Pith. sign in

REVIEW 2 major objections 4 minor 138 references

Sketches of Nonuniformly Elliptic Schauder Theory

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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

Pith's one-line read The paper asserts that Schauder regularity — solutions inherit the regularity of the coefficients — holds for nonuniformly elliptic variational problems under the sharp threshold q/p ≤ 1 + α/n, giving locally Hölder continuous gradients for

desk verdict A clear, honest survey of a major recent advance, not a research paper—the proof sketch is too thin to verify but it never pretends otherwise. read the letter →

arxiv 2509.11205 v1 pith:TN2PTMCW submitted 2025-09-14 math.AP

Cristiana De Filippis This is my paper · ORCID
classification math.AP MSC 49N6035J60
keywords nonuniformellipticitySchauderestimates(pq)-growthgradientregularityLavrentievphenomenonLSMrelaxationnonlinearpotentialtheoryHöldercontinuity
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

Schauder theory says that elliptic solutions are as regular as the coefficients driving them. This paper surveys and sketches a recently settled extension of that principle to nonuniformly elliptic variational integrals, where the classical freeze-and-compare proofs fail because the ellipticity ratio is allowed to blow up. The central claim is that under (p,q)-growth, Hölder-α coefficients, and the sharp threshold q/p ≤ 1 + α/n, every local minimizer of the Lebesgue–Serrin–Marcellini (LSM) relaxation has locally Hölder continuous gradient and simultaneously minimizes the original functional; in the nondegenerate case the gradient is C^{0,α}. The paper lays out the proof architecture: almost-Lipschitz bounds by fractional Moser iteration, a renormalized fractional Caccioppoli inequality on level sets, and nonlinear potentials that carry coefficient information through the iteration. If correct, this closes a long-standing gap: nonuniform ellipticity no longer blocks Schauder-type regularity, and no a priori Lipschitz assumption is needed.

What carries the argument

The load-bearing object is the renormalized fractional Caccioppoli inequality on level sets, (4.22), which plays the role of the classical Bernstein–Caccioppoli estimate. It bounds a fractional Sobolev seminorm of the truncated gradient (|Du|^p − κ)_+ by an L^2-type average plus a 'size' term carrying the Hölder oscillation of the coefficients, all renormalized by M^{2b}, where M bounds |Du| and b — supplied by sharp Lipschitz estimates for autonomous (p,q)-growth problems — measures the power-type growth of the ellipticity ratio and tends to 0 as q → p. Because b is sharp, M^{2b} can be reabsorbed during a De Giorgi-type iteration, converting almost-Lipschitz information from a fractional M

What would settle it

Exhibit one LSM minimizer u of an admissible functional satisfying (4.1)–(4.4) whose gradient is not locally bounded. The double-phase family with a(x)=|x|^α at the critical ratio q/p = 1 + α/n is the natural place to look, since Theorem 1 predicts a locally Hölder gradient there; a singular gradient would refute the claim. Equivalently, finding an admissible integrand for which (4.22) fails with the claimed b would break the proof at its first step.

Watch

Extended reading notes

Core claim

The paper's central claim is that Schauder theory holds for nonuniformly elliptic variational integrals: minimizers are as regular as the coefficients allow, even when the ellipticity ratio blows up as a power of the gradient. Specifically, for integrals of the form F(w,Ω)=∫F(x,Dw)dx satisfying (p,q)-growth, α-Hölder coefficients, and the bound q/p ≤ 1 + α/n, every local minimizer of the Lebesgue–Serrin–Marcellini (LSM) extension has locally Hölder continuous gradient and is at the same time a minimizer of the original functional (Theorem 1). If the integrand is nondegenerate (μ>0), the gradient belongs to C^{0,α}_{loc} (Theorem 2). The paper presents this as the settlement of a longstanding

Load-bearing premise

The proof's load-bearing premise is that the renormalized fractional Caccioppoli inequality (4.22) holds with a sharp exponent b and constants that deteriorate only polynomially; the sketch asserts this inequality rather than proving it, and if it fails the iteration that reabsorbs M^{2b} and yields Lipschitz and Hölder gradients cannot be completed.

Editorial extensions

If this is right

  • Under (4.1) and (4.4), every local minimizer of the LSM extension is automatically a minimizer of the original functional F, so the relaxation singles out regular minimizers even when the Lavrentiev gap is nonzero.
  • Any minimizer of the original functional on which the Lavrentiev gap vanishes has locally Hölder continuous gradient; if μ>0, the gradient is C^{0,α} — the same exponent as the coefficients.
  • The classical order of proof is reversed: gradient boundedness is established first, so no a priori Lipschitz assumption is needed in the nonuniformly elliptic setting.
  • The sharp threshold q/p ≤ 1 + α/n yields arbitrarily high integrability of the gradient (every L^t, t<∞) for bounded minimizers, and the exponent bounds are optimal by counterexamples.
  • For the model class a(x)F_0(Du), minimizers have locally Hölder continuous gradient without any smoothness assumption on u, and C^{1,α}_{loc} in the nondegenerate case.

