Pith. sign in

REVIEW 4 major objections 6 minor 57 references

Schauder estimates for elliptic equations degenerating on lower dimensional manifolds

T0 review · 4 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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

Pith's one-line read The paper proves explicit Hölder and Schauder estimates up to a characteristic manifold of codimension at least two, via perforation, blow-up, and a new Liouville theorem.

desk verdict Genuine extension of Schauder theory to codimension n≥2 with a new anisotropic perforation scheme; the main structure holds up, but two load-bearing estimates are deferred to a companion preprint or to 'standard' status. read the letter →

arxiv 2501.19033 v2 pith:2PMFRUBP submitted 2025-01-31 math.AP

Gabriele Cora , Gabriele Fioravanti , Stefano Vita This is my paper · ORCID
classification math.AP MSC 35B6535J7035J7535B4035B4435B4535B53
keywords weightedellipticequationsdegenerateellipticitySchauderregularityestimateslowerdimensionalboundariesLiouvilletheoremsperforateddomainscodimensionatleasttwo
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

Elliptic equations whose coefficients vanish or blow up like $|y|^a$ on a $(d-n)$-dimensional plane lose uniform ellipticity exactly on that set. This paper proves that, as long as $a+n>0$, weak solutions satisfying a natural homogeneous conormal condition are Hölder continuous up to the singular set, and in a parameter range singled out by an explicit exponent $\alpha_*$ they are $C^{1,\alpha}$ up to it. The proof replaces the singular set by a small anisotropic hole, solves a Neumann problem on the perforated domain, and shows the estimates are stable as the hole shrinks to zero. If the argument is correct, it gives the first Schauder theory for degeneracies on manifolds of codimension $n\ge 2$, plus $\varepsilon$-uniform stable estimates that are new even for the Laplacian. The theory extends to equations degenerating on smooth curved thin manifolds.

What carries the argument

The load-bearing object is the anisotropic perforation $\Sigma^A_\varepsilon = \{A_3^{-1}(x,y)y\cdot y \le \varepsilon^2\}$, an $\varepsilon$-neighborhood of $\Sigma_0$ shaped by the lower-right block $A_3$ of the coefficient matrix; on its boundary the approximating solution is given the conormal Neumann condition $(A\nabla u_\varepsilon+F)\cdot \nu=0$. The estimates are proved by contradiction: if the $\varepsilon$-uniform bound failed, a fine blow-up around points near $\partial\Sigma^A_\varepsilon$ would produce an entire limiting profile defined either on $\mathbb{R}^d$, a half-space, or $\mathbb{R}^d$ minus an unbounded cylinder, and the new Liouville theorem rules out every nonconstant or non-linear profile of the allowed growth. The quantitative engine inside the Liouville theorem is the spectral gap of a weighted spherical operator: its first nontrivial eigenvalue $\mu_1$, bounded below via Lemma 5.2, enters the explicit threshold $\alpha_*$ in (1.7), and the condition $\alpha_*>1$ is exactly what forces the limiting gradients to be constant.

What would settle it

Compute the model profile $u_i(y)=y_i|y|^{\gamma_1^+-1}$ for the isotropic case $A=I$ with $a<0$ and $n=2$, where $\gamma_1^+=\alpha_*$ as defined in (1.7). By construction this profile should be a weak solution of the homogeneous conormal problem with growth exactly at the threshold; its second derivatives behave like $|y|^{\gamma_1^+-2}$, so if the exponent $\alpha_*$ is correct the $C^{1,\alpha}$ regularity in Theorem 1.2 cannot hold for $\alpha>\gamma_1^+-1$. Checking that the computed blow-up rate matches this prediction confirms the threshold, while any discrepancy in the rate, or any failure of the profile to satisfy the conormal condition, would refute the claimed exponent.

