Pith. sign in

REVIEW 3 major objections 5 minor 53 references

Regularity results for elliptic equations on cones

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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

Pith's one-line read This paper proves that a single spectral quantity—the first nontrivial eigenvalue of a spherical domain—decides whether solutions of elliptic equations on cones have bounded gradients.

desk verdict The linear threshold result is solid and new; the p-Laplace theorems are conditional on an unverified boundary-regularity estimate applied to a domain that likely doesn't satisfy its hypotheses. read the letter →

arxiv 2608.00199 v1 pith:RLKUXQGE submitted 2026-07-31 math.AP

Carlo Alberto Antonini , Filomena Pacella , Camilla Chiara Polvara , Luigi Provenzano This is my paper · ORCID
classification math.AP MSC 35B6535J6035J9235D30
keywords ellipticregularityconicalsingularitiessphericalsectorsp-LaplaciangradientboundsspectralthresholdLaplace–Beltramieigenvaluessecond-order
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

The paper studies solutions of Poisson and p-Laplace equations in a spherical sector—a cone cut out by a domain D on the unit sphere—and asks when their gradients stay bounded at the cone tip. Its central result is that for the linear Poisson problem, bounded gradients are governed by one number: the first nontrivial eigenvalue λ1(D) of the Laplace–Beltrami operator on D. If λ1(D) = N−1 every weak solution is Lipschitz; if λ1(D) > N−1 solutions are C^{1,α} with zero gradient at the vertex; if λ1(D) < N−1 a counterexample shows gradients can blow up. Thus the spectral condition replaces convexity, which had been the standard sufficient hypothesis. For p-Laplace equations the paper proves weighted global Lipschitz and second-order regularity under a separate scaling argument.

What carries the argument

The spectral decomposition of the Laplacian in the cone writes a solution as a regular part plus finitely many singular harmonic modes P_i = r^{ξ_i}a_i(θ), where the exponents satisfy ξ(ξ+N−2)=λ_i(D). The threshold λ1(D)=N−1 becomes ξ1=1, exactly the exponent separating Lipschitz from non-Lipschitz radial behavior. For the nonlinear theorems, the load is carried by scale invariance: rescaling solutions to unit-width annuli and applying boundary regularity estimates, the paper controls |x|∇u uniformly across all scales.

What would settle it

Take a smooth nonconvex spherical domain D with λ1(D) = N−1, solve the Poisson problem with a smooth right-hand side, and check whether the gradient remains bounded at the vertex; the paper predicts Lipschitz but generally not C¹ behavior. A single example with unbounded gradient at λ1(D) = N−1 would disprove the linear theorem, while an example at λ1(D) < N−1 with bounded gradient would disprove sharpness.

Watch

Extended reading notes

Core claim

The paper establishes that the regularity of solutions at the tip of a cone is not determined by convexity of the spherical cross-section but by the position of λ1(D) relative to N−1. The mechanism is an explicit spectral decomposition: near the vertex a solution behaves like a finite sum of harmonic functions r^{ξ_i}Y_i(θ), with exponents ξ_i computed from the eigenvalues λ_i(D); the smallest exponent ξ1 crosses 1 exactly when λ1(D) crosses N−1. Lipschitz regularity holds when ξ1 ≥ 1, and C^{1,α} with vanishing gradient when ξ1 > 1. For the p-Laplacian, the paper shows via scaling that |x|∇u is bounded and |x|^{1+α}∇u is Hölder, and that the stress field has weighted square-integrability.

Load-bearing premise

The nonlinear half of the paper depends on a quantitative C^{1,α} boundary-regularity estimate for p-Laplace problems that is taken from another article and is not proved or checked here; the linear threshold theorem, by contrast, rests on classical spectral expansions.

