Pith. sign in

REVIEW 2 major objections 5 minor 76 references

Higher dimensional Sacks-Uhlenbeck-type functionals and applications

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · 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 non-trivial $C^{1,\alpha}$-regular $n$-harmonic spheres $u:(S^n,\mathrm{ground})\to(N,h)$ exist for every $n\ge 2$ when $\pi_{n+k}(N)\neq 0$, under suitable metric assumptions, and that the approximating energies…

desk verdict Significant paper, but the n=3 uniform regularity proof has a real gap that the authors acknowledge in the introduction and never close. read the letter →

arxiv 2506.17166 v1 pith:YXRCI3Q5 submitted 2025-06-20 math.AP math.DG

Gianmichele Di Matteo , Tobias Lamm This is my paper · ORCID
classification math.APmath.DG MSC 35-XX49-XX53-XX
keywords n-harmonicmapsSacks-Uhlenbeckapproximationbubblingenergyidentitymin-maxdoublephasefunctionalsdegenerateellipticPDEshomogeneousspaces
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 extends the classical Sacks–Uhlenbeck existence theory from harmonic $2$-spheres to higher-dimensional $n$-harmonic spheres ($n\ge 2$). Its central claim, Theorem 1.1, is that if the closed target manifold $N$ has $\pi_{n+k}(N)\neq 0$ for some $k\in\mathbb{N}$, then there exists a non-trivial $C^{1,\alpha}$-regular $n$-harmonic map $u:(S^n,\mathrm{ground})\to(N,h)$—for any homogeneous left-invariant metric $h$ in every dimension, and for any metric $h$ when $n=3$. The proof introduces a two-parameter family of approximating energies $E_{p,\delta}$, establishes uniform $C^{1,\alpha_0}$ regularity for small-energy critical points, and develops a bubble-tree (neck) analysis that yields an energy identity under a Struwe-type entropy condition. The authors use that identity to solve min-max problems for the $n$-energy modulo bubbling and to produce infinitely many new null-homotopic $n$-harmonic spheres. The upshot is that the Sacks–Uhlenbeck machinery now operates beyond the conformal two-dimensional case.

What carries the argument

The load-bearing object is the two-parameter approximating family (1.4), $E_{p,\delta}(u)=\frac{1}{p}\int_M[(1+(\delta+|\nabla u|^2)^{n/2})^{p/n}-(1+\delta^{n/2})^{p/n}]\,d\mu_g$, which for $p=n$, $\delta=0$ reduces to the $n$-energy $D_n$ and for $n=2$, $\delta=0$, $\alpha=p/2$ to the Sacks–Uhlenbeck functional. Its Euler–Lagrange system (1.5) is a double-phase-type, non-uniformly elliptic divergence system whose structure balances a $p$-growth phase against an $n$-growth phase; monotonicity in $p$ allows Struwe's trick to produce critical points at min-max levels. The proof's workhorses are the uniform small-energy regularity Theorem 1.3, whose three branches use Luckhaus regularity, Toro–Wang Hardy–BMO compensation for homogeneous targets, and the Cordes condition (a closeness-to-Laplacian algebraic inequality that holds exactly for $n=2,3$) in the three-dimensional case; the Brezis–Coron maximal concentration function to select bubble radii; the entropy condition (1.8) to force the sharp concentration-radius limit (4.9); and a Hopf-differential-type estimate controlling tangential gradient energy on the connecting annuli.

What would settle it

Solve the approximating system (1.5) for a sequence with $p_k\searrow n$, $\delta_k\searrow 0$ and uniformly bounded energy satisfying the entropy condition, and check whether the concentration radii obey $\lim_k (r_k)^{n-p_k}=1$; a violation of this equality, or any bounded critical sequence satisfying (1.8) without the energy identity (1.10), would refute Theorem 1.5. Alternatively, in dimension $n=3$, a closed target with $\pi_{3+k}(N)\neq 0$ for which no non-trivial $C^{1,\alpha}$ 3-harmonic sphere exists under some Riemannian metric would refute Theorem 1.1(ii).

