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Second-order boundary estimates for solutions to a class of quasilinear elliptic equations

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Pith's one-line read Quasilinear stress fields gain a square-integrable derivative up to the boundary, provided the boundary curvature lies in a sharp borderline space, with convex domains needing no smoothness at all.

desk verdict A genuinely new extension of second-order boundary regularity for quasilinear elliptic equations, but the approximation argument has a missing uniform curvature-norm estimate that needs fixing. read the letter →

arxiv 2507.16402 v1 pith:WV5NEOL7 submitted 2025-07-22 math.AP

Giuseppe Spadaro , Domenico Vuono This is my paper · ORCID
classification math.AP MSC 35B6535J2535J62
keywords quasilinearellipticequationssecond-orderregularityboundaryestimatesOrlicz-Sobolevspacesp-LaplaceequationsecondfundamentalformLorentz-Zygmundconvexdomains
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes global second-order regularity up to the boundary for solutions of $-\operatorname{div}(a(|\nabla u|)\nabla u)=f$ with either zero Dirichlet or zero Neumann data, for any coefficient $a\in C^1(0,\infty)$ whose logarithmic derivative stays above $-1$ and is bounded above. The regularity is expressed on the flux object $a(|\nabla u|)^k\nabla u$: it has a square-integrable weak derivative for a range of exponents $k$ that depends on the growth indices of $a$. The boundary condition needed is that the weak second fundamental form of $\partial\Omega$ lies in a borderline Lorentz-Zygmund space, $L^{n-1,1}$ for $n\ge 3$ and $L\log L$ for $n=2$, together with a smallness condition derived from it. In the model case $a(t)=t^{p-2}$, the result recovers the sharp boundary regularity known for the $p$-Laplacian. If the domain is merely convex, the same conclusion holds with no boundary regularity assumption at all.

What carries the argument

The central device is a global weighted Hessian estimate for the regularized equation, proved for smooth domains and smooth solutions and then transferred to weak solutions by approximation. The estimate controls $\int_\Omega \frac{a_\varepsilon(|\nabla u|)}{(\varepsilon+|\nabla u|^2)^{\alpha/2}}|D^2u|^2\,dx$ up to boundary terms in which the second fundamental form $B$ of $\partial\Omega$ appears through identity (4.44); the trace estimate (3.30) bounds these boundary terms by the capacity quotient $K_\Omega(r)$, so the smallness condition $\limsup_{r\to0^+}K_\Omega(r)<c$ is exactly what makes the boundary integrals absorbable. A pointwise linear-algebra lemma (Lemma 3.7) supplies the ellipticity of the quadratic form in the Hessian, and the approximation family $a_\varepsilon$ from Lemma 3.1 preserves the growth indices $i_a,s_a$ while regularizing the equation. The resulting uniform bounds on $a_\varepsilon(|\nabla u_\varepsilon|)^k\nabla u_\varepsilon$ in $W^{1,2}$ pass to the limit through compactness and almost-everywhere convergence of gradients.

What would settle it

Compute $K_\Omega(r)$ for a domain whose boundary second fundamental form lies in $L^{n-1}$ but not in $L^{n-1,1}$ (for $n=2$, in $L^1$ but not $L\log L$); if $\limsup_{r\to0^+}K_\Omega(r)$ is not below the threshold, solve the $p$-Laplace equation with a smooth nonzero right-hand side and test whether $|\nabla u|^{p-2+k}\nabla u\in W^{1,2}(\Omega)$ for the exponents $k$ in (1.9). A failure at the claimed exponent, or a check that the boundary term in (4.46) does not stay absorbable, would refute the theorem's boundary hypotheses; conversely, confirming the estimate on such borderline domains would support the sharpness of the Lorentz-Zygmund condition.