Watch

Extended reading notes

Core claim

For the equation $-\mathrm{div}(|y|^a A(x,y)\nabla u)=|y|^a f+\mathrm{div}(|y|^aF)$ in $B_1$, with $2\le n\le d$ and $a\in\mathbb{R}$, Theorem 1.1 says that whenever $a+n>0$ every weak solution in $H^{1,a}$ that solves across $\Sigma_0=\{y=0\}$ belongs to $C^{0,\alpha}_{\mathrm{loc}}(B_1)$ for the explicit range (1.8). Theorem 1.2 is the central claim: if the threshold $\alpha_*$ defined in (1.7), built from $a$, $n$, and the restricted ellipticity ratio $\lambda_*/\Lambda_*$, exceeds 1, then for $\alpha$ satisfying (1.9), every such solution is $C^{1,\alpha}_{\mathrm{loc}}(B_1)$, satisfies the pointwise conormal condition $(A\nabla u+F)\cdot e_{y_i}=0$ on $\Sigma_0$, and obeys estimate (1.11) with constants independent of $u$. The supporting Theorem 1.3 shows that the approximating Neumann problems on the perforated domains $B_1\setminus\Sigma^A_\varepsilon$ admit $\varepsilon$-uniform $C^{0,\alpha}$ and $C^{1,\alpha}$ bounds together with the quantitative boundary estimate (1.14), while Theorem 1.4 classifies entire solutions on the perforated space: sublinear growth forces constants and subquadratic growth forces linear functions. Corollaries 8.4 and 8.5 carry the flat result to equations whose weight degenerates on $C^{1,\alpha}$ curved manifolds.

Load-bearing premise

The stability of the estimates in the perforated domains depends on the boundary of the anisotropic hole having a well-controlled normal, which requires the lower-right coefficient block $A_3$ to be at least $C^1$ ($C^{1,\alpha}$ in the Schauder case); if $A_3$ were only continuous at that stage, the $\varepsilon$-uniform trace and Sobolev inequalities used in the blow-up contradiction would fail, and the paper later removes this extra assumption only by a separate mollification argument.

Editorial extensions

If this is right

  • For every $a+n>0$, weak solutions of the homogeneous conormal problem are Hölder continuous up to the characteristic manifold, with an explicit exponent range depending on the ellipticity ratio restricted to $\Sigma_0$.
  • When $\alpha_*>1$, weak solutions are $C^{1,\alpha}$ up to $\Sigma_0$ and satisfy the strong conormal condition $(A\nabla u+F)\cdot e_{y_i}=0$ pointwise, not just in the weighted sense of (1.3).
  • The $\varepsilon$-uniform stable estimates in perforated domains are new even for the Laplacian with Neumann holes and give stable $\alpha$-Hölder bounds for eigenfunctions in Neumann-perforated domains.
  • Sublinear entire solutions of the degenerate problem on the perforated space are constant and subquadratic ones are linear, which is the rigidity fact powering the blow-up argument.
  • The flat results transfer to equations degenerating on smooth curved thin manifolds of codimension $n\ge 2$, with the conormal condition normal to the manifold.

Reading between the lines

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

  • The explicit exponent $\alpha_*$ suggests an optimal-Hölder-exponent picture for this degeneracy analogous to known optimal exponents for bounded measurable coefficients; the paper notes it remains open whether the reduction of $\alpha_*$ by $\lambda_*/\Lambda_*$ is intrinsic or an artifact of the method.
  • Because Theorem 1.3(i) is stable in $\varepsilon$, it should yield quantitative rates for Neumann eigenvalue problems in domains with small holes, converting the known qualitative convergence into explicit control of eigenfunctions near the holes.
  • The failure of $C^{2,\alpha}$ stability under perforation documented in Remark 6.2 indicates that higher regularity, if true, needs a different regularization than hole removal; one testable route is to iterate only tangential derivatives and impose axial symmetry in $y$, under which the operator commutes enough to bootstrap.
  • The pointwise conormal condition (1.10) gives a higher-codimension analogue of the boundary condition appearing in fractional-Laplacian extension problems, so the estimates may serve as a regularity tool for very thin obstacle problems with obstacles of codimension at least two.
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 / 6 minor

Summary. The paper develops a local regularity theory for degenerate/singular elliptic equations in divergence form whose weight is a power of the distance to a lower-dimensional set Sigma_0 of codimension n>=2. The main results are C^{0,alpha} (Theorem 1.1) and C^{1,alpha} (Theorem 1.2) estimates up to Sigma_0 under explicit assumptions on the data and a critical exponent alpha* defined through the ellipticity constants restricted to Sigma_0. The proof strategy is a regularization-approximation scheme: coefficients are mollified, the domain is perforated around Sigma_0 by anisotropic holes adapted to the matrix A, and epsilon-uniform regularity estimates (Theorem 1.3) are obtained by contradiction, blow-up, and a new Liouville theorem in perforated space (Theorem 1.4). A separate a priori estimate (Proposition 7.1) removes the extra C^1 or C^{1,alpha} regularity of the coefficients needed for the perforation geometry, yielding the final theorems under the optimal assumptions on A. The paper also extends the results to curved characteristic manifolds (Corollaries 8.4 and 8.5).