Editorial extensions

If this is right

  • If the linear theorem is correct, all weak solutions of the Poisson problem on spherical sectors with λ1(D) ≥ N−1 are globally Lipschitz, even when the cross-section is nonconvex.
  • The threshold is sharp: below N−1, unbounded gradients can occur even for smooth bounded data.
  • When λ1(D) > N−1, solutions are differentiable at the vertex with gradient zero, so the singularity of the cone is fully smoothed out.
  • For p-Laplace equations, the weighted gradient bound |x|∇u ∈ L∞ holds for every p > 1 and for both Dirichlet and Neumann conditions, with a similar weighted second-order estimate for the stress field.
  • The spectral condition connects to known symmetry results, suggesting that convexity in problems like the critical exponent equation and isoperimetric inequalities may be replaceable by λ1(D) ≥ N−1.

Reading between the lines

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

  • The sharp threshold suggests a broader principle: conical-domain regularity and symmetry results known for convex cross-sections should first be tested at λ1(D) ≥ N−1; the paper explicitly conjectures such an extension for symmetry problems.
  • One can numerically probe the equality case λ1(D) = N−1 on, say, a spherical lune on S², solving the Poisson equation with smooth data and checking whether the gradient is bounded yet not C¹; the paper's Remark 4.2 predicts such exceptional domains exist.
  • The nonlinear theorems would become fully self-contained if the imported boundary-regularity estimate were independently proved; until then, the linear threshold theorem is the robust core of the paper.
  • Because the argument uses only radial scaling and spectral expansion, the same threshold may transfer to other operators or to manifolds with conic singularities, but that is an extrapolation beyond the paper.
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

3 major / 5 minor

Summary. The paper studies regularity at the vertex of spherical sectors Σ_D∩B_2 in R^N. For the Poisson problem with Dirichlet or Neumann boundary conditions, Theorem 1.1 shows that if f∈L^q and the first eigenvalue λ1(D) of the Laplace–Beltrami operator on the spherical cross-section D satisfies λ1(D)≥N−1, then weak solutions are globally Lipschitz (λ1=N−1) or C^{1,α} (λ1>N−1), with an explicit finite expansion (1.4). The proof uses the spectral decomposition of Dauge and Kozlov–Maz'ya–Rossmann, together with the exact equivalence ξ1≥1 ⟺ λ1≥N−1. For p-Laplace equations, Theorems 1.3–1.4 claim weighted gradient bounds, weighted Hölder continuity of |x|^{1+α}∇u, and W^{1,2} regularity of the stress field, via a scaling argument that relies on a quantitative C^{1,α} estimate imported from the first author's preprint [3]. The paper also cites Maz'ya's example to argue sharpness of the spectral threshold.

Significance. The linear theorem is a clean and attractive result: it replaces convexity by the spectral condition λ1(D)≥N−1, and the connection to the leading singular exponent ξ1 is exact and parameter-free. The proof is grounded in classical spectral theory and appears essentially sound. If correct, this is likely to be useful for regularity theory in non-smooth domains. The nonlinear results are more conditional: their proof depends on an unpublished estimate, and as written applies it to a domain with corners. These issues must be resolved before the p-Laplace claims can be accepted, but they do not undermine the linear part.