Watch

Extended reading notes

Core claim

On a bounded Lipschitz domain whose boundary has weak second derivatives in the Lorentz-Zygmund space $X$ with $X=L^{n-1,1}$ for $n\ge 3$ and $X=L\log L$ for $n=2$, every weak solution of the Dirichlet or Neumann problem satisfies $a(|\nabla u|)^k\nabla u\in W^{1,2}(\Omega)$ for every $k$ in the intervals (1.9), whenever $f\in W^{1,1}(\Omega)\cap L(\Omega)$. This is a global, up-to-the-boundary result for the stress field. In the special case $a(t)=t^{p-2}$ it gives the optimal $p$-Laplace boundary regularity; for $k=1$ it reproduces the known characterization $a(|\nabla u|)\nabla u\in W^{1,2}$ under the stated source condition. For bounded convex domains, the sign of the boundary curvature replaces the integrability condition and the same flux regularity follows without any regularity assumption on $\partial\Omega$ (Theorem 1.6). When the source term has a definite local sign, the inverse weight $1/a(|\nabla u|)$ is locally integrable to the powers listed in (1.11), yielding $u\in W^{2,q}(\Omega)$ for $1\le q<(s_a+1)/s_a$ in the degenerate regime.

Load-bearing premise

The load-bearing premise is that the boundary curvature is small at small scales, in the precise sense that the capacity quotient $K_\Omega(r)$ tends below a fixed constant as $r\to0^+$; this is inherited from $\partial\Omega\in W^{2,X}$, and if it fails the boundary integrals in the proof cannot be absorbed.

Editorial extensions

If this is right

  • For the $p$-Laplace operator, $a(t)=t^{p-2}$, the theorem gives exactly the optimal range of exponents $k$ for which $|\nabla u|^{p-2+k}\nabla u\in W^{1,2}(\Omega)$, matching the known sharp boundary result and its counterexamples.
  • Taking $k=1$ recovers, under the paper's domain and source assumptions, the known result that the stress field $a(|\nabla u|)\nabla u$ lies in $W^{1,2}(\Omega)$ when $f\in L^2(\Omega)$.
  • On any bounded convex domain, the conclusion holds with no boundary regularity assumption, so convexity alone guarantees the second-order flux estimate for both Dirichlet and Neumann problems.
  • If the source term has a fixed sign in a neighbourhood, the inverse weight $1/(a(|\nabla u|))^\beta$ is integrable for the exponents $\beta$ in (1.11); consequently, when $a(t)$ vanishes at $t=0$ and $s_a\ge 1$, the solution lies in $W^{2,q}(\Omega)$ for every $q<(s_a+1)/s_a$.
  • The boundary-regularity threshold is scale-critical: it requires $L^{n-1,1}$ or $L\log L$, the borderline Lorentz refinement of the natural $L^{n-1}$ integrability of the second fundamental form.

Reading between the lines

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

  • Because the argument needs only the smallness of $K_\Omega(r)$, the geometric condition could be checked numerically from boundary curvature data alone; a domain whose curvature concentrates enough to violate the smallness condition would be a candidate counterexample.
  • The convex-domain result suggests that the real mechanism is curvature sign rather than curvature magnitude; one might expect analogues for other second-order elliptic problems where a favourable curvature sign is available, though the paper does not state this.
  • The borderline $L^{n-1,1}$/$L\log L$ scale is sharp for the method, and constructing examples with curvature in $L^{n-1}$ but not $L^{n-1,1}$ would test whether the integrability assumption is necessary.
  • The inverse-gradient bounds are likely to interact with fine properties of the zero set of $\nabla u$; they could be used to quantify the size of the singular set in degenerate problems.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper proves global second-order regularity for the stress field a(|∇u|)^k ∇u for a class of quasilinear elliptic equations with homogeneous Dirichlet or Neumann boundary conditions. The main result, Theorem 1.1, asserts that a(|∇u|)^k ∇u ∈ W^{1,2}(Ω) for a range of exponents k determined by the growth indices ia, sa of a, provided the boundary belongs to a Lorentz-Zygmund class W^{2,X} with X = L^{n-1,1} (n≥3) or X = L log L (n=2). The proof is built on a global integral inequality (Theorem 4.1) for smooth approximating problems, with boundary curvature terms absorbed through the capacity quotient K_Ω(r). A sign condition on the source term yields integrability of the inverse of the gradient (Theorem 1.7), and convex domains are treated without boundary regularity assumptions (Theorems 1.6 and 1.8).