Reading between the lines

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

  • Extension: the same fractional Moser–Caccioppoli architecture is a plausible template for nonuniformly parabolic problems; a testable analogue would use parabolic fractional seminorms and a threshold replacing n by the parabolic dimension.
  • Extension: because the LSM relaxation selects regular minimizers, numerical schemes that approximate the relaxed functional should converge to the regular representative even when the Lavrentiev gap is positive — testable, for instance, on the checkerboard example.
  • Extension: the threshold formula suggests a quantitative compensation principle: coefficient smoothness of order α buys α/n of 'ellipticity budget.' Anisotropic or Orlicz-type integrands could be probed to see whether per-direction growth gaps obey an analogous weighted constraint.
  • Extension: making the constants and exponent in (4.22) explicit would turn the qualitative theorem into quantitative Schauder estimates — Hölder exponents and radii — suitable for free-boundary applications.
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

2 major / 4 minor

Summary. This paper is an expository survey of recent Schauder-type regularity results for nonuniformly elliptic variational integrals. It introduces local notions of ellipticity ratio, reviews classical uniformly elliptic Schauder theory and its perturbative 'freeze-and-compare' scheme, and surveys counterexamples showing failure of regularity in softly nonuniformly elliptic problems (Lavrentiev gap, Zhikov checkerboard, fractal singular sets). The central new material is in Sections 4.4-4.5: Theorems 1 and 2, attributed to [DM23a, DM25a], assert that under growth/ellipticity conditions (4.1) and the sharp gap bound q/p < 1 + α/n, local minimizers of the Lebesgue-Serrin-Marcellini extension have locally Hölder continuous gradients and are also minimizers of the original functional; if μ > 0 then Du ∈ C^{0,α}. The proof is sketched through a fractional Moser iteration, a renormalized fractional Caccioppoli inequality (4.22) on level sets, and nonlinear potentials.

Significance. Assuming the cited proofs are correct, these theorems settle a longstanding problem and substantially broaden Schauder theory beyond uniform ellipticity, with sharp numerology and a selection principle for regular minimizers when the Lavrentiev gap vanishes. The manuscript is valuable as an overview: it situates the results in the historical literature, gives explicit model examples, and states the main theorems and the key innovation (direct gradient boundedness without gradient Hölder estimates). Its strengths are the clear taxonomy of ellipticity notions and the careful description of the failure of perturbative methods. The limitation is that the proof sketch is not self-contained: the decisive estimate (4.22) is quoted rather than derived, and the text explicitly defers full details to [DM25a]. This is appropriate for a survey, but it means the reader cannot independently check the central step from the material provided.

major comments (2)
  1. [Section 4.5, Eq. (4.22)] Estimate (4.22) is the load-bearing step in the proof of Theorem 1. It is stated as 'holds whenever M ≥ ||Du||_L∞' with an unspecified exponent b ≡ b(n,p,q,α), and the text says it must be used 'in its sharpest possible form' to obtain the threshold (4.4). No derivation, no formula for b, and no quantitative reabsorption condition for M^{2b} are given. In particular, it is not shown that b tends to 0 fast enough as q → p, nor how the third 'size' term is controlled by the nonlinear potentials to allow the De Giorgi iteration to close under (4.4) rather than the weaker (4.19). Since the manuscript itself states that full details are in [DM25a], this is not a demonstrated error, but the survey cannot be checked at exactly the point on which the main theorem rests. Please either supply the proof or state (4.22) as a quoted theorem with a precise reference to [DM25a] (theorem/proposition num
  2. [Section 4.4, Theorem 1] The second assertion of Theorem 1 — that a minimizer of the LSM-extension ¯F is also a minimizer of F under (4.4) — is a nontrivial statement in the presence of Lavrentiev phenomena. The sketch in Section 4.5 only discusses a priori estimates for smooth approximations and says 'results then come by approximation procedures' (footnote 28). No approximation argument or pointer to the exact theorem in [DM23a, DM25a] is given. For a survey this may be acceptable, but since this assertion is the 'selection principle' highlighted as a main outcome, it should be documented with a precise reference.
minor comments (4)
  1. [Section 4.4, p. 24] Typographical issues: 'longstading' should be 'longstanding' and 'smoothingredients' should be 'smooth ingredients'.
  2. [Section 4.4, first paragraph] The phrase 'Section(4.3)' should read 'Section 4.3'.
  3. [Equation (4.22)] The notation [ (|Du|^p - κ)_+ ]^2_{β,2;B_{ρ/2}} is not defined. Since the inequality is central, please define the seminorm explicitly or state that it is the standard fractional Sobolev/Besov seminorm.
  4. [Abstract] The phrase 'problems exhibiting ellipticity' is vague; likely 'exhibiting nonuniform ellipticity' is intended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the survey transparently attributes its main theorems to the published papers [DM25a, DM23a] and to independent Bella–Schäffner estimates; no claim reduces by construction to its own input.

full rationale

This paper is a survey/sketch of nonuniformly elliptic Schauder theory. The two main theorems are explicitly introduced as results appearing in the author's own published works: 'The following results appear in [DM25a]' (Section 4.4), followed by Theorem 1 and Theorem 2. There is no claim in the text that these theorems are derived here from scratch; instead, Section 4.5 gives an outline of the proof and repeatedly defers to [DM23a, DM25a] for the full arguments, including the approximation procedures and the delicate estimates. The key renormalized fractional Caccioppoli inequality (4.22) is stated with proof deferred to the same sources and to independent regularity estimates of Bella and Schäffner [BS20, BS24, Sch24]: 'the usual atoms appearing in dyadic decompositions are replaced by solutions to nonlinear, autonomous problems, whose regularity is quantified via suitable improved forms of related a priori estimates due to Bella & Schäffner'. These are separate, published, parameter-free results with stated assumptions, not an unverified self-citation chain. The LSM relaxation (4.3) is a definition, and the fact that minimizers of the relaxed functional are regular and also minimize the original functional is a substantive theorem, not a tautology; condition (4.4) is a sharp quantitative hypothesis originating from the Lavrentiev-gap counterexamples, and it is not the same as the conclusion. No parameter is fitted to data and then renamed as a prediction, and no uniqueness theorem from the author's prior work is invoked to force a choice. The skeptic's concern that (4.22) is asserted rather than proved here is a verification gap of the survey format, not a circularity: the estimate is structurally distinct from, and stronger than, the final Hölder-continuity conclusion, and its proof is located in the cited articles. Accordingly, the derivation chain in this manuscript is self-contained in the sense required for a circularity analysis: it does not reduce any of its claims to its own inputs by construction.