major comments (3)
  1. [§3, Eq. (3.13)] The estimate (3.13) is invoked for v_R on Σ_{1/2,1}. Under Definition 2.1, Σ_{1/2,1} is not a C^{1,β} domain: its boundary has edges where the lateral boundary ∂Σ meets the spherical caps |x|=1 and |x|=1/2. Moreover, v_R satisfies a homogeneous boundary condition only on ∂Σ∩(B_1\B_{1/2}); on the caps it satisfies no boundary condition at all. Thus [3, Theorems 1.2–1.3] cannot be applied as stated. This estimate is the sole mechanism that yields the R-independent C^{1,α} bounds (3.18)–(3.22), and hence the conclusions (1.11)–(1.12). The proof must be repaired, for example by a localization that treats the caps as interior surfaces, or by a genuine regularity result for the mixed boundary value problem on the truncated cone. As written, Theorems 1.3 and 1.4 are not proven.
  2. [§3, dependency on [3]] The central quantitative ingredient (3.13) is taken from the first author's arXiv preprint [3], with no statement of the theorem's hypotheses and no proof. The scaling argument cannot close without this estimate. The authors should either include a complete proof of the imported estimate in an appendix, or state its precise hypotheses and verify them for the scaled solutions v_R. In particular, the issue of the artificial caps and the absence of boundary conditions on them is not addressed. Without access to the proof or a clear verification, the p-Laplace theorems rest on an unverified external result.
  3. [Remark 2.2] The assertion that ∂Σ_{r,R} is of class C^{1,β} is inconsistent with Definition 2.1. The boundary has dihedral-type edges at ∂Σ∩∂B_r and ∂Σ∩∂B_R. This is not merely a wording problem; it is exactly the reason why the application of (3.13) fails. The remark should be corrected to say 'piecewise C^{1,β}' or 'C^{1,β} away from the edges'.
minor comments (5)
  1. [Abstract] 'Weighted global lipschitzianity' should be 'weighted Lipschitz regularity' for clarity.
  2. [Eq. (1.4)] The Neumann/Dirichlet alternatives are typeset awkwardly; please clarify the summation index and the exponent notation.
  3. [Eq. (3.4)] The notation B_{7/4R} is ambiguous; write B_{7R/4}.
  4. [Author header] There is a typo in the header: 'POL V ARA' should be 'POLVARA'.
  5. [Lemma 3.2, Step 1] Citing [3] for the C^{1,α} regularity of v_{ε,k} on Σ_{t/4,3t/2} is not necessary; standard interior regularity (e.g., Lieberman [36]) suffices and avoids the same domain-regularity issue.

Circularity Check

2 steps flagged · score 4.0 of 10

Linear Theorem 1.1 is self-contained and non-circular; the nonlinear Theorems 1.3-1.4 rest on a load-bearing quantitative boundary-regularity estimate imported from the first author's own unpublished arXiv preprint [3].

  1. self citation load bearing [Section 3, Proof of Theorem 1.3, equation (3.13)]
    "Since Σ_{1/2,1} is a set of class C^{1,β}, owing to boundary regularity for the p-Laplace equation [3, Theorems 1.2-1.3] (see also [36]), we have that v_R ∈ C^{1,α}(Σ_{1/2,1}) for some α∈(0,1) ... together with the quantitative estimate ∥∇v_R∥_{C^{0,α}(Σ_{1/2,1})} ≤ C_1(...)."

    This is the key step that lets the annulus rescaling close: (3.14), (3.18), (3.20) and (3.22) all use the R-independent C^{1,α} bound (3.13). The paper does not prove this estimate; it cites [3], an arXiv preprint by the first author, with no machine-checked proof, code reproduction, or external verification supplied here. Thus the main p-Laplace conclusion is supported by a load-bearing self-citation rather than by an argument contained in or independently grounded in the paper.

  2. self citation load bearing [Section 3, Lemma 3.2, Step 1, equation (3.35)]
    "Next, by regularity theory for p-Laplace problems [3] combined with (3.33), (3.34), we have that v_{ε,k} ∈ C^{1,α}(Σ_{t/4,3t/2}) for every t∈(0,1), and for some α_k ∈(0,1) independent of ε, with quantitative estimate ∥v_{ε,k}∥_{C^{1,α_k}(Σ_{t/4,3t/2})} ≤ C_{k,t}."

    The second-order regularity result in Theorem 1.4 passes through Lemma 3.2, whose Step 1 needs a quantitative C^{1,α} bound on the regularized solutions v_{ε,k}. That bound is again delegated to [3] rather than proven. As with (3.13), the nonlinear chain therefore depends on an unverified same-author preprint at a load-bearing point.

