Pith. sign in

REVIEW 4 major objections 5 minor 2 cited by

Optimal Boundary Regularity for Uniformly Degenerate Elliptic Equations

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read For uniformly degenerate elliptic operators, solutions inherit exactly the Hölder regularity the boundary permits—no more, no less—and the paper proves this up to the boundary.

desk verdict A promising general Schauder theory for uniformly degenerate elliptic operators, with careful base proofs, but the higher-order induction steps the main theorem depends on are omitted. read the letter →

arxiv 2411.16418 v1 pith:MGIPH6RQ submitted 2024-11-25 math.AP

Qing Han , Jiongduo Xie This is my paper · ORCID
classification math.AP MSC 35J7035B6535J25
keywords uniformlydegenerateellipticequationsboundaryregularitySchaudertheorycharacteristicpolynomialindicialrootsHölderspacesoptimal
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 survey paper proves a complete boundary Schauder theory for uniformly degenerate elliptic operators $L=\rho^2 a^{ij}\partial_{ij}+\rho b^i\partial_i+c$, where the second-order part vanishes at the boundary like the square of the defining function $\rho$. Under the condition that the characteristic polynomial $P(\mu)$ satisfies $P(k+\alpha)<0$ on the boundary, any bounded solution of $Lu=f$ with boundary value $u=f/c$ is $C^{k,\alpha}$ up to the boundary, and the weighted derivatives $\rho\nabla^{k+1}u$ and $\rho^2\nabla^{k+2}u$ are $C^\alpha$ and vanish on the boundary, with a full estimate in terms of $|u|_{L^\infty}$ and $|f|_{C^{k,\alpha}}$. The condition is sharp: explicit solutions of the form $\psi\rho^{k+\alpha}$ (with logarithmic factors when $k+\alpha$ is an integer) have arbitrarily smooth data yet cannot be $C^{k,\beta}$ for any $\beta>\alpha$. This matters because uniformly degenerate operators arise from complete conformal metrics, complex Monge-Ampère equations, minimal hypersurfaces in hyperbolic space, and conformally compact Einstein metrics, where earlier regularity results lived in weighted Hölder spaces; the paper removes growth and decay assumptions and obtains the classical Hölder spaces directly.

What carries the argument

The load-bearing object is the characteristic polynomial $P(\mu)=\mu(\mu-1)a^{ij}\nu_i\nu_j+\mu b^i\nu_i+c$ on the boundary (with the flat-model version $Q(\mu)=\mu(\mu-1)a^{nn}+\mu b^n+c$). Its positive root is the sharp Hölder threshold: the condition $P(k+\alpha)<0$ says the desired exponent lies below that root. The argument is carried by a family of homogeneous supersolutions $\psi=t^\sigma(\varepsilon|x'-x_0'|^2+t^2)^{(\mu-\sigma)/2}+K t^\mu$, whose construction in Lemma 3.1 turns the inequalities $Q(\sigma)\le -c_\sigma$ and $Q(\mu)\le -c_\mu$ into pointwise decay of solutions near the boundary. A calculus lemma then upgrades decay to $C^\alpha$ bounds on $u$, $tDu$, and $t^2D^2u$. Tangential regularity comes from differentiating the equation along the boundary; normal regularity comes from differentiating in $t$, which produces a shifted operator $L^{(m)}$ with characteristic polynomial $Q(\mu+m)$ and with boundary values of $\partial_t^m u$ fixed by the equation.

What would settle it

In the flat model $L=t^2a^{nn}\partial_t^2+tb^n\partial_t+c$ with constant coefficients, take $a^{nn}=1$, $b^n=0$, $c=-1$, so $Q(\mu)=\mu(\mu-1)-1$ and the positive root is $(1+\sqrt5)/2$. Solve $Lu=f$ explicitly for smooth $f$ and compute the boundary Hölder exponent of the solution and of $\partial_t^2u$; the theorem predicts the solution is $C^{k,\alpha}$ exactly when $Q(k+\alpha)<0$ and no better. The same computation with $f$ chosen so the second normal derivative develops a logarithmic term would test the omitted $m=2$ induction directly.