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

No free parameters are fitted; p, q, alpha, and mu are hypotheses of the theorems, not fitted constants. The axioms list the imported regularity results on which the survey statements rest. The decisive imported estimate is (4.22), which is not derived here.

assumptions (4)
  • standard math Ural'tseva-Uhlenbeck and Campanato-Meyers estimates for uniformly elliptic p-growth problems are valid.
    Used in Section 3 (estimates (3.3) to (3.6)) to build the frozen-coefficient comparison that yields uniformly elliptic Schauder theory.
  • domain assumption Bella-Schaffner Lipschitz estimates and Hirsch-Schaffner local boundedness hold for (p,q)-growth minimizers.
    Invoked in Section 4.5 to supply local boundedness of minimizers and the sharp exponent b in inequality (4.22), which the proof of Theorem 1 needs.
  • standard math The LSM relaxation has the stated properties: extension, lower semicontinuity, and vanishing Lavrentiev gap in the described cases.
    Section 4 uses (4.3) to (4.5) to switch from minimizers of F to minimizers of the relaxed functional.
  • domain assumption The theorems of [DM23a] and [DM25a] are correct as cited.
    Theorems 1 and 2 and Corollaries 1 and 2 are explicitly attributed to these papers; no proof appears in this preprint.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Sketches of Nonuniformly Elliptic Schauder Theory." pith.science (2026). https://pith.science/paper/TN2PTMCW

@misc{pith2026250911205,
  author       = {Pith},
  title        = {Pith review of: Sketches of Nonuniformly Elliptic Schauder Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TN2PTMCW}},
  note         = {Machine review of arXiv:2509.11205}
}
read the original abstract

Schauder theory is a basic tool in the study of elliptic and parabolic PDEs, asserting that solutions inherit the regularity of the coefficients. It plays a central role in establishing higher regularity for solutions to a broad class of elliptic problems exhibiting ellipticity, including those involving free boundaries. In the linear setting, Schauder theory dates back to the 1920-30s and is now considered classical. Nonlinear extensions were developed in the 1980s. All these classical results are restricted to uniformly elliptic operators and heavily rely on perturbative techniques - freezing the coefficients and comparing the solution to that of a constant-coefficient problem. However, such methods fail in the nonuniformly elliptic setting, where homogeneous a priori estimates break down and standard iteration arguments no longer apply. Here we give a brief survey on recent progresses including the solution to the longstanding problem of proving the validity of Schauder estimates in the nonlinear, nonuniformly elliptic setting.

Figures

Figures reproduced from arXiv: 2509.11205 by the authors.

Figure 1
Figure 1. Zhikov’s checkerboard. Source [BDS20]. 4.3. Soft nonuniform ellipticity Before focusing on classical, strong nonuniform ellipticity in the sense of (1.13), let us consider the softly nonuniformly elliptic case of the double phase functional D in (1.17), where from now on we assume that 0 ≤ 𝔞 ∈ 𝐶 0,𝛼 loc (Ω), 𝛼 ∈ (0, 1]. (4.9) Behind a rather simple structure, energy D hides several interesting (ir)regularity aspects… view at source ↗
Figure 2
Figure 2. Low-regular competitor vs coefficient 𝔞 [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

138 extracted references · 2 linked inside Pith

  1. [1]

    Acerbi, G

    E. Acerbi, G. Mingione, Regularity results for a class of functionals with non-standard growth. Arch. Ration. Mech. Anal. 156, 121-140, (2001)

    2001

  2. [2]

    Agmon, A

    S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I. Comm. Pure Appl. Math., 12, 623-727 (1959) and 17 (1964) 35–92

    1959

  3. [3]

    Agmon, A

    S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Comm. Pure Appl. Math., 17, 35-92, (1964)

    1964

  4. [4]

    Baasandorj, S.-S

    S. Baasandorj, S.-S. Byun, Regularity for Orlicz phase problems. Memoirs Amer. Math. Soc. 308, 131 pp, (2025)

    2025

  5. [5]

    Balci, C

    A. Balci, C. Ortner, J. Storn, Crouzeix-Raviart finite element method for non-autonomous variational problems with Lavrentiev gap. Numer. Math. 151, 779-805, (2022)

    2022

  6. [6]

    Balci, L

    A. Balci, L. Diening, M. Surnachev, New examples on Lavrentiev gap using fractals. Calc. Var. & PDE 59:180, (2020)

    2020

  7. [7]

    Balci, L

    A. Balci, L. Diening, M. Surnachev, Scalar minimizers with maximal singular sets and lack of Meyers property. In: Friends in Partial Differential Equations. The Nina N. Uraltseva 90th Anniversary Volume, D. Apushkinskaya, A. Laptev, A. I. Nazarov, H. Shahgholian eds. EMS Press, pp.1-43, 2025

    2025

  8. [8]

    J. M. Ball, V. Mizel, One dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation. Arch. Ration. Mech. Anal. 90, 325-388, (1985)

    1985

Show all 138 references
  1. [9]

    J. M. Ball, G. Knowles, A numerical method for detecting singular minimizers. Numer. Math. 51, 181-197, (1987)

    1987

  2. [10]