full rationale

The paper's linear result, Theorem 1.1, is essentially non-circular: its proof reduces to the classical spectral asymptotics of Dauge [17] and Kozlov-Maz'ya-Rossmann [27], together with the algebraic equivalence ξ1≥1 ⇔ λ1(D)≥N−1. There is no fitted parameter, no prediction forced by construction, and no renaming of a known result. The sharpness discussion via Maz'ya's example is external. The main circularity concern is confined to the nonlinear theorems 1.3 and 1.4. The rescaling argument is sound in structure, but it cannot close without the quantitative boundary C^{1,α} estimate (3.13), which is imported from the first author's own unpublished preprint [3]; a second use of [3] appears in Lemma 3.2 at (3.35). These citations are load-bearing because the uniformity in R and the final weighted bounds depend on them, and no independent verification of [3] is supplied. A separate hypothesis-mismatch issue is the application of (3.13) to Σ_{1/2,1}, whose boundary has edges and on whose caps v_R has no imposed boundary condition; that is a correctness risk rather than a circularity. Overall, the linear theorem gives the paper substantial independent content, so the appropriate score is moderate rather than high.

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

The paper introduces no free parameters or invented entities. It rests on: (i) classical spectral-expansion theory for linear problems on cones (Dauge/Kozlov-Maz'ya-Rossmann), (ii) boundary C^{1,α} regularity for p-Laplace type operators — with the crucial quantitative form imported from the authors' own preprint [3], (iii) standard Sobolev/trace machinery. No numbers are fitted; all constants are structural.

assumptions (5)
  • standard math Dauge's spectral decomposition: solutions to −Δu=f in a cone can be expanded as a regular part plus a finite sum of r^{ξ_i}Y_i(θ) (Theorem 4.1, based on [17, Thm 6.4] and Properties (1)-(3)).
    Theorem 4.1 reduces the linear problem to the spectrum of −Δ_{S^{N−1}} on D; this is the load-bearing expansion behind Theorem 1.1 and is taken from the published literature (Dauge, Kozlov–Maz'ya–Rossmann).
  • domain assumption Quantitative C^{1,β} boundary regularity for p-Laplace operators on C^{1,β} annuli (estimate (3.13)), from [3, Theorems 1.2-1.3].
    The proof of Theorem 1.3's weighted Lipschitz bound needs a scale-invariant C^{1,α} estimate on Σ_{1/2,1}; this is imported from an arXiv preprint by the first author, not independently verified here.
  • standard math C^{1,α} and stress-field regularity machinery for p-Laplace in convex/nice domains ([4,5,11,12,13]) and trace inequalities ([5, Prop 6.2]).
    Lemma 3.2 and Theorem 1.4 rely on established (or authors'-prior) second-order estimates for the stress field A(∇u); these are quoted with references.
  • standard math Maz'ya's counterexample: existence of D with λ1(D)<N−1 and unbounded gradient for f∈L∞ (cited [40]).
    Used to justify that the condition λ1≥N−1 is sharp; no proof reproduced in the paper.
  • standard math Stein extension theorem and Sobolev embeddings for u0 ∈ W^{2,q}, q>N, implying u0∈C^{1,β} and the vertex conditions u0(0)=0, ∇u0(0)=0.
    Used in proof of Theorem 1.1(2)-(4).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Regularity results for elliptic equations on cones." pith.science (2026). https://pith.science/paper/RLKUXQGE

@misc{pith2026260800199,
  author       = {Pith},
  title        = {Pith review of: Regularity results for elliptic equations on cones},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RLKUXQGE}},
  note         = {Machine review of arXiv:2608.00199}
}
abstract

We study global regularity of solutions to Dirichlet or Neumann elliptic problems in spherical sectors $S_{D,R}$ of radius $R>0$ in $\mathbb{R}^N, N\ge 2$, where $D$ is the bounded domain on the unit sphere $\mathbb{S}^{N-1}$ which spans the spherical sector. One of the main results shows that boundedness of the gradient of the solutions of Poisson equations holds whenever $\lambda_1(D)\ge N-1$, where $\lambda_1(D)$ is the first nontrivial eigenvalue of the Laplace Beltrami operator $-\Delta_{\mathbb{S}^{N-1}}$ on the domain $D$ with Dirichlet or Neumann boundary conditions on $\partial D$. As an example of Maz'ya shows, the condition on the eigenvalue is sharp. For general spherical sectors and for $p$-Laplacian equations, $p>1$ we prove weighted global lipschitzianity of the solutions, as well as second order regularity.