Watch

Extended reading notes

Core claim

The central claim is that the degeneracy does not cause any loss of differentiability beyond the amount already encoded in the characteristic polynomial. For $L=\rho^2 a^{ij}\partial_{ij}+\rho b^i\partial_i+c$ with $c\le -c_0$ and $P(k+\alpha)\le -c_{k+\alpha}$ on $\partial\Omega$, the Dirichlet problem forces $u=f/c$ on the boundary, and the paper proves that $u\in C^{k,\alpha}(\bar\Omega)$, $\rho\nabla^{k+1}u\in C^\alpha(\bar\Omega)$, and $\rho^2\nabla^{k+2}u\in C^\alpha(\bar\Omega)$, with these weighted derivatives equal to zero on the boundary and the estimate (2.10). The proof works locally near the boundary and treats tangential and normal derivatives by the same elliptic arguments; normal differentiations shift the characteristic polynomial to $Q^{(m)}(\mu)=Q(\mu+m)$, and the boundary values of $\partial_t^m u$ are determined by the equation. Example 2.1 shows optimality: when the positive characteristic exponent is $k+\alpha$, one can make $f$ as smooth as desired while $u$ remains only $C^{k,\alpha}$.

Load-bearing premise

The argument rests on the claimed but omitted induction that repeated differentiation in the normal and tangential directions keeps the equation in the same degenerate class with shifted characteristic polynomials, together with the explicit boundary conditions $c\le -c_0$ and $P(k+\alpha)<0$.

Editorial extensions

If this is right

  • For $c\le 0$ in $\Omega$ and $c<0$, $P(k+\alpha)<0$ on $\partial\Omega$, the operator $L$ is an isomorphism from the weighted Hölder space $C^{k,\alpha}_2(\bar\Omega)$ onto the usual space $C^{k,\alpha}(\bar\Omega)$, so the degenerate Schauder theory matches the nondegenerate one with weighted control at the boundary.
  • The regularity is optimal: even with arbitrarily smooth coefficients and data, a solution can be $C^{k,\alpha}$ but not $C^{k,\beta}$ for any $\beta>\alpha$ when the positive characteristic exponent is $k+\alpha$, and integer exponents introduce logarithmic factors instead of extra differentiability.
  • The boundary data are not free: both $u=f/c$ and the normal derivatives $\partial_t^i u$ up to order $k$ on the boundary are determined by the equation, so the well-posed Dirichlet problem for this class has a single boundary function, namely the ratio of the right-hand side to $c$.
  • For $k=0$ and any $\alpha\in(0,1)$, the regularization argument produces a unique solution for every $f\in C^\alpha(\bar\Omega)$, with $\rho\nabla u$ and $\rho^2\nabla^2u$ continuous up to the boundary and vanishing there.

Reading between the lines

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

  • A direct test of the omitted inductions is to compute the $m=2$ normal-derivative equation in the flat constant-coefficient model; if the shifted polynomial $Q(\mu+2)$ and the boundary values $\partial_t^2u_0$ do not control the right-hand side, the full $C^{k,\alpha}$ statement for $k\ge2$ would require a different argument.
  • The same scheme should extend to fully nonlinear uniformly degenerate equations whose linearization has this structure, since the proof of the linear estimates is the only additional input beyond standard interior Schauder theory; nonlinear boundary blow-up problems linearize exactly to the model equation (1.4).
  • Because the proof never assumes the positive characteristic root is constant along the boundary, the local estimates should carry over to geometric settings where the indicial root varies from point to point on the conformal boundary, a case the paper explicitly notes its PDE method is designed to handle.
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.