Significance. If correct, this is the first Schauder theory for elliptic equations degenerating on characteristic manifolds of codimension n>=2, and it gives a quantitative conormal boundary condition on Sigma_0 that is new even for the Laplacian in the appropriate parameter range. The epsilon-stable estimates in perforated domains, with holes whose shape follows the anisotropy of A, are of independent interest, and the connection to spectral stability for Neumann-perforated domains is a valuable byproduct. The paper is carefully written and contains a substantial amount of original technical work, including weighted functional inequalities, geometry of the perforated domains, and a Liouville theorem for entire solutions in perforated spaces. The main weakness is not an internal inconsistency but the deferral of a few key estimates to companion papers or standard references without full proofs.

major comments (4)
  1. [Sec. 4.2, Prop. 4.6] Proposition 4.6 (epsilon-stable L-infinity bounds) is stated without proof, with the text saying the proof is carried out in [29, Sec. 2.4]. This estimate is used in an essential way in Step 1 of the proofs of Theorem 1.3(i) and (ii), and it feeds into (6.22), which is then used throughout the C^{1,alpha} proof and in Proposition 7.1. Because Theorem 1.3 is the engine of the paper, this omission leaves a load-bearing gap. The authors should provide a self-contained proof or state precisely which statements of [29] they use and confirm that [29] does not itself rely on results of the present paper.
  2. [Sec. 5, Prop. 5.1] Proposition 5.1, which asserts that an entire solution to (1.15) with growth |u(z)| <= c(1+|z|^gamma) is a polynomial of degree at most floor(gamma) in the x-variables, is used essentially in the proof of Theorem 1.4, both in the gamma<1 and gamma<2 cases. Its proof is omitted with a reference to a standard difference-quotients technique and to [52, Corollary 4.2, Lemma 4.3]. Since Theorem 1.4 is one of the paper's new Liouville theorems and is invoked in every blow-up contradiction argument, this proof should be included, or the exact statement from [52] should be quoted and the hypotheses verified for the present weighted, perforated setting.
  3. [Sec. 6.2, Step 5 of the proof of Thm. 1.3(ii)] In Step 5, when assuming r_k -> \bar{r} > 0, the paper claims that "Omega_infty = B_{1/\bar{r}}(ς) \setminus \Sigma_0 for some ς". This is not correct in general: if \bar{\varepsilon} = lim \varepsilon_k/r_k > 0, the limiting hole is the cylinder \Sigma^{\bar{A}}_{\bar{\varepsilon}} = { \bar{A}_3^{-1} y \cdot y \leq \bar{\varepsilon}^2 }, not the set \Sigma_0. The subsequent contradiction that \bar{v} is linear follows from the estimates already established and does not require the exact domain characterization, so the claim can be corrected or removed; as written it is a flaw in the proof of r_k -> 0.
  4. [Sec. 7, Prop. 7.1(i) and Thm. 1.1] Theorem 1.1 and Proposition 7.1(i) are stated with the proof omitted, with the text saying "the proof for the other case follows the same argument and is easier to establish." Since Theorem 1.1 is one of the two main theorems of the paper, a proof or a detailed reduction to the estimates already established should be included; otherwise the Holder regularity result rests on an unproved a priori estimate that is not available in the existing literature in this form.
minor comments (6)
  1. [Sec. 3.3, Lemma 3.6] In the definition of G_epsilon, the expression "epsilon log epsilon" for the case a+n=2 should be "epsilon |log epsilon|" (or the absolute value should be noted), since the constant must be positive.
  2. [Sec. 3.1.1] The notation C^\infty_c(B_R \setminus \Sigma^A_\varepsilon) is defined twice with different support conditions; please use distinct symbols (e.g., different subscripts) to avoid ambiguity.
  3. [Sec. 7.2, proof of Thm. 1.2] In Step 1, the sentence "By continuity, we can extend v in the whole B_{3/4}" should read "extend u" rather than "extend v", as the letter v is not otherwise introduced there.
  4. [Sec. 6.2, Eq. (6.15)] In the change of variables leading to (6.15), the Jacobian determinant factor is not written explicitly; adding it would improve clarity and make the estimate easier to follow.
  5. [Sec. 2.2, Prop. 2.3] Inequality (3.3) in Proposition 3.5 is missing the volume element dz in the integrand on the left-hand side.
  6. [Sec. 5, Lemma 5.2] The lower bound for mu_1 in the case n=2 is cited to [47, Lemma 1]; please confirm that the formula used in (5.4) exactly matches the statement of that lemma, since a small mismatch in the exponent |a|/4 would propagate into the definition of alpha*.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main Schauder and Holder estimates are proved by internal blow-up/compactness arguments, and the deferred standard lemmas and companion-paper citations do not reduce the result to its own inputs.

