The most serious is in Theorem 2.10, the Hölder estimate that drives everything. After rescaling, the author assumes |{v ≥ 1/2} ∩ B1| ≥ 1/2 and then asserts a lower bound on the L^ε norm of v on B1/4. That does not follow: the large set could sit entirely outside B1/4. What is needed is the standard expansion-of-positivity or covering argument that converts mass in B1 into mass in B1/4. The paper does not supply it. This is not a fatal conceptual flaw—this is exactly where the Krylov-Safonov machinery usually appears—but as written the main analytic ingredient is incomplete. The nearby application of Corollary 2.11 to v_k without checking nonnegativity is patchable by shifting, so I count that as minor.

The application to sigma_k is more doubtful. Lemma 4.2 is cited to Chaudhuri-Trudinger, but their Alexandrov-type theorem is for k > n/2; for 2-convex functions in n ≥ 5 the distributional Hessian need not be a matrix of Radon measures, and the paper does not derive that representation from the equation. If the lemma fails, the Radon-Nikodym step in Proposition 1.10(i) has no basis. The a.e. twice differentiability claim may still be true—Shankar-Yuan gives evidence in the sigma_2 case—but this proof does not establish it.

Who this is for: PDE people working on fully nonlinear regularity, especially those interested in locally uniformly elliptic operators and Hessian equations. They will want to read the sliding-paraboloid weak Harnack section and check whether the propagation step can be filled.

Recommendation: send to peer review. The main theorem is important and the proof strategy is worth referee time. Expect major revision, and the referee should be explicitly asked to verify the missing propagation step and to either prove Lemma 4.2 in the stated generality or restrict the application accordingly.

Referee Report

3 major / 7 minor

Summary. The paper proves a nonhomogeneous generalization of Savin's small perturbation theorem: if u is a viscosity solution to F(D^2u,Du,u,x)=f on B1, F is locally uniformly elliptic, F(0,0,0,x)=0, F has Lipschitz dependence on (p,z) and Hölder-small x-dependence, and both ||u||_L∞ and ||f||_C^{0,α} are sufficiently small, then u∈C^{2,α}(B_{1/2}) with a universal estimate (Theorem 1.7). The proof develops weak Harnack and Hölder estimates for locally uniformly elliptic equations via the method of sliding paraboloids (Section 2), then uses a compactness/improvement-of-flatness argument and Caffarelli iteration (Section 3). As an application, the paper claims a Lebesgue-a.e. twice differentiability result and a partial regularity result for σ_k(D^2u)=f with Lipschitz positive f (Proposition 1.10).

Significance. If the proof is completed, Theorem 1.7 is a meaningful extension of Savin's theorem to nonhomogeneous equations with a genuinely different technical route that avoids rescaling the ellipticity constants. The sliding-paraboloid weak Harnack machinery is of independent interest. Proposition 1.10 would be an attractive application connecting ε-regularity theory with Hessian equations. The paper is largely self-contained and contains no fitted parameters or circular reasoning. However, the current manuscript has several load-bearing gaps, in particular in the Hölder estimate of §2 and in the Hessian-measure lemma used for the application, so the advertised results are not yet fully established.

major comments (3)