Referee Report

4 major / 5 minor

Summary. The paper develops a global Schauder-type regularity theory for uniformly degenerate elliptic operators of the form L = ρ^2 a^{ij} ∂_{ij} + ρ b^i ∂_i + c in bounded domains, where ρ is a defining function. The central result, Theorem 2.2, asserts that under C^{k+1,α} boundary and C^{k,α} coefficients, with c ≤ −c0 and P(k+α) ≤ −c_{k+α} on the boundary, every bounded solution of Lu = f with boundary value u = f/c belongs to C^{k,α}(Ω̄), with ρ∇^{k+1}u and ρ^2∇^{k+2}u in C^α(Ω̄), and with estimate (2.10). The proof is organized through a flat-boundary local theory: decay estimates (Lemmas 3.1, 3.2), Hölder regularity (Theorem 4.3), tangential regularity (Theorem 5.2), and normal regularity (Theorem 6.2), followed by a regularization argument for existence (Theorem 2.3). Sharpness is illustrated by explicit monomial and logarithmic solutions in Example 2.1.

Significance. If the full statement holds, Theorem 2.2 is a substantial and useful result: it gives optimal boundary regularity in standard Hölder spaces for uniformly degenerate elliptic equations without imposing growth or decay assumptions on the data, and it includes the sharp boundary condition u = f/c and the vanishing of weighted first and second derivatives. The paper has notable strengths: the characteristic polynomial P(μ) is derived directly from the operator, the sharpness examples in Section 2 are explicit constructions rather than re-statements of the theorem, and the base cases of the local theory (Lemmas 3.1, 3.2, 4.2, Theorem 4.3, Theorem 5.2 for ℓ = 1, Lemma 6.1, and Theorem 6.2 for m = 1) are proved in detail. The main weakness is that the higher-order inductions that are essential for the global C^{k,α} statement for k ≥ 1 are not displayed, so the claimed full generality is not verifiable from the given arguments.