Figures

Figures reproduced from arXiv: 2608.00199 by the authors.

Figure 1
Figure 1. The boundary adapted balls Br. Under this notation, we now establish a boundary Sobolev inequality on annuli, which will follow by combining Lemma 2.3 with a flattening-reflection argument. Lemma 2.4 (Sobolev inequality on boundary annuli). Let Ω ⊂ R N be a Lipschitz domain, with Lipschitz characteristics LΩ = (LΩ, RΩ). Let x0 ∈ ∂Ω, R ≤ RΩ/ [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

53 extracted references · 24 canonical work pages

  1. [3]

    Antonini,Local and globalC 1,β-regularity for uniformly elliptic quasilinear operators ofp-Laplace and Orlicz-Laplace type, ArXiv Preprint https://arxiv.org/abs/2601.07140, (2026)

    C.A. Antonini,Local and globalC 1,β-regularity for uniformly elliptic quasilinear operators ofp-Laplace and Orlicz-Laplace type, ArXiv Preprint https://arxiv.org/abs/2601.07140, (2026). 24 ANTONINI, FILOMENA PACELLA, CAMILLA CHIARA POL V ARA, AND LUIGI PROVENZANO

    arXiv 2026

  2. [1]

    Yu. A. Alkhutov and V. G. Maz’ya,L 1,p-Coercitivity and Estimates of the Green Function of the Neu- mann Problem in a Convex Domain, Journal of Mathematical Sciences, 196 (2014), no. 3, 245–261. doi:10.1007/s10958-014-1656-y

    work page doi:10.1007/s10958-014-1656-y 2014

  3. [2]

    Antonini,Smooth approximation of Lipschitz domains, weak curvatures and isocapacitary estimates, Calc

    C.A. Antonini,Smooth approximation of Lipschitz domains, weak curvatures and isocapacitary estimates, Calc. Var. Partial Differ. Equ., vol. 63, no. 4, 34 pages, (2024), doi = 10.1007/s00526-024-02711-x

    work page doi:10.1007/s00526-024-02711-x 2024

  4. [4]

    Antonini, A

    C.A. Antonini, A. Cianchi,Global Lipschitz regularity in anisotropic elliptic problems with natural gradient growth, ArXiv preprint, (2025), arXiv:2507.14606

    arXiv 2025

  5. [5]

    Antonini, A

    C.A. Antonini, A. Cianchi, G. Ciraolo, A. Farina, V.G. Maz’ya,Global second order estimates in anisotropic el- liptic problems, Proc. Lond. Math. Soc. 130, (2025), no. 3, Paper No. e70034, 60 pp., doi = 10.1112/plms.70034

    work page doi:10.1112/plms.70034 2025

  6. [6]

    Antonini, G

    C.A. Antonini, G. Ciraolo, F. Pagliarin,Second-order regularity for degeneratep-Laplace type equations with log-concave weights, J. Lond. Math. Soc., II. Ser., vol. 112, no. 3, 42 pages, (2025), doi = 10.1112/jlms.70299

    work page doi:10.1112/jlms.70299 2025

  7. [7]

    Auchmuty,Steklov eigenproblems and the representation of solutions of elliptic boundary value problems, Numer

    G. Auchmuty,Steklov eigenproblems and the representation of solutions of elliptic boundary value problems, Numer. Funct. Anal. Optim.,vol. 25, no. 3-4, pp. 321–348, (2004)

    2004

  8. [8]

    Bögelein, F

    V. Bögelein, F. Duzaar, P. Marcellini, C. Scheven,Boundary regularity for elliptic systems withp, q-growth, J. Math. Pures Appl. (9), vol. 159, pp. 250-293, (2022), doi = 10.1016/j.matpur.2021.12.004

    work page doi:10.1016/j.matpur.2021.12.004 2022

Show all 53 references
  1. [9]

    Cabré, X

    X. Cabré, X. Ros-Oton and J. Serra,Sharp isoperimetric inequalities via the ABP method,J. Eur. Math. Soc., 18 (2016), 2971–2998