[§2, Theorem 2.10, proof] The induction step asserts c0 ≤ ||v_{r_k^2ρ/2}||_{L^{ε0}(B1/4)} after assuming |{v≥1/2}∩B1|≥1/2. This lower bound does not follow: the large set {v≥1/2} may be essentially disjoint from B1/4, in which case the L^ε norm on B1/4 is arbitrarily small. Corollary 2.8 is a distribution-function upper bound, so it cannot produce c0. A measure-propagation or covering argument (e.g. using Lemma 2.6) is needed to convert mass in B1 into mass in B1/4, or a separate treatment of the complementary case is needed. As written, the Hölder estimate — and hence the compactness step in Lemma 3.1 — is not proved.
[§3, Lemma 3.1, Step 1] Corollary 2.11 is applied to the rescaled functions v_k = (u_k(r_k x)-P_k(r_k x))/r_k^{2+α}. Corollary 2.11 explicitly requires the function to be nonnegative in B1, but v_k may change sign because P_k is subtracted. This is not a vacuous hypothesis; the Hölder equicontinuity of {v_k} is the basis for passing to the limit v0. The gap is likely repairable by applying the corollary to a shifted, normalized version such as (v_k+2)/3, but the repair must be written.
[§4, Lemma 4.2 and Proposition 1.10] Lemma 4.2 asserts that every 2-convex function on B1 has a distributional Hessian that is a matrix of Radon measures, citing [CT05]. However, [CT05] is an Alexandrov-type theorem for k-convex functions with k>n/2; for k=2 in dimensions n≥5 this hypothesis is not met. The manuscript itself notes that a.e. twice differentiability may fail for k≤n/2 without an extra equation. Since the proof of Proposition 1.10(i) relies on Lemma 4.2 to obtain the Lebesgue-Radon-Nikodym decomposition of [D^2u], the application is not supported at present. The author must either prove the measure representation for solutions of σ_k(D^2u)=f or replace this step.

minor comments (7)

[§2, Lemma 2.3] Typo: 'area forluma' should be 'area formula'.
[§1, Remark 1.4] Typo: 'ellipricity' should be 'ellipticity'.
[§2, Theorem 2.10 and Corollary 2.11] The displayed range 'r 2ρ0/ρ ≤ r ≤ 1' is missing an inequality sign; it should read 'r ≥ 2ρ0/ρ'.
[§3, Lemma 3.1 statement] In the condition F(D2P0, DP(0), P(0), 0)=f(0), the arguments P(0) and DP(0) should be P0(0) and DP0(0).
[§3, Lemma 3.1, Step 1] The bound on f^*_k is stated for the C^{0,α} norm, but the argument controls only the L∞ norm; the term r_k^2|v_k| is not shown to be Hölder small without a priori information. Since Corollary 2.11 needs only L∞ smallness, the estimate should be stated for ||·||_{L∞}.
[§2, Theorem 2.10] The formula for v(y) is missing parentheses; it should be v(y)=(u(r_k y)-m_k)/(M_k-m_k).
[§2, Theorem 2.9] The proof divides by B := inf_{B1/4} u + ||f||_{L∞}/8. If B=0 the scaling v=u/B is undefined; a separate sentence treating this case (or taking a limiting argument) is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation steps: Theorem 1.7 is proved by compactness against constant-coefficient linear theory, and the application uses independent external theorems; flagged analytic gaps are correctness issues, not circularity.

full rationale

The paper's central claim, Theorem 1.7, is not derived from its own conclusion. Its proof proceeds by contradiction/compactness: flat normalized solutions are shown to converge to a solution of the constant-coefficient linearized equation a_ij D_ij v_0 = 0, and the regularity of such linear equations is an external classical benchmark rather than an input. The weak Harnack and Hölder estimates in Section 2 are built from Pucci-class measure estimates and the sliding-paraboloid method, not from the C^{2,alpha} conclusion. Lemma 3.1 is an improvement-of-flatness argument whose contradiction uses only the linearized solution and uniform ellipticity, and Lemma 3.2 iterates it. There are no fitted parameters renamed as predictions, and the paper does not rely on any self-citation: all load-bearing cited results (Savin, Caffarelli-Cabré, Chaudhuri-Trudinger, Trudinger, Imbert-Silvestre, etc.) are independent external theorems. The application to sigma_k equations imports Lemma 4.2 from Chaudhuri-Trudinger and gradient estimates from Trudinger; this is external support, not a self-referential reduction. I do flag two correctness gaps that are not circularity. First, in the proof of Theorem 2.10, after assuming |{v >= 1/2} ∩ B1| >= 1/2, the paper asserts without a covering/propagation argument that c0 <= ||v_{r_k^2 rho/2}||_{L^{epsilon0}(B1/4)}; Corollary 2.8 gives only an upper bound on superlevel sets and does not by itself force positive L^epsilon mass in B1/4. Second, Lemma 4.2 is stated for all 2-convex functions, while the cited [CT05] Alexandrov-type theorem is normally stated for k > n/2; for k=2 with n>=5 the measure representation is not derived in the paper. These are missing-support/gap issues in the proof chain, not equivalences between inputs and outputs. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data; the constants delta, C, r0, eta, Gamma, M, mu are universal and exist by compactness. No new entities are introduced.