Significance. If the proof is completed, the results are significant: they extend sharp second-order boundary regularity for the p-Laplacian to general quasilinear operators with (p-1)-type growth, recover the Cianchi-Maz'ya stress-regularity theorem as the case k=1, and provide new information on the inverse of the gradient. The algebraic core in Theorem 4.1 is carefully presented, and the lower bound (4.39) for the quadratic form is sound. The convex-domain results are attractive because they require no boundary regularity. The main caveat is that the approximation step from smooth to W^{2,X} domains is not fully justified, as explained below.

major comments (2)
  1. [§5, Step 2 of Theorem 1.1 (and §6, Step 2 of Theorem 1.7)] The uniform-in-m bound (5.80) is not established. The constant in (5.75) depends, through the global Lipschitz estimate (5.55), on the strong boundary norms ∥tr B∥_{L^{n-1,1}(∂Ω)} for n≥3 and the L log L norm for n=2. Lemma 4.2 only provides (4.51)-(4.52), namely uniform Lipschitz characteristics, diameters, and the capacity quotient K_{Ω_m}(r) ≤ C K_Ω(r). The approximation property explicitly invoked in Step 2 preserves only the weaker L^{n-1,∞} and L^{1,∞} log L norms. These controlled quantities do not imply uniform bounds on the strong L^{n-1,1} or L log L norms of tr B_m, so the constant in (5.80) may depend on m and the subsequential limit argument does not yield a finite bound on Ω. The authors need to supply an approximation lemma preserving the strong curvature norms, or replace (5.55) by a global Lipschitz estimate whose constant is controlled by the quantities supplied by Lemma 4.2.
  2. [§5, Eqs. (5.63) and (5.68)] The displayed inequalities (5.63) and (5.68) are incorrect as written. In Case 1, where 0 ≤ ia < sa, condition (3.23) implies aε(t)^γ ≤ a(1)^γ (1+∥∇u∥_{L∞}^2)^{saγ/2} for γ ≥ 0; it does not give the stated lower bound with exponent ia, and the direction of the inequality is reversed. The same problem occurs in Case 2 at (5.68) for γ ≤ 0. The desired estimates (5.65) and (5.70) are nevertheless true and follow from the corrected upper bound on aε^γ, so this is a local but necessary correction in the proof of the central estimate.
minor comments (4)
  1. [§4, Eq. (4.42)] The denominator in the two displayed formulas of (4.42) should be (ε + |∇u|²)^{α/2}, not (ε − |∇u|²)^{α/2}.
  2. [§6, Lemma 6.1] In the statement of Lemma 6.1, u is a scalar function, not a vector field; the wording should be corrected.
  3. [§5, Step 2 of Theorem 1.1] The sentence introducing the approximation properties says 'see [21]' but the actual approximation lemma used is Lemma 4.2, which is attributed to [1]; the attribution should be made precise to avoid confusion.
  4. [§1, Theorem 1.7] The interval for β in (1.11) in the mixed case ia < 0 < sa is displayed with parentheses that may be read as open endpoints; please clarify which endpoints are included, especially since the subsequent estimates use closed intervals in places.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the main estimates are derived from independent a priori inequalities and external published results; the only self-citation is methodological and non-essential.

full rationale

The derivation of Theorem 1.1 does not assume its own conclusion. The key estimate is Theorem 4.1, an a priori inequality for smooth solutions on C^2 domains, proved in this paper from integration by parts, the pointwise algebraic identity (4.39), Lemma 3.7, and the boundary capacity estimate (4.46); it does not use the desired regularity of a(|∇u|)^k∇u. The passage to weak solutions is a standard three-step regularization: Step 1 uses the external global Lipschitz bound (5.55) from Cianchi-Mazya [19] and the C^{1,alpha} bound from Lieberman [42]; Step 2 approximates the domain via Lemma 4.2 (Antonini) and the smooth-domain estimates; Step 3 removes smoothness of f by density. The exponent ranges in (1.9) are computed algebraically from the growth bounds (3.23) and the bounds on theta_epsilon, not fitted to data. The convex-domain theorem is justified by the sign of the boundary curvature, a structurally distinct geometric input. The self-citations are routine literature mentions; [56], which overlaps with the present authors, is cited only as methodological inspiration, and no load-bearing theorem is imported from it. The reviewer concern about uniform control of the strong norm ||tr B_m||_{L^{n-1,1}} in Step 2 is a potential gap in the approximation argument, not a circularity, because the missing bound is not assumed as the conclusion.