    2016

  2. [10]

    Cianciaruso, L

    F. Cianciaruso, L. Muglia, B. Sciunzi,Second-order boundary regularity for degenerate elliptic problems, J. Fixed Point Theory Appl. 28, 23 (2026), doi=https://doi.org/10.1007/s11784-025-01270-8

    2026 doi

  3. [11]

    Cianchi, V.G

    A. Cianchi, V.G. Maz’ya,Global Lipschitz regularity for a class of quasilinear elliptic equations, Commun. Partial Differ. Equations, vol. 36, pp. 100-133 (2011), doi=10.1080/03605301003657843

    2011 doi

  4. [12]

    Cianchi, V.G

    A. Cianchi, V.G. Maz’ya,Global boundedness of the gradient for a class of nonlinear elliptic systems, Arch. Ration. Mech. Anal., vol. 212, no. 1, pp. 129-177, (2014), doi = 10.1007/s00205-013-0705-x

    2014 doi

  5. [13]

    Cianchi, V.G

    A. Cianchi, V.G. Maz’ya,Second-order two-sided estimates in nonlinear elliptic problems, Arch. Ration. Mech. Anal. 229 (2018), no. 2, 569-599, doi = 10.1007/s00205-018-1223-7

    2018 doi

  6. [14]

    Ciraolo, A

    G. Ciraolo, A. Figalli and A. Roncoroni,Symmetry results for critical anisotropicp-Laplacian equations in convex cones,Geom. Funct. Anal., 30 (2020), 770–803

    2020

  7. [15]

    Ciraolo, F

    G. Ciraolo, F. Pacella and C. C. Polvara,Symmetry breaking and instability for semilinear elliptic equations in spherical sectors and cones,J. Math. Pures App. 187 (2024), 138–170. doi:10.1016/j.matpur.2024.05.004

    2024 doi

  8. [16]

    Cozzi, A

    M. Cozzi, A. Farina, E. Valdinoci,Monotonicity formulae and classification results for singular, degenerate, anisotropic PDEs, Adv. Math., vol. 293, pp. 343-381, (2016), doi = 10.1016/j.aim.2016.02.014

    2016 doi

  9. [17]

    Dauge,Neumann and mixed problems on curvilinear polyhedra, Integral Equations and Operator Theory, vol

    M. Dauge,Neumann and mixed problems on curvilinear polyhedra, Integral Equations and Operator Theory, vol. 15, no.2, pp. 227-261, (1992), doi=10.1007/BF01204238

    1992 doi

  10. [18]

    De Filippis, M

    C. De Filippis, M. Piccinini,Borderline global regularity for nonuniformly elliptic systems, Int. Math. Res. Not., vol. 20, pp. 17324-17376, (2023), doi = 10.1093/imrn/rnac283

    2023 doi

  11. [19]

    Pure Appl

    J.F.Escobar,Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities and an eigenvalue estimate,Commun. Pure Appl. Math., XLIII (1990), 857–883

    1990

  12. [20]

    Figalli and E

    A. Figalli and E. Indrei,A sharp stability result for the relative isoperimetric inequality inside convex cones, J. Geom. Anal., 23 (2013), 938–969

    2013

  13. [21]

    Gilbarg, N.S

    D. Gilbarg, N.S. Trudinger,Elliptic partial differential equations of second order, Classics in Mathematics, Berlin: Springer, Reprint of the 1998 ed., (2001)

    1998

  14. [22]

    Grisvard,Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24, Pitman (Advanced Publishing Program), Boston, MA, 1985, xiv+410

    P. Grisvard,Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24, Pitman (Advanced Publishing Program), Boston, MA, 1985, xiv+410

    1985

  15. [23]

    Giusti,Direct methods in the calculus of variations, Singapore: World Scientific, (2003)

    E. Giusti,Direct methods in the calculus of variations, Singapore: World Scientific, (2003)

    2003

  16. [24]