major comments (4)
  1. [Theorem 6.2, proof for general m] The proof of Theorem 6.2 for m ≥ 2 is omitted after the m = 1 case, with the sentence 'The proof for general m is based on induction and hence omitted.' This is a load-bearing gap for Theorem 2.2 when k ≥ 1: the C^{k,α} regularity of u and the estimates for ρ∇^{k+1}u and ρ^2∇^{k+2}u require control of normal derivatives ∂_t^m u for m up to k. The induction is not a routine repetition of the m = 1 argument, because for m = 2 one must differentiate (6.1) in t, isolate the ∂_t^2 u terms so that the new operator has the form t^2a^{ij}∂_{ij} + t(b^i + 4a^{in})∂_i + (c + 2b^n + 2a^{nn}), verify Q^{(2)}(μ) = Q(μ+2), prove that the new right-hand side f_2 lies in C^α(Ω̄), and, crucially, express the boundary value u_2 = f_2/(c + 2b^n + 2a^{nn}) using only f, coefficients, and lower-order derivatives of u. Without this displayed recursion, the argument could be circular if ∂_t^2 u appears in f_2. The same recursive construction is needed for the 'more generally' claim after Lemma 6.1 and for the step 'Theorem 6.2 implies Theorem 2.2.' I therefore cannot verify the k ≥ 1 part of the main theorem from the submitted text.
  2. [Theorem 5.2, proof for general ℓ] The proof of Theorem 5.2 for ℓ ≥ 2 is also omitted after the ℓ = 1 case, stated as 'The proof for the general ℓ is based on induction and hence omitted.' This induction is needed to obtain tangential regularity D_{x'}^τ u for all τ ≤ ℓ and hence for the full C^{k,α} conclusion in Theorem 2.2. The ℓ = 1 case uses the boundedness lemma (Lemma 5.1) and Theorem 4.3 applied to (5.1); the higher-order version requires an analogous construction in which differentiating (5.1) in a tangential direction produces a right-hand side in C^α(Ω̄) with estimates that depend only on the data and on the previously established lower-order regularity. The omitted induction is not a trivial formal exercise because the boundary value of D_{x'}^τ u is determined by the equation through u0 = f/c, and one must check that the recursion preserves that structure at every step.
  3. [Section 7, proof of Theorem 2.3] The proof of Theorem 2.3 is only sketched. Steps 2 and 3 assert, respectively, a boundary decay estimate and a weighted C^{2,α} estimate for the regularized solutions u_δ, with constants independent of δ, but no details are given. This is not a routine application of the flat-boundary theory because the regularized operator L_δ = L + δΔ has coefficients that are only C^α(Ω̄), the boundary is only C^{1,α}, and the uniform-in-δ estimates must control the passage to the limit in a way that preserves the boundary value u = f/c and the vanishing of ρ∇u and ρ^2∇^2u. Since Theorem 2.3 is the existence half of the claimed isomorphism L : C^{k,α}_2(Ω̄) → C^{k,α}(Ω̄), the absence of a complete argument is a load-bearing gap.
  4. [Sections 4–6, localization to the global theorem] The transition from the flat-boundary local results to the global statement of Theorem 2.2 is compressed into the sentence 'Theorem 6.2 implies Theorem 2.2 easily.' This transition requires a boundary-coordinate change that preserves the structure of the operator, a partition of unity, and a verification that the weighted tangential and normal estimates combine into the stated C^{k,α}(Ω̄) norms and the boundary conditions ρ∇^{k+1}u = 0 and ρ^2∇^{k+2}u = 0 on ∂Ω. Given that the higher-order normal regularity is the delicate part of the paper, the global assembly should be written out rather than left as an exercise.
minor comments (5)
  1. [Lemma 3.2, proof] In the display after 'A simple computation yields Q^{(−κ)}(μ) = Q(μ − κ)', the symbol Q1 should be G1.
  2. [Section 6, heuristic discussion before (6.1)] The text 'we write 4 anβ∂tβu = 2anβ∂tβu + 2anβ∂tβu' contains a typo: the first factor should likely be 2, not 4, to match the displayed operator L^{(1)}.
  3. [Lemma 4.1] The proof of Lemma 4.1 is declared 'standard and hence omitted.' This is acceptable for a calculus lemma, but since the lemma is used to convert pointwise boundary decay into global Hölder regularity, a one-sentence indication of the covering argument would improve readability.
  4. [Section 6, statement of Lemma 6.1] The hypotheses state Q(1 + α) ≤ −c_{1+α} in G1, while the text also uses Q(1) = b^n + c < 0; it would be clearer to state explicitly that Q(1) < 0 follows from the assumed negativity of Q(0) and Q(1+α), since that is why the denominator in (6.4) is nonzero.
  5. [General presentation] The paper is labeled a survey, but Theorem 2.2 and Theorem 2.3 are presented as new results with proofs; the relation to existing literature (e.g., Graham–Lee [25] and Mazzeo's theory) could be clarified regarding which parts are new and which are expository, but this does not affect the mathematical content.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the regularity theorem is derived from the operator's characteristic polynomial and maximum-principle estimates, not from its own conclusion.

full rationale

After tracing the derivation chain, I find no circular step. The hypotheses of Theorem 2.2 (C^{k,α} coefficients, c≤−c0, P(k+α)≤−c_{k+α}, and the compatibility boundary value u=f/c) are analytic conditions, and the conclusion (u, ρ∇^{k+1}u, ρ^2∇^{k+2}u in C^α with estimates) is not embedded in them. The characteristic polynomial Q(μ)=μ(μ−1)a_{nn}+μb_n+c is computed directly from the operator in Section 3, and the supersolutions in Lemma 3.1 are constructed from the assumed negativity of Q(σ) and Q(μ), not from the desired regularity. Lemma 3.2 derives boundary Hölder decay by maximum principle, and Lemmas 4.2, 5.1, and 6.1 successively upgrade regularity using already-established lower-order information; no fitted parameter is renamed as a prediction. The cited prior work is contextual (Loewner–Nirenberg, Graham–Lee, Mazzeo–Melrose), and the proof does not rest on a self-citation chain. The explicit omissions—'The proof for general m is based on induction and hence omitted' (Section 6, proof of Theorem 6.2) and the analogous statement in Theorem 5.2—are proof-completeness gaps rather than circularity, because the unstated induction would have to produce the higher normal-derivative estimates rather than assume them. Sharpness in Example 2.1 is an explicit construction with ψρ^s or ψρ^s log ρ, not a restatement of the theorem. The derivation is therefore self-contained as far as circularity is concerned.