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

No empirically fitted parameters appear; all constants are explicit and depend only on dimension, the structure constants ia and sa, and norms of f and Ω. The axioms are standard tools from the regularity literature, plus the stated boundary regularity assumption. The paper introduces no new particles, forces, or abstract entities.

assumptions (4)
  • standard math Orlicz-Sobolev space theory: the Young function B and its conjugate satisfy the Δ2 condition, reflexivity, and the Sobolev embedding (2.13), as stated in Theorems 2.1-2.2 and Proposition 2.3.
    Provides the function space setting and weak solution theory for operators with general growth a; the results are cited from Cianchi-Maz'ya [19].
  • standard math Uniform C^{1,α} regularity up to the boundary of the regularized solutions, via Lieberman [42, Theorem 1.7], and global L∞ gradient bounds via Cianchi-Maz'ya [19, Theorem 1.3].
    These estimates are used in Step 1 of Theorem 1.1 to obtain compactness of the family u_ε and to pass to the limit (see (5.55) and the C^{1,α} claim).
  • standard math Boundary trace and capacity estimates: Lemma 3.5 and Lemma 3.6, which control boundary integrals of the second fundamental form by the capacity quotient K_{Ω,ρ}.
    The key inequality (3.30) is used in (4.46) to absorb the boundary curvature terms; these lemmas are taken from Antonini-Cianchi-Ciraolo-Farina-Maz'ya [3] and Antonini [1].
  • domain assumption The weak second fundamental form B of ∂Ω is well-defined and belongs to L^{n-1,∞} (n≥3) or L^{1,∞} log L (n=2), with K_Ω(r) finite and satisfying the smallness condition (3.28).
    This is the domain regularity hypothesis ∂Ω ∈ W^{2,X} in Theorem 1.1; it guarantees the curvature absorption condition (4.31)-(4.32) used in Theorem 4.1.

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Pith. "Pith review of Second-order boundary estimates for solutions to a class of quasilinear elliptic equations." pith.science (2026). https://pith.science/paper/WV5NEOL7

@misc{pith2026250716402,
  author       = {Pith},
  title        = {Pith review of: Second-order boundary estimates for solutions to a class of quasilinear elliptic equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WV5NEOL7}},
  note         = {Machine review of arXiv:2507.16402}
}
read the original abstract

We prove global second-order regularity for a class of quasilinear elliptic equations, both with homogeneous Dirichlet and Neumann boundary conditions. A condition on the integrability of the second fundamental form on the boundary of the domain is required. As a consequence, with the additional assumption that the source term has a sign, we obtain integrability properties of the inverse of the gradient of the solution. Assuming convexity of the domain, no boundary regularity is required.

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Works this paper leans on

57 extracted references · 56 canonical work pages

  1. [21]

    Cianchi, V .G

    A. Cianchi, V .G. Maz’ya. Second-order two-sided estimates in nonlinear elliptic problems. Archive for Rational Mechanics and Analysis, 229 (2018), 569–599

    work page 2018

  2. [1]

    C. A. Antonini. Smooth approximation of Lipschitz domains, weak curvatures and isocapacitary estimates. Calc. Var. Partial Differential Equations, 63 (2024), 91(1–34)

    work page 2024

  3. [2]

    A NTONINI , G

    C.A. A NTONINI , G. C IRAOLO , A. F ARINA . Interior regularity results for inhomogeneous anisotropic quasilinear equations. Mathematische Annalen 387 (3), 1745-1776

    work page

  4. [3]