    Guarnotta, S

    U. Guarnotta, S. Mosconi,A general notion of uniform ellipticity and the regularity of the stress field for elliptic equations in divergence form, Anal. PDE, vol. 16, no. 9, pp. 1955-1988, (2023), doi = 10.2140/apde.2023.16.1955

    1955 doi

  17. [25]

    Iacopetti, F

    A. Iacopetti, F. Pacella and T. Weth,Existence of nonradial domains for overdetermined and isoperimetric problems in nonconvex cones,Arch. Ration. Mech. Anal., 245 (2022), 1005–1058

    2022

  18. [26]

    Kondrat’ev,Boundary-value problems for elliptic equations in domains with conical or angular points, Trans

    V.A. Kondrat’ev,Boundary-value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc. vol. 16, pp. 227-313 (1967)

    1967

  19. [27]

    V. A. Kozlov, V.G. Maz’ya, J. Rossmann,Elliptic boundary value problems in domains with point singularities, Mathematical Surveys and Monographs, vol. 52, American Mathematical Society, Providence, RI, (1997), pp. x+414, doi= 10.1090/surv/052

    1997 doi

  20. [28]

    Kuusi, G

    T. Kuusi, G. Mingione,Universal potential estimates, J. Funct. Anal., vol. 262, no. 10, pp. 4205-4269, (2012), doi = 10.1016/j.jfa.2012.02.018

    2012 doi

  21. [29]

    Kuusi, G

    T. Kuusi, G. Mingione,Linear potentials in nonlinear potential theory, Arch. Ration. Mech. Anal., vol. 207, no.1, pp. 215-246, (2013), doi = 10.1007/s00205-012-0562-z

    2013 doi

  22. [30]

    Kuusi, G

    T. Kuusi, G. Mingione, A nonlinear Stein theorem, Calc. Var. Partial Differ. Equ., vol. 51, no. 1-2, pp. 45-86, (2014), doi = 10.1007/s00526-013-0666-9

    2014 doi

  23. [31]

    Kuusi, G

    T. Kuusi, G. Mingione,Guide to nonlinear potential estimates, Bull. Math. Sci., vol. 4, no. 1, pp. 1-82, (2014), doi = 10.1007/s13373-013-0048-9

    2014 doi

  24. [32]

    Kuusi, G

    T. Kuusi, G. Mingione,Vectorial nonlinear potential theory, J. Eur. Math. Soc. (JEMS), vol. 20, no. 4, pp. 929-1004, (2018), doi = 10.4171/JEMS/780

    2018 doi

  25. [33]

    O. A. Ladyzhenskaya,The Mixed Problem for a Hyperbolic Equation, Gosudarstv. Izdat. Tekhn.-Teor. Lit., Moscow, 1953. REGULARITY RESULTS FOR ELLIPTIC EQUATIONS ON CONES 25

    1953

  26. [34]

    Ladyzhenskaya, N.N

    O.A. Ladyzhenskaya, N.N. Uralt’seva,Linear and quasilinear equations, Math. Sci. Eng., vol. 64, (1968)

    1968

  27. [35]

    G.Leoni,A First Course in Sobolev Spaces, vol.105, Graduatestudiesinmathematics, AmericanMathematical Soc. (2009)

    2009

  28. [36]

    Lieberman,Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., Theory Methods Appl., vol

    G.M. Lieberman,Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., Theory Methods Appl., vol. 12, no. 11, pp. 1203-1219, 1988, doi = 10.1016/0362-546X(88)90053-3

    1988 doi

  29. [37]

    Lieberman,Oblique derivative problems for elliptic equations, 2013 Hackensack, NJ: World Scientific, 2013, doi = 10.1142/8679

    G.M. Lieberman,Oblique derivative problems for elliptic equations, 2013 Hackensack, NJ: World Scientific, 2013, doi = 10.1142/8679

    2013 doi

  30. [38]

    Lions and F

    P. Lions and F. Pacella,Isoperimetric inequalities for convex cones,Proc. Am. Math. Soc., 109 (1990), no. 2, 477–485

    1990

  31. [39]