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes three connected results. First, uniform regularity (Theorem 1.3): for $p$ in a narrow interval $(n,P_0)$ and $\delta>0$ small, solutions of the double-phase Euler–Lagrange system associated to $E_{p,\delta}$ are $C^{1,\alpha_0}$ with constants independent of $p$ and $\delta$, provided the rescaled energy is small and one of three conditions holds—local minimality, a homogeneous target with left-invariant metric, or a three-dimensional domain. Second, quantization (Theorem 1.5): a bounded sequence of such critical maps with $p_k\searrow n$, $\delta_k\searrow 0$ and satisfying the entropy bound (1.8) converges, up to extraction, to a weak $n$-harmonic base map plus finitely many $n$-harmonic bubbles $\omega_{i,j}:S^n\to N$, and the $E_{p,\delta}$ and $D_n$ energies decompose exactly as $D_n(u_n)+\sum D_n(\omega_{i,j})$. Third, existence and min-max (Theorems 1.1 and 1.6): the quantized bubble tree realises the min-max value for the $n$-energy, and whenever $\pi_{n+k}(N)\neq 0$ the construction yields a non-trivial regular $n$-harmonic sphere, null-homotopic in infinitely many explicit cases.

Load-bearing premise

The energy-identity theorem (Theorem 1.5) rests on the Struwe-type entropy condition (1.8); the paper itself notes, citing [35], that without such a condition there are sequences of critical maps for which the energy identity fails, so any application of Theorem 1.5 to a general critical sequence must verify this bound.

Editorial extensions

If this is right

  • For any closed target with $\pi_{n+k}(N)\neq 0$ and any homogeneous left-invariant metric, the theorem yields a non-trivial $C^{1,\alpha}$ $n$-harmonic sphere, including cases where $\pi_n(N)=0$ and hence energy minimizers are necessarily trivial.
  • In dimension $n=3$, the same conclusion holds for every Riemannian metric on such targets, so arbitrary-metric existence is settled there.
  • By Corollary 6.4, whenever Theorem 1.1 applies in dimension $n$, it automatically yields non-trivial $i$-harmonic spheres for every $2\le i<n$, since $\pi_{i+(n-i+k)}(N)\neq 0$.
  • The min-max value of the $n$-energy over a homotopy class of maps from $M$ to $N$ is attained by a bubble tree: a regular $n$-harmonic base map plus finitely many $n$-harmonic spheres, with total energy equal to $\beta$ (Theorem 1.6).
  • The generating-set theorem (6.3) gives a subset of homotopy classes that generate $\pi_n(N)$ and each contain a minimizing $n$-harmonic map, extending the classical Sacks–Uhlenbeck picture.

Reading between the lines

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

  • The biggest open structural gap is the regularity branch: the homogeneity assumption in Theorem 1.1(i) enters only through Theorem 1.3(b), so a small-energy regularity theorem for arbitrary targets would automatically extend the existence result to every Riemannian metric in all dimensions; the paper explicitly says such a result is out of reach.
  • If the entropy condition could be shown to follow from compensation phenomena for homogeneous targets, as the paper conjectures, then the energy identity would hold for every bounded critical sequence, not just min-max sequences, and the same min-max conclusions would follow without invoking Struwe's monotonicity trick.
  • The double-phase nature of the approximating energies connects the geometric existence problem to the regularity theory for $(p,q)$-growth functionals; a natural quantitative test is whether the threshold $P_0$ for uniform regularity matches the Cordes-interval bound in dimension $3$, where Cordes' condition is exactly what makes the argument work.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper develops a higher-dimensional analogue of the Sacks-Uhlenbeck construction for n-harmonic maps. It introduces a two-parameter family of approximating functionals E_{p,δ}, studies their Euler-Lagrange systems, and proves uniform C^{1,α} regularity under three alternative hypotheses: local minimality, homogeneous targets, and three-dimensional domains. It then constructs bubble trees for sequences of critical maps, proves an energy identity under a Struwe-type entropy condition, adapts Struwe's monotonicity trick to produce min-max sequences satisfying that entropy condition, and derives existence theorems for non-trivial n-harmonic spheres, including null-homotopic examples. The main claims are Theorem 1.1 (existence of n-harmonic spheres, with arbitrary metrics in the n=3 case), Theorem 1.5 (energy identity), and Theorem 1.6 (min-max modulo bubbling).