assumptions (4)

standard math Viscosity solution framework and standard elliptic regularity background (Krylov-Safonov, Evans-Krylov, Schauder, Arzela-Ascoli).
Used throughout; for example, classical C^{∞} estimates for constant-coefficient linear equations in Lemma 3.1, Step 3.
domain assumption Local uniform ellipticity with structure condition (1.2) preserves membership in Pucci classes after the sliding paraboloid renormalization.
Theorems 2.9 and 2.10 are stated for S_rho classes; Lemma 2.2 transfers solutions of F to these classes, requiring 2rho-uniform ellipticity and b0,c0 control.
domain assumption Every 2-convex function has a distributional Hessian that is a matrix of Radon measures (Lemma 4.2).
Load-bearing for Proposition 1.10(i); cited to [CT05], but that source's Alexandrov theorem is for k>n/2, so for 2≤k≤n/2 the assertion is not justified in the text.
standard math Trudinger gradient estimate and TW99 interpolation inequality for k-convex solutions of sigma_k=psi^k with Lipschitz psi^{1/k}.
Used in Lemma 4.3 to turn L1 approximation into local Lipschitz control of the error h.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A generalization of Savin's small perturbation theorem for fully nonlinear elliptic equations and applications." pith.science (2026). https://pith.science/paper/GJ2KGD3W

@misc{pith2026250901138,
  author       = {Pith},
  title        = {Pith review of: A generalization of Savin's small perturbation theorem for fully nonlinear elliptic equations and applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GJ2KGD3W}},
  note         = {Machine review of arXiv:2509.01138}
}

abstract

In this note, we generalize Savin's small perturbation theorem to nonhomogeneous fully nonlinear equations $F(D^2u, Du, u,x)=f$ provided the coefficients and the right-hand side terms are H\"older small perturbations. As an application, we establish a partial regularity result for the sigma-$k$ Hessian equation $\sigma_{k}(D^2u)=f$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

38 extracted references · 37 canonical work pages

[1]

Allard, On the first variation of a varifold, Ann

William K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417--491. 307015

work page 1972
[2]

Armstrong, Luis E

Scott N. Armstrong, Luis E. Silvestre, and Charles K. Smart, Partial regularity of solutions of fully nonlinear, uniformly elliptic equations, Comm. Pure Appl. Math. 65 (2012), no. 8, 1169--1184. 2928094

work page 2012
[3]

249 (2024), Paper No

Sumiya Baasandorj, Sun-Sig Byun, and Jehan Oh, Second derivative L^ -estimates for a class of singular fully nonlinear elliptic equations , Nonlinear Anal. 249 (2024), Paper No. 113630, 15. 4784015

work page 2024
[4]

Sun-Sig Byun, Hongsoo Kim, and Jehan Oh, Interior W^ 2, type estimates for degenerate fully nonlinear elliptic equations with L^ n data , J. Funct. Anal. 289 (2025), no. 6, Paper No. 111007, 37. 4897643

work page 2025
[5]

Pure Appl

Xavier Cabr\'e, Nondivergent elliptic equations on manifolds with nonnegative curvature, Comm. Pure Appl. Math. 50 (1997), no. 7, 623--665. 1447056

work page 1997
[6]