    Baroni, M

    P. Baroni, M. Colombo, G. Mingione, Regularity for general functionals with double phase. Calc. Var. & PDE 57:62, (2018)

    2018

  3. [11]

    L. Beck, G. Mingione, Lipschitz bounds and nonuniform ellipticity. Comm. Pure Appl. Math. 73, 944-1034, (2020)

    2020

  4. [12]

    L. Beck, T. Schmidt, On the Dirichlet problem for variational integrals in BV. J. Reine Angew. Math. 674, 113-194, (2013)

    2013

  5. [13]

    L. Beck, T. Schmidt, Interior gradient regularity for BV minimizers of singular variational problems. Nonlinear Anal. 120, 86-106, (2015)

    2015

  6. [14]

    Bella, M

    P. Bella, M. Sch\"affner, On the regularity of minimizers for scalar integral functionals with (p,q) -growth. Anal. & PDE 13, 2241-2257, (2020)

    2020

  7. [15]

    Bella, M

    P. Bella, M. Sch\"affner, Local Boundedness and Harnack Inequality for Solutions of Linear Nonuniformly Elliptic Equations. Comm. Pure Appl. Math. 74, 453-477, (2021)

    2021

  8. [16]

    Bella, M

    P. Bella, M. Sch\"affner, Lipschitz bounds for integral functionals with (p,q) -growth conditions. Adv. Calc. Var. 17, 373-390, (2024)

    2024

  9. [17]

    Bildhauer, M

    M. Bildhauer, M. Fuchs, Convex variational problems with linear growth. Geometric analysis and nonlinear partial differential equations, Springer, Berlin 327-344, (2003)

    2003

  10. [18]

    Bombieri, E

    E. Bombieri, E. De Giorgi, M. Miranda, Una maggiorazione a priori relativa alle ipersuperfici minimali non parametriche. Arch. Ration. Mech. Anal. 32, 255-267, (1969)

    1969

  11. [19]

    Bouchitté, I

    G. Bouchitté, I. Fonseca, J. Malý, The effective bulk energy of the relaxed energy of multiple integrals below the growth exponent. Proc. R. Soc. Edinb. Sect. A, Math. 128, 463-479, (1998)

    1998

  12. [20]

    Bousquet, L

    P. Bousquet, L. Brasco, Lipschitz regularity for orthotropic functionals with nonstandard growth conditions. Rev. Math. Iberoam. 36, 1989-2032, (2020)

    1989

  13. [21]

    B\"ogelein, F

    V. B\"ogelein, F. Duzaar, N. Liao, G. Molica Bisci, R. Servadei, Regularity for the fractional p -Laplace equation. J. Funct. Anal. 289, 111078, (2025)

    2025

  14. [22]

    Brasco, E

    L. Brasco, E. Lindgren, Higher Sobolev regularity for the fractional p -Laplace equation in the superquadratic case. Adv. Math. 304, 300-354, (2017)

    2017

  15. [23]

    Brasco, E

    L. Brasco, E. Lindgren, A. Schikorra, Higher H\"older regularity for the fractional p -Laplacian in the superquadratic case. Adv. Math. 338, 782-846, (2018)

    2018

  16. [24]

    Bul\'i c ek, P

    M. Bul\'i c ek, P. Gwiazda, J. Skrzeczkowski, On a range of exponents for absence of Lavrentiev phenomenon for double phase functionals. Arch. Ration. Mech. Anal. 246, 209-240, (2022)

    2022

  17. [25]

    Buttazzo, V

    G. Buttazzo, V. Mizel, Interpretation of the Lavrentiev Phenomenon by relaxation. J. Functional Anal. 110, 434-460, (1992)

    1992

  18. [26]

    S.-S. Byun, J. Oh, Regularity results for generalized double phase functionals. Anal. PDE 13, 1269-1300, (2020)

    2020

  19. [27]

    S.-S. Byun, J. Ok, K. Song, Hölder regularity for weak solutions to nonlocal double phase problems. J. Math. Pures Appl. 9, 168, 110-142, (2022)

    2022

  20. [28]

    Caccioppoli, Sulle equazioni ellittiche a derivate parziali con n variabili indipendenti

    R. Caccioppoli, Sulle equazioni ellittiche a derivate parziali con n variabili indipendenti. Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl. 19, 83-89, (1934)

    1934

  21. [29]

    Caffarelli, X

    L. Caffarelli, X. Cabré, Fully nonlinear elliptic equations. Colloquium Publications. 43. American Math. Soc. 104 p. (1995)

    1995

  22. [30]

    Caffarelli, C.-H

    L. Caffarelli, C.-H. Chan, A. Vasseur, Regularity theory for parabolic nonlinear integral operators. J. Amer. Math. Soc. 24, 849-869, (2011)

    2011

  23. [31]

    Campanato, Equazioni ellittiche del II ordine e spazi L ^ 2,

    S. Campanato, Equazioni ellittiche del II ordine e spazi L ^ 2, . Ann. Mat. Pura Appl. 69, 321-381, (1965)

    1965

  24. [32]

    Carozza, J

    M. Carozza, J. Kristensen, A. Passarelli di Napoli, Higher differentiability of minimizers of convex variational integrals. Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire 28, 395-411, (2011)

    2011

  25. [33]

    H. J. Choe, Interior behaviour of minimizers for certain functionals with nonstandard growth. Nonlinear Anal. 19, 933-945 (1992)

    1992

  26. [34]

    Colombo, G

    M. Colombo, G. Mingione, Regularity for double phase variational problems. Arch. Ration. Mech. Anal. 215, 443-496, (2015)

    2015

  27. [35]

    Cozzi, Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: A unified approach via fractional De Giorgi classes