    C. A. Antonini, A. Cianchi, G. Ciraolo, A. Farina, and V . Maz’ya. Global second-order estimates in anisotropic elliptic problems. Proc. Lond. Math. Soc. (3) 130 (2025), no. 3, Paper No. e70034, 60 pp

    work page 2025

  5. [4]

    C. A. Antonini, G. Ciraolo, F. Pagliarin Second order regularity for degenerate p-Laplace type equations with log-concave weights. Preprint arXiv:2501.02106

    work page Pith review arXiv

  6. [5]

    Karppinen, S

    A. Karppinen, S. Sarsa. Local second order regularity of solutions to elliptic Orlicz-Laplace equa- tion. Nonlinear Anal. 253 (2025), Paper No. 113737, 20 pp

    work page 2025

  7. [6]

    Avelin, T

    B. Avelin, T. Kuusi, G. Mingione . Nonlinear Calder ´on-Zygmund theory in the limiting case. Arch. Ration. Mech. Anal., 227(2), (2018), 663-714

    work page 2018

  8. [7]

    Balci, A

    A. Balci, A. Cianchi, L. Diening, V . Maz’ya. A pointwise differential inequality and second-order regularity for nonlinear elliptic systems. Math. Ann., 383 (2022), no. 3-4, 1775–1824

    work page 2022

Show all 57 references
  1. [8]

    B ARATTA , L

    D. B ARATTA , L. M UGLIA , D. V UONO . Second order regularity for solutions to anisotropic de- generate elliptic equations. J. Differential Equations 435 (2025), Paper No. 113250, 29 pp

    2025

  2. [9]

    Baratta, B

    D. Baratta, B. Sciunzi, D. Vuono. Third order estimates and the regularity of the stress field for solutions to p-Laplace equations. Accepted on Commun. Contemp. Math. https://doi.org/10.1142/S0219199725500658

    doi

  3. [10]

    P . Baroni. Riesz potential estimates for a general class of quasilinear equations.Calculus of Varia- tions and Partial Differential Equations. 53, (2015), 803-846

    2015

  4. [11]

    B´enilan, L

    P . B´enilan, L. Boccardo, T. Gallou ¨et, R. Gariepy, M. Pierre, J.L. V ´azquez. An L1-theory of exis- tence and uniqueness of solutions of nonlinear elliptic equations. Ann. Sc. Norm. Super. Pisa. 22, (1995), 241–273

    1995

  5. [12]

    H. Brezis. Functional Analysis, Sobolev Spaces and Partial Diffrential Equations. Springer, New York, 2010

    2010

  6. [13]

    Castorina, G

    D. Castorina, G. Riey, B. Sciunzi. Hopf Lemma and regularity results for quasilinear anisotropic elliptic equations. Calculus of Variations and PDE, 58 (2019), 1–18

    2019

  7. [14]

    A. Cellina. The regularity of solutions to some variational problems, including the p-Laplace equation for 2 ≤ p < 3. ESAIM: Control, Optimisation and Calculus of Variations , 23 (2017), no.4, 1543–1553

    2017

  8. [15]

    Cianciaruso, L

    F. Cianciaruso, L. Muglia, B. Sciunzi. Second Order Boundary Regularity for Degenerate Elliptic Problems. Available at SSRN 4750137

  9. [16]

    Maximizing the L∞ norm of the gradient of solutions to the Poisson equation

    A.Cianchi. Maximizing the L∞ norm of the gradient of solutions to the Poisson equation. J. Geom. Anal. 2 (1992), 499-515

    1992

  10. [17]

    A. Cianchi. A sharp embedding theorem for Orlicz–Sobolev spaces. Indiana Univ. Math. J. (1996) 39-65

    1996

  11. [18]

    A. Cianchi. Boundedness of solutions to variational problems under general growth conditions. Comm. Partial Differential Equations 22 (1997) 1629–1646

    1997

  12. [19]

    Cianchi, V

    A. Cianchi, V . Maz’ya. Global Lipschitz regularity for a class of quasilinear elliptic equations. Comm. Partial Differential Equations. 36 (2011) 100-133