full rationale

The central claims (Theorems 1.1, 1.2, and 1.3) are established by an internal regularization-perforation-blow-up scheme. The Liouville theorem used in the blow-up step is proved in Section 5 via spectral decomposition, and the codimension-1 half-space Liouville input is cited from the independent prior work [52, Theorem 1.6], whose assumptions concern codimension 1 and do not include the present codimension n >= 2 conclusions. The omitted proofs of Proposition 4.6 (epsilon-stable L-infinity bounds) and Proposition 5.1 (polynomial growth reduction in x) are explicitly deferred to [29] and to standard difference-quotient/Caccioppoli arguments respectively; these are auxiliary regularity ingredients, not the target Schauder statement, and they are not fitted to the data or derived from the conclusion. The only regularity gap flagged by the reader, namely the C^1 requirement on A_3 for the perforation geometry, is bypassed by the two-step argument: one first proves estimates for mollified coefficients and then applies the a priori estimate Proposition 7.1 under merely C^{0,alpha} assumptions, so the perforation geometry is not load-bearing for Theorems 1.1 and 1.2. No parameter is fitted to a subset of data and renamed as a prediction, and the exponent alpha_* is defined by an explicit spectral formula independent of any solution. The paper is self-contained modulo standard functional inequalities and published companion results; the self-citations are not load-bearing reductions.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No parameters are fitted to data. The paper introduces the perforation parameter ε as a regularization parameter, not as a fitted constant, and the exponent α* is derived from an eigenvalue estimate. No new physical entities are postulated; the perforated domain Σ^A_ε is a mathematical construction.

assumptions (7)
  • domain assumption The weight |y|^a is locally integrable at Σ0, i.e. a+n>0, and the codimension satisfies n≥2.
    This is the standing assumption for Theorems 1.1, 1.2, 1.3, and 1.4; it makes the H=W property and Poincaré-Wirtinger inequality valid (Section 2).
  • domain assumption Uniform ellipticity of A in the whole ball and restricted ellipticity of the lower-right block A_3 on Σ0, with 0<λ≤λ*≤Λ*≤Λ.
    Assumptions (1.5) and (1.6); the restricted ratio λ*/Λ* enters the exponent α* in (1.7).
  • domain assumption A is C0, C0,α, C1, or C1,α according to the theorem being used.
    C1/C1,α is needed for the perforated domain geometry (Remark 3.2), C0/C0,α for the final statements.
  • domain assumption Integrability conditions p>(d+a+)/2 and q>d+a+ for the data.
    Used for L∞ estimates and for vanishing of the right-hand side in blow-up; see Theorem 1.1 and Theorem 1.2.
  • standard math Classical weighted functional inequalities: Hardy-Poincaré, Poincaré, Sobolev, Caffarelli-Kohn-Nirenberg, and compact embedding for the weight |y|^a.
    Proved in Sections 2 and 3, often with references for the unweighted or known cases.
  • standard math Liouville theorems for uniform elliptic equations in the whole space and half-space with homogeneous conormal condition, including [52, Theorem 1.6].
    Used to rule out nonconstant blow-up profiles in Cases 1 and 2 of Theorem 1.3; [52] is cited as external but overlaps with the authors.
  • domain assumption In the curved case, the defining function δ must satisfy Definition 8.1, i.e. be comparable to the distance to Γ and have a C0,α ratio after the change of variables.
    Needed in Corollaries 8.4 and 8.5 to bring the curved problem to the flat model.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Schauder estimates for elliptic equations degenerating on lower dimensional manifolds." pith.science (2026). https://pith.science/paper/2PMFRUBP