    M. Cozzi, Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: A unified approach via fractional De Giorgi classes. J. Funct. Anal. 272, 4762-4837, (2017)

    2017

  28. [36]

    Cupini, P

    G. Cupini, P. Marcellini, E. Mascolo, Regularity for Nonuniformly Elliptic Equations with p,q -Growth and Explicit x,u -Dependence. Arch. Ration. Mech. Anal. 248:60, (2024)

    2024

  29. [37]

    Dacorogna, Direct methods in the Calculus of Variations

    B. Dacorogna, Direct methods in the Calculus of Variations. Applied Mathematical Sciences, Springer 78, (2008)

    2008

  30. [38]

    De Filippis, Quasiconvexity and partial regularity via nonlinear potentials

    C. De Filippis, Quasiconvexity and partial regularity via nonlinear potentials. J. Math. Pures Appl. 163, 11-82, (2022)

    2022

  31. [39]

    De Filippis, -ellipticity and nonautonomous integrals

    C. De Filippis, -ellipticity and nonautonomous integrals. EMS Magazine, to appear

  32. [40]

    De Filippis, F

    C. De Filippis, F. De Filippis, M. Piccinini, Bounded minimizers of double phase problems at nearly linear growth. Preprint (2024). arXiv:2411.14325

    2024 arXiv

  33. [41]

    De Filippis, L

    C. De Filippis, L. Koch, J. Kristensen, Quantified Legendreness and the regularity of minima. Arch. Ration. Mech. Anal. 248:69, (2024)

    2024

  34. [42]

    De Filippis, G

    C. De Filippis, G. Mingione, On the regularity of minima of non-autonomous functionals. J. Geom. Anal. 30, 1584-1626, (2020)

    2020

  35. [43]

    De Filippis, G

    C. De Filippis, G. Mingione, Lipschitz bounds and nonautonomous integrals. Arch. Ration. Mech. Anal. 242, 973-1057, (2021)

    2021

  36. [44]

    De Filippis, G

    C. De Filippis, G. Mingione, Nonuniformly elliptic Schauder theory. Invent. math. 234, 1109-1196, (2023)

    2023

  37. [45]

    De Filippis, G

    C. De Filippis, G. Mingione, Regularity for double phase problems at nearly linear growth. Arch. Ration. Mech. Anal. 247:85, (2023)

    2023

  38. [46]

    De Filippis, G

    C. De Filippis, G. Mingione, The sharp growth rate for nonuniformly elliptic Schauder estimates. Duke Math. J. 174, 1775-1848, (2025)

    2025

  39. [47]

    De Filippis, G

    C. De Filippis, G. Mingione, Nonuniform ellipticity in variational problems and regularity. Notices Amer. Math. Soc. 72, No.9, (2025)

    2025

  40. [48]

    De Filippis, B

    C. De Filippis, B. Stroffolini, Singular multiple integrals and nonlinear potentials. J. Funct. Anal. 285, 109952, (2023)

    2023

  41. [49]

    De Filippis, M

    F. De Filippis, M. Piccinini, Regularity for multi-phase problems at nearly linear growth. J. Diff. Equ. 410, 832-868, (2024)

    2024

  42. [50]

    De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico

    E. De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico. Boll. Unione Mat. Ital. IV:1, 135-137, (1968)

    1968

  43. [51]

    DiBenedetto, C^ 1+ -local regularity of weak solutions of degenerate elliptic equations

    E. DiBenedetto, C^ 1+ -local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7, 827-850, (1983)

    1983

  44. [52]

    Di Castro, T

    A. Di Castro, T. Kuusi, G. Palatucci, Local behavior of fractional p -minimizers. Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire 33, 1279-1299, (2016)

    2016

  45. [53]

    Diening, K

    L. Diening, K. Kim, H.-S. Lee, S. Nowak, Higher differentiability for the fractional p -Laplacian. Math. Ann. 391, 5631-5693, (2025)

    2025

  46. [54]

    Diening, A

    L. Diening, A. Verde, B. Stroffolini: Everywhere regularity of functionals with -growth. manuscripta math. 129, 449-481, (2009)

    2009

  47. [55]

    Domokos, Differentiability of solutions for the non-degenerate p -Laplacian in the Heisenberg group

    A. Domokos, Differentiability of solutions for the non-degenerate p -Laplacian in the Heisenberg group. J. Diff. Equ. 204, 439-470, (2004)

    2004

  48. [56]

    Eleuteri, P

    M. Eleuteri, P. Marcellini, E. Mascolo, Lipschitz estimates for systems with ellipticity conditions at infinity. Ann. Mat. Pura Appl. 195, 1575-1603, (2016)

    2016

  49. [57]

    Eleuteri, P

    M. Eleuteri, P. Marcellini, E. Mascolo, Regularity for scalar integrals without structure conditions. Adv. Calc. Var. 13, 279-300, (2020)

    2020

  50. [58]

    Esposito, F

    L. Esposito, F. Leonetti, G. Mingione, Sharp regularity for functionals with (p,q) growth. J. Diff. Equ. 204, 5-55, (2004)

    2004

  51. [59]

    Ferreri, L

    L. Ferreri, L. Spolaor, B. Velichkov, Unique continuation for nonlinear variational problems. Preprint (2024). arXiv:2408.00405

    2024 arXiv

  52. [60]

    Fonseca, J

    I. Fonseca, J. Mal\'y, Relaxation of multiple integrals below the growth exponent. Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire 14, 309-338, (1997)

    1997

  53. [61]

    Fonseca, J

    I. Fonseca, J. Mal\'y, G. Mingione, Scalar minimizers with fractal singular sets. Arch. Ration. Mech. Anal. 172, 295-307, (2004)