Caffarelli and Xavier Cabr\'e, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol

Luis A. Caffarelli and Xavier Cabr\'e, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. 1351007

work page 1995
[7]

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
[8]

Caffarelli, L

L. Caffarelli, L. Nirenberg, and J. Spruck, The D irichlet problem for nonlinear second-order elliptic equations. III . F unctions of the eigenvalues of the H essian , Acta Math. 155 (1985), no. 3-4, 261--301. 806416

work page 1985

Show all 38 references

[9]

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
[10]

Trudinger, An A lexsandrov type theorem for k -convex functions , Bull

Nirmalendu Chaudhuri and Neil S. Trudinger, An A lexsandrov type theorem for k -convex functions , Bull. Austral. Math. Soc. 71 (2005), no. 2, 305--314. 2133413

2005
[11]

Caffarelli and Yu Yuan, A priori estimates for solutions of fully nonlinear equations with convex level set, Indiana Univ

Luis A. Caffarelli and Yu Yuan, A priori estimates for solutions of fully nonlinear equations with convex level set, Indiana Univ. Math. J. 49 (2000), no. 2, 681--695. 1793687

2000
[12]

Ennio De Giorgi, Frontiere orientate di misura minima, Editrice Tecnico Scientifica, Pisa, 1961, Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960-61. 179651

1961
[13]

Teixeira, Asymptotics and regularity of flat solutions to fully nonlinear elliptic problems, Ann

Disson dos Prazeres and Eduardo V. Teixeira, Asymptotics and regularity of flat solutions to fully nonlinear elliptic problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 15 (2016), 485--500. 3495436

2016
[14]

Evans and Ronald F

Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, revised ed., Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2015. 3409135

2015
[15]

Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm

Lawrence C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 35 (1982), no. 3, 333--363. 649348

1982
[16]

80, Birkh\"auser Verlag, Basel, 1984

Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkh\"auser Verlag, Basel, 1984. 775682

1984
[17]

Alessandro Goffi, High-order estimates for fully nonlinear equations under weak concavity assumptions, J. Math. Pures Appl. (9) 182 (2024), 223--252. 4677158

2024
[18]

Qingbo Huang, On the regularity of solutions to fully nonlinear elliptic equations via the L iouville property , Proc. Amer. Math. Soc. 130 (2002), no. 7, 1955--1959. 1896027

2002
[19]

Cyril Imbert and Luis Silvestre, Estimates on elliptic equations that hold only where the gradient is large, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 6, 1321--1338. 3500837

2016
[20]

N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), no. 1, 75--108. 688919

1983
[21]

N. V. Krylov and M. V. Safonov, An estimate for the probability of a diffusion process hitting a set of positive measure, Dokl. Akad. Nauk SSSR 245 (1979), no. 1, 18--20. 525227

1979
[22]

, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 1, 161--175, 239. 563790

1980
[23]

Dongsheng Li and Zhisu Li, Global W^ 2, estimates for a type of singular fully nonlinear elliptic equations , Math. Z. 285 (2017), no. 3-4, 1167--1179. 3623745

2017
[24]

Yuanyuan Lian and Kai Zhang, Pointwise regularity for locally uniformly elliptic equations and applications, preprint, arXiv:2405.07199 (2024)

2024 arXiv
[25]

Partial Differential Equations 44 (2019), no

Connor Mooney, A proof of the K rylov- S afonov theorem without localization , Comm. Partial Differential Equations 44 (2019), no. 8, 681--690. 3952774

2019
[26]

Pure Appl

Louis Nirenberg, On nonlinear elliptic partial differential equations and H \"older continuity , Comm. Pure Appl. Math. 6 (1953), 103--156; addendum, 395. 64986

1953
[27]

Nascimento and Eduardo V

Thialita M. Nascimento and Eduardo V. Teixeira, On the sharp H essian integrability conjecture in the plane , J. Differential Equations 414 (2025), 890--903. 4808803