@misc{pith2026250119033,
  author       = {Pith},
  title        = {Pith review of: Schauder estimates for elliptic equations degenerating on lower dimensional manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2PMFRUBP}},
  note         = {Machine review of arXiv:2501.19033}
}
abstract

In this paper we begin exploring a local regularity theory for elliptic equations having coefficients which are degenerate or singular on some lower dimensional manifold $$ -\mathrm{div}(|y|^aA(x,y)\nabla u)=|y|^af+\mathrm{div}(|y|^aF)\qquad\mathrm{in \ } B_1\subset\mathbb R^d, $$ where $z=(x,y)\in\mathbb R^{d-n}\times\mathbb R^n$, $2\leq n\leq d$ are two integers and $a\in\mathbb R$. Such equations are a prototypical example of elliptic equations spoiling their uniform ellipticity on the (possibly very) thin characteristic manifold $\Sigma_0=\{|y|=0\}$ of dimension $0\leq d-n\leq d-2$, having $$\lambda|y|^a|\xi|^2\leq |y|^aA(x,y)\xi\cdot\xi\leq\Lambda|y|^a|\xi|^2.$$ Whenever $a+n>0$, the weak solutions with a homogeneous conormal boundary condition at $\Sigma_0$ are provided to be $C^{0,\alpha}$ or even $C^{1,\alpha}$ regular up to $\Sigma_0$. Our approach relies on a regularization-approximation scheme which employs domain perforation, very fine blow-up procedures, and a new Liouville theorem in the perforated space. Our theory extends to the case of equations degenerating on suitably smooth curved manifolds.

Figures

Figures reproduced from arXiv: 2501.19033 by the authors.

Figure 1
Figure 1. This image describes Case 1 when n = d = 2. In this case we have dk/rk → ∞ and z˜k = zk. ẑk zk rk |yk| 0 ∂ΣA εk dk ẑk zk rk |yk| 0 ∂ΣA εk dk z 0 k z 0 k [PITH_FULL_IMAGE:figures/full_fig_p046_1.png] view at source ↗
Figure 2
Figure 2. The images on the left and on the right describe respectively Case 2 and Case 3 when n = d = 2. In Case 2 we have dk/rk ≤ c, |yk|/rk → ∞ and z˜k = z 0 k . In Case 3 we have |yk|/rk ≤ c and z˜k = 0. We introduce the sequence of rescaled domains Ωk = B1 \ Σ A εk − z˜k rk = n z = (x, y) ∈ R d | |z˜k + rkz| < 1, A−1 3 (˜zk + rkz)(˜yk + rky) · (˜yk + rky) ≥ ε 2 k o , and the limit blow-up domain (6.5) Ω∞ = {z = (x, y) ∈ … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

57 extracted references · 54 canonical work pages

  1. [29]

    G.Fioravanti, TheDirichletproblemonlowerdimensionalboundaries: Schauderestimatesviaperforateddomains , Preprint (2024) arXiv:2412.11294

    work page arXiv 2024

  2. [52]

    Terracini, G

    S. Terracini, G. Tortone and S. Vita,Higher order boundary Harnack principle via degenerate equations, Arch. Ration. Mech. Anal. 248 (2024), no. 2, Paper No. 29, 44 pp

    work page 2024

  3. [1]

    Abatangelo, A

    L. Abatangelo, A. Ferrero and P. Luzzini,On solutions to a class of degenerate equations with the Grushin operator, Preprint (2024), arXiv:2410.12637

    arXiv 2024

  4. [2]

    Audrito, G

    A. Audrito, G. Fioravanti and S. Vita,Schauder estimates for parabolic equations with degenerate or singular weights, Calc. Var. Partial Differential Equations63 (2024), no. 8, Paper No. 204, 46 pp

    work page 2024

  5. [3]