    2004

  54. [62]

    Fonseca, P

    I. Fonseca, P. Marcellini, Relaxation of multiple integral in subcritical Sobolev spaces. J. Geom. Anal. 7, 57-81, (1997)

    1997

  55. [63]

    Fragalà, F

    I. Fragalà, F. Gazzola, B. Kawohl, Existence and nonexistence results for anisotropic quasilinear elliptic equations. Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire 21, 715-734, (2004)

    2004

  56. [64]

    Frehse, G

    J. Frehse, G. Seregin, Regularity of solutions to variational problems of the deformation theory of plasticity with logarithmic hardening, Proc. St. Petersburg Math. Soc. V, 127-152; Amer. Math. Soc. Transl., Amer. Math. Soc., Providence, RI 193, Ser. 2, (1999)

    1999

  57. [65]

    Fuchs, G

    M. Fuchs, G. Mingione, Full C^ 1, -regularity for free and constrained local minimizers of elliptic variational integrals with nearly linear growth. manuscripta math. 102, 227-250, (2000)

    2000

  58. [66]

    Garain, E

    P. Garain, E. Lindgren, Higher H\"older regularity for mixed local and nonlocal degenerate elliptic equations. Calc. Var. & PDE 62:67, (2023)

    2023

  59. [67]

    Garain, E

    P. Garain, E. Lindgren, Higher H\" o lder regularity for the fractional p -Laplace equation in the subquadratic case. Math. Ann. 390, 5753-5792, (2024)

    2024

  60. [68]

    Giaquinta, Growth conditions and regularity, a counterexample

    M. Giaquinta, Growth conditions and regularity, a counterexample. manuscripta math. 59, 245-248, (1987)

    1987

  61. [69]

    Giaquinta, E

    M. Giaquinta, E. Giusti, On the regularity of the minima of variational integrals. Acta Math. 148, 31-46, (1982)

    1982

  62. [70]

    Giaquinta, E

    M. Giaquinta, E. Giusti, Differentiability of minima of nondifferentiable functionals. Invent. Math. 72, 285-298, (1983)

    1983

  63. [71]

    Giaquinta, E

    M. Giaquinta, E. Giusti, Global C^ 1, -regularity for second order quasilinear elliptic equations in divergence form. J. Reine Angew. Math. (Crelle J.) 351, 55-65, (1984)

    1984

  64. [72]

    Giaquinta, G

    M. Giaquinta, G. Modica, J. Souček, Functionals with linear growth in the calculus of variations I. Comm. Univ. Carolinae 20, 143-156, (1979)

    1979

  65. [73]

    Giaquinta, G

    M. Giaquinta, G. Modica, J. Souček, Functionals with linear growth in the calculus of variations II. Comm. Univ. Carolinae 20, 157-172, (1979)

    1979

  66. [74]

    Gilbarg, N

    D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order. Grundlehren der Mathematischen Wissenschaften, 224. Springer-Verlag, Berlin xiii+513 pp, (1983)

    1983

  67. [75]

    Giraud, Sur le problème de Dirichlet généralisé (Deuxième mémoire)

    G. Giraud, Sur le problème de Dirichlet généralisé (Deuxième mémoire). Ann. Sci. Eötvös Nomin. 46, 131-245, (1929)

    1929

  68. [76]

    Giusti, On the equation of surfaces of prescribed mean curvature

    E. Giusti, On the equation of surfaces of prescribed mean curvature. Invent. Math. 46, 111-137, (1978)

    1978

  69. [77]

    Gmeineder, J

    F. Gmeineder, J. Kristensen, Partial regularity for BV minimizers. Arch. Ration. Mech. Anal. 232, 1429-1473, (2019)

    2019

  70. [78]

    Gmeineder, J

    F. Gmeineder, J. Kristensen, Quasiconvex functionals of (p,q) -growth and the partial regularity of relaxed minimizers. Arch. Ration. Mech. Anal. 248:80, (2024)

    2024

  71. [79]

    H\"ast\"o, J

    P. H\"ast\"o, J. Ok, Maximal regularity for local minimizers of non-autonomous functionals. J. Eur. Math. Soc. 24, 1285-1334, (2022)

    2022

  72. [80]

    H\"ast\"o, J

    P. H\"ast\"o, J. Ok, Regularity theory for non-autonomous partial differential equations without Uhlenbeck structure. Arch. Ration. Mech. Anal. 245, 1401-1436, (2022)

    2022

  73. [81]

    H\"ast\"o, J

    P. H\"ast\"o, J. Ok, Regularity theory for non-autonomous problems with a priori assumptions. Calc. Var. & PDE 62:251, (2023)

    2023

  74. [82]

    Havin, V

    M. Havin, V. Maz'ya, A nonlinear potential theory. Russ. Math. Surveys 27, 71-148, (1972)

    1972

  75. [83]

    A. C. Heinricher, V. Mizel, A stochastic control problem with different value functions for singular and absolutely continuous control. Proceedings 25th IEEE Conference on Decision and Control, Athens 134-139, (1986)

    1986

  76. [84]

    Hirsch, M

    J. Hirsch, M. Sch\"affner, Growth conditions and regularity, an optimal local boundedness result. Comm. Cont. Math. 23, 2050029, (2021)

    2021

  77. [85]

    Hopf, Bemerkungen zu einem satze von S

    E. Hopf, Bemerkungen zu einem satze von S. Bernstein aus del' theorie del' elliptischen differential-gleichungen. Math. Z. 29, 744-745, (1928)

    1928

  78. [86]