    2011

  13. [20]

    Cianchi, V

    A. Cianchi, V . Maz’ya. Global boundedness of the gradient for a class of nonlinear elliptic sys- tems. Arch. Ration. Mech. Anal. 212 (2014) 129–177

    2014

  14. [22]

    Cianchi, V

    A. Cianchi, V . Maz’ya. Optimal second-order regularity for the p-Laplace system. J. Math. Pures Appl. (9), 132 (2019), 41–78

    2019

  15. [23]

    Damascelli, B

    L. Damascelli, B. Sciunzi. Regularity, monotonicity and symmetry of positive solutions of m- Laplace equations. J. Differential Equations, 206 (2004), no. 2, 483–515

    2004

  16. [24]

    De Filippis, G

    C. De Filippis, G. Mingione. A borderline case of Calder ´on-Zygmund estimates for nonuni- formly elliptic problems. st. petersburg mathematical journal, 31, (2020) 455–477

    2020

  17. [25]

    De Filippis, G

    C. De Filippis, G. Mingione. On the regularity of minima of non-autonomous functionals. The Journal of Geometric Analysis, 30 (2020), 1584–1626

    2020

  18. [26]

    De Filippis, G

    C. De Filippis, G. Mingione. Regularity for Double Phase Problems at Nearly Linear Growth Arch. Rational Mech. Anal., 247(2023), no.5, Paper No. 85, 50 pp

    2023

  19. [27]

    De Filippis, G

    C. De Filippis, G. Mingione. Nonuniformly elliptic Schauder theory. Invent. Math., 234(2023), no.3, 1109–1196. 34 GIUSEPPE SPADARO AND DOMENICO VUONO

    2023

  20. [28]

    De Filippis, G

    C. De Filippis, G. Mingione. Gradient regularity in mixed local and nonlocal problems. Mathe- matische annalen, 388, (2024), 261–328.doi 10.1007/s00208-022-02512-7

    2024 doi

  21. [29]

    De Filippis, M

    C. De Filippis, M. Piccinini. Borderline global regularity for nonuniformly elliptic systems, Int. Math. Res. Not., (20) (2023), 17324–17376

    2023

  22. [30]

    D I BENEDETTO

    E. D I BENEDETTO . C 1+α local regularity of weak solutions of degenerate elliptic equations.Non- linear Anal. 7(8), 1983, pp. 827–850

    1983

  23. [31]

    D ONG , F

    H. D ONG , F. PENG , Y. ZHANG , Y. ZHOU. Hessian estimates for equations involving p-Laplacian via a fundamental inequality Adv. Math. 370 (2020) 107212

    2020

  24. [32]

    D. T. Donaldson, N. S. Trudinger. Orlicz-Sobolev spaces and imbedding theorems.J. Funct. Anal., 8 (1971), 52-75

    1971

  25. [33]

    Duzaar, G

    F. Duzaar, G. Mingione. Gradient continuity estimates. Calc. Var. Partial Differential Equations,39 (2010), no. 3-4, 379–418

    2010

  26. [34]

    Duzaar, G

    F. Duzaar, G. Mingione. Gradient estimates via non-linear potentials. Amer. J. Math., 133 (2011), no. 4, 1093–1149

    2011

  27. [35]

    Esposito, B

    F. Esposito, B. Sciunzi, A. Trombetta Regularity and symmetry results for nonlinear degenerate elliptic equations. J. Differential Equations 336 (2022), 315–333

    2022

  28. [36]

    E. Giusti. Direct Methods in the Calculus of Variations. River Edge, NJ: World Scientific., 2003

    2003

  29. [37]

    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, 16(8), (2023), 1955–1988

    2023

  30. [38]

    Grisvard,Elliptic Problems in Nonsmooth Domains

    P . Grisvard,Elliptic Problems in Nonsmooth Domains. Pitman, Boston, MA, 1985

    1985

  31. [39]

    Iandoli, D

    F. Iandoli, D. Vuono. Local behaviour of the second order derivatives of solutions to p-Laplace equations arXiv preprint arXiv:2502.06273