    Audrito, G

    A. Audrito, G. Fioravanti and S. Vita,Higher order Schauder estimates for degenerate or singular parabolic equations, Rev. Mat. Iberoam.41 (2025), 42 pp

    work page 2025

  6. [4]

    Bakry, T

    D. Bakry, T. Coulhon, M. Ledoux and L. Saloff-Coste,Sobolev inequalities in disguise, Indiana Univ. Math. J.44 (1995), no. 4, 1033–1074

    work page 1995

  7. [5]

    Caffarelli, R

    L. Caffarelli, R. V. Kohn and L. Nirenberg,First order interpolation inequalities with weights, Compositio Math.53 (1984), no. 3, 259–275

    work page 1984

  8. [6]

    7-9, 1245–1260

    L.CaffarelliandL.Silvestre, AnextensionproblemrelatedtothefractionalLaplacian , Comm.PartialDifferentialEquations 32 (2007), no. 7-9, 1245–1260

    work page 2007

Show all 57 references
  1. [7]

    Caffarelli and P

    L. Caffarelli and P. Stinga,Fractional elliptic equations, Caccioppoli estimates and regularity, Ann. Inst. H. Poincaré C Anal. Non Linéaire33 (2016), no. 3, 767–807

    2016

  2. [8]

    Castro,Hardy-Sobolev-type inequalities with monomial weights, Ann

    H. Castro,Hardy-Sobolev-type inequalities with monomial weights, Ann. Mat. Pura Appl. (4)196 (2017), no. 2, 579–598

    2017

  3. [9]

    S.-Y. A. Chang and M. M. González,Fractional Laplacian in conformal geometry, Adv. Math.226 (2011), no. 2, 1410–1432

    2011

  4. [10]

    J. L. Chern and C. S. Lin,Minimizers of Caffarelli-Kohn-Nirenberg inequalities with the singularity on the boundary, Arch. Ration. Mech. Anal.197 (2010), no. 2, 401–432. ELLIPTIC EQUATIONS DEGENERATING ON LOWER DIMENSIONAL MANIFOLDS 79

    2010

  5. [11]

    G. Cora, G. Fioravanti and S. Vita,Remarks on elliptic equations degenerating on lower dimensional manifolds, Preprint (2025)

    2025

  6. [12]

    Coulhon, A

    T. Coulhon, A. A. Grigor’yan and D. Levin,On isoperimetric profiles of product spaces, Comm. Anal. Geom.11 (2003), no. 1, 85–120

    2003

  7. [13]

    Dahlberg,Estimates of harmonic measure, Arch

    B. Dahlberg,Estimates of harmonic measure, Arch. Rational Mech. Anal.65 (1977), no. 3, 275–288

    1977

  8. [14]

    Z. Dai, J. Feneuil and S. Mayboroda,The regularity problem in domains with lower dimensional boundaries, J. Funct. Anal. 284 (2023), no. 11, Paper No. 109903, 72 pp

    2023

  9. [15]

    David, J

    G. David, J. Feneuil and S. Mayboroda,Dahlberg’s theorem in higher co-dimension, J. Funct. Anal.276 (2019), no. 9, 2731–2820

    2019

  10. [16]

    David, J

    G. David, J. Feneuil and S. Mayboroda,Elliptic theory for sets with higher co-dimensional boundaries, Mem. Amer. Math. Soc. 274 (2021), no. 1346, vi+123 pp

    2021

  11. [17]

    David and S

    G. David and S. Mayboroda,Approximation of Green functions and domains with uniformly rectifiable boundaries of all dimensions, Adv. Math.410 (2022), Paper No. 108717, 52 pp

    2022

  12. [18]

    De Silva and O

    D. De Silva and O. Savin,A note on higher regularity boundary Harnack inequality, Discrete Contin. Dyn. Syst.35 (2015), no. 12, 6155–6163

    2015

  13. [19]

    Dieci, A

    L. Dieci, A. Papini and A. Pugliese, Decompositions and coalescing eigenvalues of symmetric definite pencils depending on parameters, Numer. Algorithms91 (2022), no. 4, 1879–1910