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

No free parameters are fitted; the only spectral quantities, the roots of the characteristic polynomial, are determined by the operator and serve as the sharpness threshold. The paper introduces no new entities. The unstated background consists of standard elliptic PDE machinery plus the theorem's structural assumptions on the defining function and coefficients.

assumptions (5)
  • standard math Maximum principle holds for L despite degeneracy at the boundary.
    Invoked in Lemma 3.2 and Lemma 5.1; the authors assert in Section 7 that the degeneracy is only on the boundary, so the maximum principle remains valid.
  • standard math Interior Schauder estimates for uniformly elliptic operators with Holder coefficients hold and scale appropriately.
    Used in Lemma 4.2 and in the proofs of Lemmas 5.1 and 6.1 for interior C^{2,alpha} and C^1 estimates.
  • standard math A C^{k+1,alpha} domain can be locally straightened so the defining function is t and the operator takes the form (3.1).
    Assumed at the start of Section 3 and used throughout the proof; the global-to-local localization is not written out.
  • standard math The regularized operator L_delta = L + delta Delta has a unique classical solution by the Schauder existence theorem.
    Invoked in Section 7, proof of Theorem 2.3, to obtain the approximating solutions u_delta.
  • domain assumption A defining function rho with rho grad^{k+2}rho in C^alpha(Omega bar) and rho grad^{k+2}rho = 0 on the boundary exists under the stated boundary regularity.
    Assumed in Theorem 2.2; the existence of such a defining function is not proven in the paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Optimal Boundary Regularity for Uniformly Degenerate Elliptic Equations." pith.science (2026). https://pith.science/paper/MGIPH6RQ

@misc{pith2026241116418,
  author       = {Pith},
  title        = {Pith review of: Optimal Boundary Regularity for Uniformly Degenerate Elliptic Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MGIPH6RQ}},
  note         = {Machine review of arXiv:2411.16418}
}
read the original abstract

In this survey paper, we study the optimal regularity of solutions to uniformly degenerate elliptic equations in bounded domains and establish the H\"older continuity of solutions and their derivatives up to the boundary.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Schauder estimates for elliptic equations degenerating on lower dimensional manifolds

    math.AP 2025-01 conditional novelty 7.0 of 10

    Weak solutions to weighted elliptic equations with a distance-to-a-manifold weight are shown to be C^{0,α} or C^{1,α} up to the characteristic manifold under a homogeneous conormal condition.

  2. Sobolev estimates for parabolic and elliptic equations in divergence form with degenerate coefficients

    math.AP 2024-12 conditional novelty 7.0 of 10

    For degenerate parabolic and elliptic equations in divergence form on the upper half space, the paper establishes well-posedness and weighted mixed-norm Sobolev estimates under partially BMO coefficients.

Reference graph

Works this paper leans on

58 extracted references · 3 canonical work pages · cited by 2 Pith papers

  1. [1]

    Andersson, Elliptic systems on manifolds with asymptotically negativ e curvature, Indiana Univ

    L. Andersson, Elliptic systems on manifolds with asymptotically negativ e curvature, Indiana Univ. Math. J., 42(1993), 1359-1388

    1993

  2. [2]

    hyperboloidal boundary conditions

    L. Andersson, P. Chru´ sciel, Solutions of the constraint equations in general relativit y satisfying “hyperboloidal boundary conditions” , Dissertationes Math. (Rozprawy Mat.) 355(1996), 1-100

    1996

  3. [3]

    Andersson, P

    L. Andersson, P. Chru´ sciel, H. Friedrich, On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for E instein ’s field equations, Comm. Math. Phys., 149(1992), 587-612

    1992

  4. [4]

    Aviles, R

    P. Aviles, R. C. McOwen, Complete conformal metrics with negative scalar curvature in compact Riemannian manifolds , Duke Math. J., 56(1988), 395-398

    1988

  5. [5]

    Biquard, M´ etriques d’Einstein asymptotiquement sym´ etriques, Ast´ erisque, 265(2000)