    A. V. Ivanov, Local estimates of the maximum modulus of the first derivatives of the solutions of quasilinear nonuniformly elliptic and nonuniformly parabolic equations and of systems of general form. Proc. Steklov Inst. Math. 110, 48-71, (1970); Am. Math. Soc., Providence (1972)

    1970

  79. [87]

    A. V. Ivanov, Quasilinear degenerate and nonuniformly elliptic and parabolic equations of second order. Proc. Steklov Inst. Math. 1:160, xi+287 pp., (1984)

    1984

  80. [88]

    V. V. Jhikov, S. M. Kozlov, O. A. Ole i nik, Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin xii+570 pp., (1994)

    1994

  81. [89]

    Karppinen, M

    A. Karppinen, M. Lee, H\"older continuity of the minimizer of an obstacle problem with generalized Orlicz growth. Inter. Math. Res. Notices 19, 15313-15354, (2022)

    2022

  82. [90]

    Kichenassamy, Schauder-type estimates and applications

    S. Kichenassamy, Schauder-type estimates and applications. Handbook of Differential Equations: Stationary Partial Differential Equations 3, 401-464, (2006)

    2006

  83. [91]

    Kilpel\"ainen, J

    T. Kilpel\"ainen, J. Mal\'y, The Wiener test and potential estimates for quasilinear elliptic equations. Acta Math. 172, 137-161, (1994)

    1994

  84. [92]

    Kristensen, G

    J. Kristensen, G. Mingione, The singular set of -minima. Arch. Ration. Mech. Anal. 177, 93-114, (2005)

    2005

  85. [93]

    Kristensen, G

    J. Kristensen, G. Mingione, The singular set of minima of integral functionals. Arch. Ration. Mech. Anal. 180, 331-398, (2006)

    2006

  86. [94]

    Kuusi, G

    T. Kuusi, G. Mingione, Universal potential estimates. J. Funct. Anal. 262, 4205-4269, (2012)

    2012

  87. [95]

    Kuusi, G

    T. Kuusi, G. Mingione, Linear potentials in nonlinear potential theory. Arch. Ration. Mech. Anal. 207, 215-246, (2013)

    2013

  88. [96]

    Kuusi, G

    T. Kuusi, G. Mingione, A nonlinear Stein theorem. Calc. Var. & PDE 51, 45-86, (2014)

    2014

  89. [97]

    Kuusi, G

    T. Kuusi, G. Mingione, Vectorial nonlinear potential theory. J. Eur. Math. Soc. 20, 929-1004, (2018)

    2018

  90. [98]

    O. A. Ladyzhenskaya, N. N. Ural'tseva, Quasilinear elliptic equations and variational problems with many independent variables. Usp. Mat. Nauk 16, 19-92, (1961)

    1961

  91. [99]

    O. A. Ladyzhenskaya, N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations. Izdat. Nauka, Moscow (1964); Academic Press, New York (1968)

    1964

  92. [100]

    O. A. Ladyzhenskaya, N. N. Ural'tseva, Local estimates for gradients of solutions of nonuniformly elliptic and parabolic equations. Comm. Pure Appl. Math. 23, 677-703, (1970)

    1970

  93. [101]

    Lavrentiev, Sur quelques probl\'emes du calcul des variations

    M. Lavrentiev, Sur quelques probl\'emes du calcul des variations. Ann. Mat. Pura Appl. 4, 7-28, (1926)

    1926

  94. [102]

    Lebesgue, Intégrale, longouer, aire

    H. Lebesgue, Intégrale, longouer, aire. Ann. Mat. Pura Appl. 7, 231-359, (1902)

    1902

  95. [103]

    G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations. Comm. PDE 16, 311-361, (1991)

    1991

  96. [104]

    J. J. Manfredi, Regularity for minima of functionals with p -growth. J. Diff. Equ. 76, 203-212, (1988)

    1988

  97. [105]

    Manià, Sopra un esempio di Lavrentieff

    B. Manià, Sopra un esempio di Lavrentieff. Boll. UMI 13, 147-153, (1934)

    1934

  98. [106]

    Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals

    P. Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals. Ann. IHP-AN 3, 391-409, (1986)

    1986

  99. [107]

    Marcellini, Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions

    P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions. Arch. Ration. Mech. Anal. 105, 267-284, (1989)

    1989

  100. [108]

    Marcellini, The stored-energy for some discontinuous deformations in nonlinear elasticity

    P. Marcellini, The stored-energy for some discontinuous deformations in nonlinear elasticity. Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston 2, (1989)

    1989

  101. [109]

    Marcellini, Regularity and existence of solutions of elliptic equations with p,q -growth conditions

    P. Marcellini, Regularity and existence of solutions of elliptic equations with p,q -growth conditions. J. Diff. Equ. 90, 1-30, (1991)

    1991

  102. [110]

    Marcellini, Everywhere regularity for a class of elliptic systems without growth conditions

    P. Marcellini, Everywhere regularity for a class of elliptic systems without growth conditions. Ann. Scuola Norm. Sup. Pisa Cl. Sci.(IV) 23, 1-25, (1996)

    1996

  103. [111]

    V. G. Maz'ya, Examples of nonregular solutions of quasilinear elliptic equations with analytic coefficients. Funct. Anal. Appl. 2, 230-234, (1968)

    1968

  104. [112]

    Min-Chun, Some remarks on the minimizers of variational integrals with non standard growth conditions

    H. Min-Chun, Some remarks on the minimizers of variational integrals with non standard growth conditions. Boll. Un. Mat. Ital. A 6, 91-101, (1992)

    1992

  105. [113]