    2022

  14. [20]

    H. Dong, S. Jeon and S. Vita,Schauder type estimates for degenerate or singular elliptic equations with DMO coefficients, Calc. Var. Partial Differential Equations63 (2024), no. 9, Paper No. 239, 42 pp

    2024

  15. [21]

    Dong and T

    H. Dong and T. Phan,Parabolic and elliptic equations with singular or degenerate coefficients: the Dirichlet problem, Trans. Amer. Math. Soc.374 (2021), no. 9, 6611–6647

    2021

  16. [22]

    Dong and T

    H. Dong and T. Phan,On parabolic and elliptic equations with singular or degenerate coefficients, Indiana Univ. Math. J. 72 (2023), no. 4, 1461–1502

    2023

  17. [23]

    L.Evans, Partialdifferentialequations , secondedition, GraduateStudiesinMathematics, 19, Amer.Math.Soc., Providence, RI, 2010

    2010

  18. [24]

    Fabes, D

    E. Fabes, D. Jerison and C. Kenig,The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier (Grenoble)32 (1982), no. 3, vi, 151–182

    1982

  19. [25]

    Fabes, D

    E. Fabes, D. Jerison and C. Kenig,Boundary behavior of solutions to degenerate elliptic equations, In Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, III., 1981), 577–589, Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983

    1981

  20. [26]

    Fabes, C

    E. Fabes, C. Kenig and R. Serapioni,The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), no. 1, 77–116

    1982

  21. [27]

    Felli, L

    V. Felli, L. Liverani and R. Ognibene,Quantitative spectral stability for the Neumann Laplacian in domains with small holes, J. Funct. Anal.288 (2025), no. 6, Paper No. 110817

    2025

  22. [28]

    Fernández-Real and Y

    X. Fernández-Real and Y. Jhaveri,On the singular set in the thin obstacle problem: higher order blow-ups and the very thin obstacle problem, Anal. PDE14 (2021), no. 5, 1599–1669

    2021

  23. [30]

    Garofalo,Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension, J

    N. Garofalo,Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension, J. Differential Equations104 (1993), no. 1, 117–146

    1993

  24. [31]

    A. A. Grigor’yan,Isoperimetric inequalities for Riemannian products, Math. Notes38 (1985), 849-854

    1985

  25. [32]

    Han and J

    Q. Han and J. Xie, Optimal boundary regularity for uniformly degenerate elliptic equations , Preprint (2024) arXiv:2411.16418

    2024 arXiv

  26. [33]

    Han and J

    Q. Han and J. Xie, Uniformly degenerate elliptic equations with varying characteristic exponents , Preprint (2024) arXiv:2411.17016

    2024 arXiv

  27. [34]

    Heinonen, T

    J. Heinonen, T. Kilpeläinen and O. Martio,Nonlinear potential theory of degenerate elliptic equations, Dover, Mineola, NY, 2006

    2006

  28. [35]

    Y. Hu, W. Wu, D. Yang and J. Zhong,Approximation of elliptic equations with interior single-point degeneracy and its application to weak unique continuation property, Preprint (2025) arXiv:2501.10923. 80 GABRIELE CORA, GABRIELE FIORA V ANTI, STEF ANO VITA

    2025 arXiv

  29. [36]

    Kilpeläinen,Weighted Sobolev spaces and capacity, Ann

    T. Kilpeläinen,Weighted Sobolev spaces and capacity, Ann. Acad. Sci. Fenn. Ser. A I Math.19 (1994), no. 1, 95–113

    1994

  30. [37]

    Li and G

    Y.-Y. Li and G. Tian,Nonexistence of axially symmetric, stationary solution of Einstein Vacuum Equation with discon- nected symmetric event horizon, Manuscripta Math.73 (1991), no. 1, 83–89

    1991

  31. [38]

    Li and G

    Y.-Y. Li and G. Tian,Regularity of harmonic maps with prescribed singularities, Comm. Math. Phys.149 (1992), no. 1, 1–30

    1992

  32. [39]

    Li and G

    Y.-Y. Li and G. Tian, Harmonic maps with prescribed singularities, inDifferential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990), 317–326, Proc. Sympos. Pure Math., 54, Part 1, Amer. Math. Soc., Providence, RI