    O. Biquard, M´ etriques d’Einstein asymptotiquement sym´ etriques, Ast´ erisque, 265(2000)

    2000

  6. [6]

    Biquard, M´ etriques autoduales sur la boule, Invent

    O. Biquard, M´ etriques autoduales sur la boule, Invent. Math., 148(2002), 545-607

    2002

  7. [7]

    Biquard, D´ esingularisation de m´ etriques d’Einstein II, Invent

    O. Biquard, D´ esingularisation de m´ etriques d’Einstein II, Invent. Math., 204(2016), 473-504

    2016

  8. [8]

    Caffarelli, X

    L. Caffarelli, X. Cabr´ e, Fully Nonlinear Elliptic Equations , Amer. Math. Soc., Providence, 1995

    1995

Show all 58 references
  1. [9]

    R. Chen, S. Li, and J. Luo, On the asymptotic expansions of the proper harmonic maps bet ween balls in Bergman metrics , J. Geom. Anal., 33(2023), Paper No. 114, 48

    2023

  2. [10]

    Cheng, S.-T

    S.-Y. Cheng, S.-T. Yau, Differential equations on Riemannian manifolds and their geo metric ap- plications, Comm. Pure Appl. Math., 28(1975), 333-354

    1975

  3. [11]

    Cheng, S.-T

    S.-Y. Cheng, S.-T. Yau, On the regularity of the Monge-Amp` ere equation det(∂ 2u/∂x i∂x j) = F (x, u ), Comm. Pure Appl. Math., 30(1977), 41-68

    1977

  4. [12]

    Cheng, S.-T

    S.-Y. Cheng, S.-T. Yau, On the existence of a complete K¨ ahler metric on non-compact complex manifolds and the regularity of Fefferman ’s equation , Comm. Pure Appl. Math., 33(1980), 507-544

    1980

  5. [13]

    Chru´ sciel, E

    P. Chru´ sciel, E. Delay, J. Lee, D. Skinner, Boundary regularity of conformally compact Einstein metrics, J. Diff. Geom., 69(2005), 111-136

    2005

  6. [14]

    Donnelly, Dirichlet problem at infinity for harmonic maps: rank one sym metric spaces, Trans

    H. Donnelly, Dirichlet problem at infinity for harmonic maps: rank one sym metric spaces, Trans. Amer. Math. Soc., 344(1994), 713-735

    1994

  7. [15]

    Fefferman, The Bergman kernel and biholomorphic mappings of pseudocon vex domains , Invent

    C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudocon vex domains , Invent. Math., 26(1974), 1-65

    1974

  8. [16]

    Fefferman, Monge-Amp` ere equation, the Bergman kernel, and geometry o f pseudoconvex do- mains, Ann

    C. Fefferman, Monge-Amp` ere equation, the Bergman kernel, and geometry o f pseudoconvex do- mains, Ann. Math., 103(1976), 395-416

    1976

  9. [17]

    Fefferman, C

    C. Fefferman, C. R. Graham, Conformal invariants , in The mathematical heritage of ´Elie Cartan , Ast´ erisque 1985, 95-116

    1985

  10. [18]

    Fefferman, C

    C. Fefferman, C. R. Graham, Q-curvature and Poincar´ e metrics, Math. Res. Lett., 9(2002), 139- 151

    2002

  11. [19]

    Fefferman, C

    C. Fefferman, C. R. Graham, The Ambient Metric , Annals of Mathematics Studies, 178, Princeton University Press, Princeton, 2012

    2012

  12. [20]

    Gilbarg, N

    D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Elliptic Type , Springer, Berlin, 1983

    1983

  13. [21]

    M. d. M. Gonz´ alez, Y.-Y. Li, L. Nguyen, Existence and uniqueness to a fully nonlinear version of the Loewner-Nirenberg problem, Commun. Math. Stat., 6(2018), 269-288

    2018

  14. [22]

    C. R. Graham, The Dirichlet problem for the Bergman Laplacian. I , Comm. P. D. E., 8(1983), 433-476

    1983

  15. [23]