    Mingione, The Calderón-Zygmund theory for elliptic problems with measure data

    G. Mingione, The Calderón-Zygmund theory for elliptic problems with measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. V 6, 195-261, (2007)

    2007

  106. [114]

    Mingione, Gradient potential estimates

    G. Mingione, Gradient potential estimates. J. Europ. Math. Soc. 13, 459-486, (2011)

    2011

  107. [115]

    Mooney, O

    C. Mooney, O. Savin, Some singular minimizers in low dimensions in the calculus of variations. Arch. Ration. Mech. Anal. 221:1-22, (2016)

    2016

  108. [116]

    Ne c as, Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity

    J. Ne c as, Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity. Theory of nonlinear operators, Proc. Fourth Internat. Summer School, Acad. Sci., Berlin (1975), Akademie-Verlag, Berlin 197-206, (1977)

    1975

  109. [117]

    Schauder, Über lineare elliptische differentialgleichungen zweiter ordnung

    J. Schauder, Über lineare elliptische differentialgleichungen zweiter ordnung. Math. Z. 38, 257-282, (1934)

    1934

  110. [118]

    Schauder, Numerische abshätzunger in elliptischen linearen differentialgleichungen equations

    J. Schauder, Numerische abshätzunger in elliptischen linearen differentialgleichungen equations. Stud. Math. 5, 34-42, (1934)

    1934

  111. [119]

    Sch\"affner, Lipschitz bounds for nonuniformly elliptic integral functionals in the plane

    M. Sch\"affner, Lipschitz bounds for nonuniformly elliptic integral functionals in the plane. Proc. Am. Math. Soc. 152, 4717-4727, (2024)

    2024

  112. [120]

    Schmidt, Regularity of relaxed minimizers of quasiconvex variational integrals with (p,q) -growth

    T. Schmidt, Regularity of relaxed minimizers of quasiconvex variational integrals with (p,q) -growth. Arch. Ration. Mech. Anal. 193, 311-337, (2009)

    2009

  113. [121]

    Serrin, On the definition and properties of certain variational integrals

    J. Serrin, On the definition and properties of certain variational integrals. Trans. Amer. Math. Soc. 101, 139-167, (1961)

    1961

  114. [122]

    Serrin, Pathological solutions of elliptic differential equations

    J. Serrin, Pathological solutions of elliptic differential equations. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat. (III) 18, 385-387, (1964)

    1964

  115. [123]

    Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables

    J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables. Philos. Trans. R. Soc. Lond. Ser. A 264, 413-496, (1969)

    1969

  116. [124]

    Simon, Interior gradient bounds for nonuniformly elliptic equations of divergence form, Ph

    L. Simon, Interior gradient bounds for nonuniformly elliptic equations of divergence form, Ph. D. Thesis University of Adelaide, (1971)

    1971

  117. [125]

    Simon, Interior gradient bounds for non-uniformly elliptic equations

    L. Simon, Interior gradient bounds for non-uniformly elliptic equations. Indiana Univ. Math. J. 25, 821-855, (1976)

    1976

  118. [126]

    Simon, Schauder estimates by scaling

    L. Simon, Schauder estimates by scaling. Calc. Var. & PDE 5, 391-407, (1997)

    1997

  119. [127]

    Sverák, X

    V. Sverák, X. Yan, Non-Lipschitz Minimizers of Smooth Uniformly Convex Functionals. PNAS 99, 15269-15276, (2002)

    2002

  120. [128]

    H. Triebel. The structure of functions. Monographs in Mathematics, Birkhäuser Verlag, Basel 97, (2001)

    2001

  121. [129]

    Trudinger, The Dirichlet problem for nonuniformly elliptic equations

    N. Trudinger, The Dirichlet problem for nonuniformly elliptic equations. Bull. Am. Math. Soc. 73, 410-413, (1967)

    1967

  122. [130]

    Trudinger, On the regularity of generalized solutions of linear, non-uniformly elliptic equations

    N. Trudinger, On the regularity of generalized solutions of linear, non-uniformly elliptic equations. Arch. Ration. Mech. Anal. 42, 50-62, (1971)

    1971

  123. [131]

    Trudinger, Harnack inequalities for nonuniformly elliptic divergence structure equations

    N. Trudinger, Harnack inequalities for nonuniformly elliptic divergence structure equations. Invent. Math. 64, 517-531, (1981)

    1981

  124. [132]

    Trudinger, A new approach to the Schauder estimates for linear elliptic equations

    N. Trudinger, A new approach to the Schauder estimates for linear elliptic equations. Proc. Centre Math. Appl. 52-59, (1986)

    1986

  125. [133]

    Uhlenbeck, Regularity for a class of non-linear elliptic systems

    K. Uhlenbeck, Regularity for a class of non-linear elliptic systems. Acta Math. 138, 219-240, (1977)

    1977

  126. [134]

    N. N. Ural'tseva, Degenerate quasilinear elliptic systems. Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov (LOMI) 7, 184-222, (1968)

    1968

  127. [135]

    N. N. Ural’tseva, A. B. Urdaletova, The boundedness of the gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations. Vestn. Leningr. Univ. Math. 16, 263-270, (1984)

    1984

  128. [136]

    V. V. Zhikov, Averaging of functionals of the Calculus of Variations and elasticity theory. Math. USSR Izv. 29, 33-66, (1987)

    1987

  129. [137]

    V. V. Zhikov, On Lavrentiev's Phenomenon. Russian J. Math. Phys. 3, 249-269, (1995)

    1995

  130. [138]

    V. V. Zhikov, On some variational problems. Russian J. Math. Phys. 5, 105-116, (1997)

    1997

Pith tools

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