  32. [40]

    O. A. Ladyzhenskaya, N. N. Ural’tseva. Linear and Quasilinear Elliptic Equations. New York: Academic Press., 1968

    1968

  33. [41]

    G. M. Lieberman. Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal., 12(11), 1988, 1203–1219

    1988

  34. [42]

    G. M. Lieberman. The natural generalization of the natural conditions of Ladyzenskaya and Ural’ceva for elliptic equations. Comm. Part. Diff. Eqs.,J. 16, 311–361 (1991)

    1991

  35. [43]

    M ERCURI , G

    C. M ERCURI , G. R IEY, AND B. S CIUNZI . A regularity result for the p-Laplacian near uniform ellipticity. SIAM J. Math. Anal. 48:3 (2016), 2059–2075

    2016

  36. [44]

    Montoro, L

    L. Montoro, L. Muglia, B. Sciunzi. Optimal second order boundary regularity for solutions to p-Laplace equations. Calc. Var. Partial Differential Equations, 64 (2025), no. 2, Paper No. 50

    2025

  37. [45]

    Montoro, L

    L. Montoro, L. Muglia, B. Sciunzi, D. Vuono. Regularity and symmetry results for the vectorial p-Laplacian. Nonlinear Anal. 251 (2025), Paper No. 113700, 16 pp

    2025

  38. [46]

    Necas Les m ´ethodes directes en th´eorie des ´equations elliptiques

    J. Necas Les m ´ethodes directes en th´eorie des ´equations elliptiques. Masson et Cie, ´Editeurs, Paris; Academia, ´Editeurs, Prague, 1967

    1967

  39. [47]

    Kuusi, G

    T. Kuusi, G. Mingione. A nonlinear Stein theorem. Calc. Var. Partial Differential Equations , 51 (2014), no. 1-2, 45–86

    2014

  40. [48]

    Kuusi, G

    T. Kuusi, G. Mingione. Vectorial nonlinear potential theory. J. Eur. Math. Soc. (JEMS) 20 (2018), no.4, 929–1004

    2018

  41. [49]

    E.H. Lieb, M. Loss. Analysis, Vol. 14 of Graduate Studies in Mathematics, American Mathematical Society, Providence, 1997

    1997

  42. [50]

    H. Lou. On singular sets of local solutions to p-Laplace equations, Chinese Annals of Mathematics, Series B 29 (2008), no.5, 521–530

    2008

  43. [51]

    Mingione

    G. Mingione. Calder ´on-Zygmund estimates for measure data problems, C. R. Math. Acad. Sci. Paris, 344(7), (2007), 437–442

    2007

  44. [52]

    Mingione

    G. Mingione. The Calder ´on-Zygmund theory for elliptic problems with measure data, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 6(2), (2007), 195–261

    2007

  45. [53]

    Mi ´skiewicz

    M. Mi ´skiewicz. Fractional differentiability for solutions of the inhomogeneousp-Laplace system. Proc. Amer. Math. Soc., 146 (2018), no.7, 3009–3017

    2018

  46. [54]

    S CIUNZI

    B. S CIUNZI . Some results on the qualitative properties of positive solutions of quasilinear elliptic equations. NoDEA. Nonlinear Differential Equations and Applications, 14(3-4), 2007, 315–334

    2007

  47. [55]

    S CIUNZI

    B. S CIUNZI . Regularity and comparison principles for p-Laplace equations with vanishing source term. Comm. Cont. Math., 16(6), 2014, 1450013, 20

    2014

  48. [56]

    Sciunzi, G

    B. Sciunzi, G. Spadaro, D. Vuono. Global second order optimal regularity for the vectorial p- Laplacian. Preprint arXiv:2502.17067

    arXiv

  49. [57]

    Tolksdorf

    P . Tolksdorf. Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations, 51 (1984), 126–150. Email address, Giuseppe Spadaro: giuseppe.spadaro@unical.it Email address, Domenico Vuono: domenico.vuono@unical.it SECOND-ORDER BOUNDARY ESTIMA...

    1984

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.