    C. R. Graham, The Dirichlet problem for the Bergman Laplacian. II , Comm. P. D. E., 8(1983), 563-641

    1983

  16. [24]

    C. R. Graham, Volume and area renormalizations for conformally compact E instein metrics, Rend. Circ. Mat. Palermo, II. Ser., Suppl. 63(2000), 31-42

    2000

  17. [25]

    C. R. Graham, J. M. Lee, Einstein metrics with prescribed conformal infinity on the b all, Adv. Math., 87(1991), 186-225. 24 HAN AND XIE

    1991

  18. [26]

    C. R. Graham, E. Witten, Conformal anomaly of submanifold observables in AdS/CFT co rrespon- dence, Nuclear Physics B, 546(1999), 52-64

    1999

  19. [27]

    Q. Han, X. Jiang, Boundary regularity of minimal graphs in the hyperbolic spa ce, Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal), Vol. 20 23, no. 801(2023), 239-272

    2023

  20. [28]

    Q. Han, X. Jiang, Asymptotics and convergence for the complex Monge-Amp` ere equation, Ann. PDE, 10(2024), Paper No. 8, 64

    2024

  21. [29]

    Q. Han, W. Shen, Y. Wang, Optimal regularity of minimal graphs in the hyperbolic spac e, Cal. Var. & P. D. E., 55(2016), no. 1, Art. 3, 19pp

    2016

  22. [30]

    Q. Han, Y. Wang, Boundary expansions for constant mean curvature surfaces i n the hyperbolic space, Math. Z., 300(2022), 851-879

    2022

  23. [31]

    Helliwell, Boundary regularity for conformally compact Einstein metr ics in even dimensions , Comm

    D. Helliwell, Boundary regularity for conformally compact Einstein metr ics in even dimensions , Comm. P.D.E., 33(2008), 842-880

    2008

  24. [32]

    H. Jian, J. Lu, and X.-J. Wang, A boundary expansion of solutions to nonlinear singular ell iptic equations, Sci. China Math., 65(2022), 9-30

    2022

  25. [33]

    Jian, X.-J

    H. Jian, X.-J. Wang, Bernstein theorem and regularity for a class of Monge-Amp` e re equations, J. Diff. Geom., 93(2013), 431-469

    2013

  26. [34]

    Jian, X.-J

    H. Jian, X.-J. Wang, Optimal boundary regularity for nonlinear singular ellipt ic equations , Adv. Math., 251(2014), 111-126

    2014

  27. [35]

    Jian, X.-J

    H. Jian, X.-J. Wang, Y. Zhao, Global smoothness for a singular Monge-Amp` ere equation , J. Diff. Eq., 263(2017), 7250-7262

    2017

  28. [36]

    Kichenassamy, Boundary blow-up and degenerate equations , J

    S. Kichenassamy, Boundary blow-up and degenerate equations , J. Funct. Anal., 215(2004), 271-289

    2004

  29. [37]

    Kichenassamy, Boundary behavior in the Loewner-Nirenberg problem , J

    S. Kichenassamy, Boundary behavior in the Loewner-Nirenberg problem , J. Funct. Anal., 222(2005), 98-113

    2005

  30. [38]

    Kichenassamy, Fuchsian Reduction

    S. Kichenassamy, Fuchsian Reduction. Applications to Geometry, Cosmology, and Mathematical Physics, Progress in Nonlinear Differential Equations and their App lications, 71, Birkh¨ auser, Boston, 2007

    2007

  31. [39]

    Lasry, P.-L

    J.-M. Lasry, P.-L. Lions, Nonlinear elliptic equations with singular boundary condi tions and sto- chastic control with state constraints. I. The model proble m, Math. Ann., 283(1989), 583-630

    1989

  32. [40]

    Lee, The spectrum of an asymptotically hyperbolic Einstein mani fold, Comm

    J. Lee, The spectrum of an asymptotically hyperbolic Einstein mani fold, Comm. Anal. Geom., 3(1995), 253-271

    1995

  33. [41]

    Lee, Fredholm Operators and Einstein Metrics on Conformally Com pact Manifolds, Mem