2025
[28]

Nikolai Nadirashvili and Serge Vl a du t , Singular solutions of H essian elliptic equations in five dimensions , J. Math. Pures Appl. (9) 100 (2013), no. 6, 769--784. 3125267

2013
[29]

Ovidiu Savin, Phase transitions: R egularity of flat level sets , ProQuest LLC, Ann Arbor, MI, 2003, Thesis (Ph.D.)--The University of Texas at Austin. 2705495

2003
[30]

Partial Differential Equations 32 (2007), no

, Small perturbation solutions for elliptic equations, Comm. Partial Differential Equations 32 (2007), no. 4-6, 557--578. 2334822

2007
[31]

, Regularity of flat level sets in phase transitions, Ann. of Math. (2) 169 (2009), no. 1, 41--78. 2480601

2009
[32]

3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983

Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. 756417

1983
[33]

Ravi Shankar and Yu Yuan, Hessian estimates for the sigma-2 equation in dimension four, Ann. of Math. (2) 201 (2025), no. 2, 489--513. 4880431

2025
[34]

Trudinger, Weak solutions of H essian equations , Comm

Neil S. Trudinger, Weak solutions of H essian equations , Comm. Partial Differential Equations 22 (1997), no. 7-8, 1251--1261. 1466315

1997
[35]

Trudinger and Xu-Jia Wang, Hessian measures

Neil S. Trudinger and Xu-Jia Wang, Hessian measures. II , Ann. of Math. (2) 150 (1999), no. 2, 579--604. 1726702

1999
[36]

Yu Wang, Small perturbation solutions for parabolic equations, Indiana Univ. Math. J. 62 (2013), no. 2, 671--697. 3158522

2013
[37]

Partial Differential Equations 42 (2017), no

Hui Yu, Small perturbation solutions for nonlocal elliptic equations, Comm. Partial Differential Equations 42 (2017), no. 1, 142--158. 3605294

2017
[38]

Yu Yuan, A priori estimates for solutions of fully nonlinear special L agrangian equations , Ann. Inst. H. Poincar\'e C Anal. Non Lin\'eaire 18 (2001), no. 2, 261--270. 1808031

2001

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Allard, On the first variation of a varifold, Ann

William K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417--491. 307015

work page 1972

[2] [2]

Armstrong, Luis E

Scott N. Armstrong, Luis E. Silvestre, and Charles K. Smart, Partial regularity of solutions of fully nonlinear, uniformly elliptic equations, Comm. Pure Appl. Math. 65 (2012), no. 8, 1169--1184. 2928094

work page 2012

[3] [3]

249 (2024), Paper No

Sumiya Baasandorj, Sun-Sig Byun, and Jehan Oh, Second derivative L^ -estimates for a class of singular fully nonlinear elliptic equations , Nonlinear Anal. 249 (2024), Paper No. 113630, 15. 4784015

work page 2024

[4] [4]

Sun-Sig Byun, Hongsoo Kim, and Jehan Oh, Interior W^ 2, type estimates for degenerate fully nonlinear elliptic equations with L^ n data , J. Funct. Anal. 289 (2025), no. 6, Paper No. 111007, 37. 4897643

work page 2025

[5] [5]

Pure Appl

Xavier Cabr\'e, Nondivergent elliptic equations on manifolds with nonnegative curvature, Comm. Pure Appl. Math. 50 (1997), no. 7, 623--665. 1447056

work page 1997

[6] [6]

Caffarelli and Xavier Cabr\'e, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol

Luis A. Caffarelli and Xavier Cabr\'e, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. 1351007

work page 1995

[7] [7]

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

[8] [8]

Caffarelli, L

L. Caffarelli, L. Nirenberg, and J. Spruck, The D irichlet problem for nonlinear second-order elliptic equations. III . F unctions of the eigenvalues of the H essian , Acta Math. 155 (1985), no. 3-4, 261--301. 806416

work page 1985