    P. L. Lions, F. Pacella and M. Tricarico,Best constants in Sobolev inequalities for functions vanishing on some part of the boundary and related questions,Indiana Univ. Math. J., 37 (1988), 301–324

    1988

  32. [40]

    Maz’ya,Boundedness of the gradient of a solution to the Neumann-Laplace problem in a convex domain, C

    V.G. Maz’ya,Boundedness of the gradient of a solution to the Neumann-Laplace problem in a convex domain, C. R., Math., Acad. Sci. Paris, vol. 247, no. 9-10, pp- 517-520, (2009), doi = 10.1016/j.crma.2009.03.001

    2009 doi

  33. [41]

    V. G. Maz’ya,Solvability in ˚W 2 2 of the Dirichlet problem in a region with smooth irregular boundary, Vestnik Leningrad. Univ. 22 (1967), no. 7, 87–95

    1967

  34. [42]

    V. G. Maz’ya,The boundedness of the first derivatives of the solution of the Dirichlet problem in a region with smooth nonregular boundary,Vestnik Leningrad. Univ., 24 (1969), no. 1, 72–79

    1969

  35. [43]

    V. G. Maz’ya,On weak solutions of the Dirichlet and Neumann problems, Trans. Moscow Math. Soc., 20 (1969), 135–172

    1969

  36. [44]

    V. G. Maz’ya, B. A. Plamenevski˘ ı,The coefficients in the asymptotic expansion of the solutions of elliptic boundary value problems in a cone, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), vol. 52, (1975), pp. 110-127 (RU)

    1975

  37. [45]

    V. G. Maz’ya, B. A. Plamenevski˘ ı,The coefficients in the asymptotics of solutions of elliptic boundary value problems with conical points, Math. Nachr., vol. 76, (1977), pp. 29-60, doi = 10.1002/mana.19770760103, English translation in Amer. Math. Soc. Transl. 123 (1984), 57-88

    1977 doi

  38. [46]

    V. G. Maz’ya, B. A. Plamenevski˘ ı,Estimates inLp and in Hölder classes, and the Miranda-Agmon maximum principle for the solutions of elliptic boundary value problems in domains with singular points on the boundary, Math. Nachr., vol. 81, (1978), pp. 25-82, doi = 10.1002/mana....

    1978 doi

  39. [47]

    Montoro, L

    L. Montoro, L. Muglia, B. Sciunzi,Optimal second order boundary regularity for solutions top-Laplace equa- tions, Calc. Var. Partial Differential Equations, vol. 64, no. 2, Paper No. 50, 22, (2025), doi = 10.1007/s00526- 024-02902-6

    2025 doi

  40. [48]

    Pacella, C.C

    F. Pacella, C.C. Polvara, L. Provenzano,Bifurcation from bubbles in nonconvex cones, ArXiv preprint (2025), arXiv:2512.05766

    2025

  41. [49]

    Pacella and G

    F. Pacella and G. Tralli,Overdetermined problems and constant mean curvature surfaces in cones,Rev. Mat. Iberoam., 36 (2020), no. 3, 841–867

    2020

  42. [50]

    Ritoré and C

    M. Ritoré and C. Rosales,Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones,Trans. Am. Math. Soc., 356 (2004), no. 11, 4601–4622

    2004

  43. [51]

    Sciunzi, G

    B. Sciunzi, G. Spadaro, D. Vuono,Global second order optimal regularity for the vectorial p-Laplacian, ArXiv preprint (2025), arXiv:2502.17067

    2025 arXiv

  44. [52]

    Spadaro, D

    G. Spadaro, D. Vuono,Second-order boundary estimates for solutions to a class of quasilinear elliptic equations, Annali di Matematica Pura ed Applicata (1923-), (2026), doi=https://doi.org/10.1007/s10231-026-01672-6

    1923 doi

  45. [53]

    Federico Enriques

    P. Tolksdorf,On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations, vol. 8, no. 7, (1983), pp. 773-817, doi = 10.1080/03605308308820285 Carlo Alberto Antonini, Dipartimento di Matematica “Federico Enriq...

    1983 doi

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.