    J. Lee, Fredholm Operators and Einstein Metrics on Conformally Com pact Manifolds, Mem. A. M. S., Providence, 2006

    2006

  34. [42]

    J. Lee, R. Melrose, Boundary behavior of the complex Monge-Amp` ere equation , Acta Math., 148(1982), 159-192

    1982

  35. [43]

    Leonori, A

    T. Leonori, A. Porretta, Gradient bounds for elliptic problems singular at the bound ary, Arch. Ration. Mech. Anal., 202(2011), 663-705

    2011

  36. [44]

    Li, L.-F

    P. Li, L.-F. Tam, The heat equation and harmonic maps of complete manifolds , Invent. Math., 105(1991), 1-46

    1991

  37. [45]

    Li, L.-F

    P. Li, L.-F. Tam, Uniqueness and regularity of proper harmonic maps , Ann. of Math., 137(1993), 167-201

    1993

  38. [46]

    Li, L.-F

    P. Li, L.-F. Tam, Uniqueness and regularity of proper harmonic maps. II , Indiana Univ. Math. J., 42(1993), 591-635

    1993

  39. [47]

    S. Li, L. Ni, On the holomorphicity of proper harmonic maps between unit b alls with the Bergman metrics, Math. Ann., 316(2000), 333-354

    2000

  40. [48]

    S. Li, E. Simon, On proper harmonic maps between strictly pseudoconvex doma ins with K¨ ahler metrics of Bergman type , Asian J. Math., 11(2007), 251-275

    2007

  41. [49]

    Y. Li, L. Nguyen, J. Xiong, Regularity of viscosity solutions of the σ k-Loewner-Nirenberg problem, Proc. Lond. Math. Soc., 127(2023), 1-34

    2023

  42. [50]

    Lin, On the Dirichlet problem for minimal graphs in hyperbolic sp ace, Invent

    F.-H. Lin, On the Dirichlet problem for minimal graphs in hyperbolic sp ace, Invent. Math., 96(1989), 593-612. UNIFORMLY DEGENERATE ELLIPTIC EQUATIONS 25

    1989

  43. [51]

    Lin, Erratum: On the Dirichlet problem for minimal graphs in hype rbolic space , Invent

    F.-H. Lin, Erratum: On the Dirichlet problem for minimal graphs in hype rbolic space , Invent. Math., 187(2012), 755-757

    2012

  44. [52]

    Loewner, L

    C. Loewner, L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, Contributions to Analysis, 245-272, Academic Press, New Y ork, 1974

    1974

  45. [53]

    Mazzeo, Elliptic theory of differential edge operators I , Comm

    R. Mazzeo, Elliptic theory of differential edge operators I , Comm. P.D.E., 16(1991), 1615-1664

    1991

  46. [54]

    Mazzeo, Regularity for the singular Yamabe problem , Indiana Univ

    R. Mazzeo, Regularity for the singular Yamabe problem , Indiana Univ. Math. Journal, 40(1991), 1277-1299

    1991

  47. [55]

    Mazzeo, R

    R. Mazzeo, R. Melrose, Meromorphic extension of the resolvent on complete spaces w ith asymptot- ically negative curvature , J. Funct. Anal., 75(1987), 260-310

    1987

  48. [56]

    Melrose, Transformation of boundary problems , Acta Math., 147(1981), 149-236

    R. Melrose, Transformation of boundary problems , Acta Math., 147(1981), 149-236

    1981

  49. [57]

    Tonegawa, Existence and regularity of constant mean curvature hypers urfaces in hyperbolic space, Math

    Y. Tonegawa, Existence and regularity of constant mean curvature hypers urfaces in hyperbolic space, Math. Z., 221(1996), 591-615

    1996

  50. [58]

    Yin, Boundary regularity of harmonic maps from hyperbolic space into nonpositively curved manifolds, Pacific J

    H. Yin, Boundary regularity of harmonic maps from hyperbolic space into nonpositively curved manifolds, Pacific J. Math., 232(2007), 491-509. Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA Email address : qhan@nd.edu Beijing International Center ...

    2007

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.