    1990

  33. [40]

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

    F.-H. Lin,On the Dirichlet problem for minimal graphs in hyperbolic space, Invent. Math.96 (1989), no. 3, 593–612

    1989

  34. [41]

    Maggi,Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory, Cambridge Studies in Advanced Mathematics, 135, Cambridge Univ

    F. Maggi,Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory, Cambridge Studies in Advanced Mathematics, 135, Cambridge Univ. Press, Cambridge, 2012

    2012

  35. [42]

    V. G. Maz’ya,Sobolev spaces with applications to elliptic partial differential equations, second, revised and augmented edition, Grundlehren der mathematischen Wissenschaften, 342, Springer, Heidelberg, 2011

    2011

  36. [43]

    Mazzeo,Elliptic theory of differential edge operators I, Comm

    R. Mazzeo,Elliptic theory of differential edge operators I, Comm. Partial Differential Equations16 (1991), no. 10, 1615– 1664

    1991

  37. [44]

    Mazzeo and B

    R. Mazzeo and B. Vertman,Elliptic theory of differential edge operators, II: boundary value problems, Indiana Univ. Math. J. 63 (2014), no. 6, 1911–1955

    2014

  38. [45]

    Nguyen,Singular harmonic maps and applications to general relativity, Comm

    L. Nguyen,Singular harmonic maps and applications to general relativity, Comm. Math. Phys.301 (2011), no. 2, 411–441

    2011

  39. [46]

    Ozawa,Asymptotic property of an eigenfunction of the Laplacian under singular variation of domains - the Neumann condition, Osaka J

    S. Ozawa,Asymptotic property of an eigenfunction of the Laplacian under singular variation of domains - the Neumann condition, Osaka J. Math.22 (1985), 639–655

    1985

  40. [47]

    L. C. Piccinini and S. Spagnolo,On the Hölder continuity of solutions of second order elliptic equations in two variables, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3)26 (1972), 391–402

    1972

  41. [48]

    Rauch and M

    J. Rauch and M. Taylor,Potential and scattering theory on wildly perturbed domains, J. Funct. Anal.18 (1975), 27–59

    1975

  42. [49]

    Simon,Schauder estimates by scaling, Calc

    L. Simon,Schauder estimates by scaling, Calc. Var. Partial Differential Equations5 (1997), no. 5, 391–407

    1997

  43. [50]

    Y. Sire, S. Terracini and S. Vita,Liouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions, Comm. Partial Differential Equations46 (2021), no. 2, 310–361

    2021

  44. [51]

    Y. Sire, S. Terracini and S. Vita,Liouville type theorems and regularity of solutions to degenerate or singular problems part II: odd solutions, Math. Eng.3 (2021), no. 1, Paper No. 5, 50 pp

    2021

  45. [53]

    Terracini, G

    S. Terracini, G. Tortone and S. Vita,A priori regularity estimates for equations degenerating on nodal sets, Preprint (2024) arXiv:2404.06980

    2024

  46. [54]

    Weinstein, On rotating black holes in equilibrium in general relativity, Comm

    G. Weinstein, On rotating black holes in equilibrium in general relativity, Comm. Pure Appl. Math. 43 (1990), no. 7, 903–948

    1990

  47. [55]

    Weinstein,The stationary axisymmetric two-body problem in general relativity, Comm

    G. Weinstein,The stationary axisymmetric two-body problem in general relativity, Comm. Pure Appl. Math.45 (1992), no. 9, 1183–1203

    1992

  48. [56]

    Zhang,On higher order boundary Harnack and analyticity of free boundaries, Nonlinear Anal.240 (2024), Paper No

    C. Zhang,On higher order boundary Harnack and analyticity of free boundaries, Nonlinear Anal.240 (2024), Paper No. 113471, 16 pp

    2024

  49. [57]

    G. Peano

    V. V. Zhikov,Weighted Sobolev spaces, Sb. Math.189 (1998), no. 7-8, 1139–1170. G. Cora, Département de Mathématique, Université Libre de Bruxelles, Boulev ard du Triomphe 155, 1050, Brussels, Belgium Email address: gabriele.cora@ulb.be G. Fiora v anti, Dipartimento di Matemati...

    1998

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.