Significance. If the regularity results are correct, this is a substantial contribution. The paper extends Sacks-Uhlenbeck's existence theory from n=2 to general n≥2, introduces a new double-phase-type approximation whose degeneracy is handled analytically, provides a bubble-tree construction with energy identity under an explicit entropy hypothesis that the authors verify for their min-max sequences, and derives new existence results including null-homotopic n-harmonic spheres. The careful separation of the entropy condition from the existence statements, together with the counterexample reference [35], shows an appropriate awareness of the limits of the energy-identity result. The regularity work under hypotheses (a), (b), and (c) is the analytic core; hypothesis (c) is the key to the n=3 arbitrary-metric existence theorem and is currently not proven at the claimed uniformity.

major comments (2)
  1. [Section 3.4, proof of Theorem 3.7, after Eq. (3.8)] The claimed uniform C^{1,α0} regularity in Theorem 1.3(c) is not established. From (3.8), Sobolev embedding gives only a uniform C^{0,α} bound for u with α>1/2, not a bound for ∇u. The proof then invokes Corollary 3.4 to deduce local minimality and applies Theorem 1.3(a) to upgrade to uniform C^{1,α0}. However, Corollary 3.4 is explicitly non-uniform: its scale τ1 depends on p, and its proof uses a Sobolev embedding whose constant degenerates as p↘3. The Introduction itself acknowledges that this local minimality 'works on a scale τR0 such that τ→0 as p↘n.' Consequently, Theorem 1.3(a) can only be applied on balls of radius τ0τ1R0, which shrink as p approaches 3, rather than on the fixed ball B_{τ0R0}(x0) required for the bound (1.6). No alternative bootstrap to a fixed scale is supplied. This gap affects Theorem 1.1(ii), the n=3 case of Theorem 1.2, and the case-(c) versions of Theorems 1.5 and 1.6. Please provide a direct argument for the uniform C^{1,α} bound (for example, by proving that in Corollary 2.4 one may choose q1>3 for n=3, or by another iteration of the estimate (3.8)), or revise the statements accordingly.
  2. [End of Section 3.2 and Corollary 3.4] The proof of Theorem 1.3(b) for homogeneous targets relies on the same passage from range-restricted minimality to unrestricted local minimality. The text states that once uniform Hölder regularity is obtained, the uniqueness result in the next subsection gives minimality and then Theorem 1.3(a) applies. But the only written mechanism for removing the range restriction is Corollary 3.4, whose scale degenerates as p↘n. Since the target geometry in the homogeneous case does not remove the dependence on the Sobolev embedding constant in the proof of Corollary 3.4, the uniform C^{1,α0} estimate on a fixed ball is not justified as written. This affects part (i) of Theorem 1.1. Please clarify whether a uniform unrestricted-minimality argument is intended and supply it, or state the theorem at the smaller scale that is actually obtained.
minor comments (5)
  1. [Theorem 1.5, Eq. (1.8)] The entropy condition is written with the limit 'lim_{k→n}'; it should be 'lim_{k→∞}'.
  2. [Corollary 3.4] In the last line of the statement, 'B^N_{ρ2}(P1)' appears to be a typo and should read 'B^N_{ρ2}(y0)'.
  3. [Lemma 4.5] The proof of Lemma 4.5 is compressed: the verification of the boundary term estimate after (4.13) and the telescoping summation over the dyadic annuli are only sketched. Please expand these steps so that the constant dependence in (4.11) can be checked.
  4. [Theorem 2.9] Theorem 2.9 is presented as a sketch of a known analogue of Sacks-Uhlenbeck's result; if it is not proved in detail, it would be clearer to state it as a reference to [58] with the necessary uniform modifications, rather than as a proof sketch in the main text.
  5. [Section 4.1] The notation E^{(s_k)}_{p_k,δ_k} is used before the parameter s is formally introduced in the rescaled system (4.2); adding a one-sentence reminder of the s-dependence would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the derivation is self-contained, and the in-text regularity caveat is a correctness concern, not a circular step.

full rationale

The paper's claimed results are not reached by assuming them. The approximating functionals (1.4) are introduced independently of the target statements; Theorem 1.3(a) is proved from Luckhaus's Proposition 2.15 and Duzaar-Mingione's arguments, with the uniformity checks carried out in the paper rather than imported from the conclusion. Theorem 1.5's energy identity is conditional on the entropy hypothesis (1.8), which is a genuine external condition; the paper explicitly cites [35] for sequences where the identity fails without it, and Lemma 5.1 later proves the condition for min-max critical points rather than imposing it, so the existence theorems do not reduce to the identity. The self-citation to Lamm [33] is a methodological template ('the proof of this result resembles the 2-dimensional analogue treated in [33]'), and the present proof re-implements the estimates in the n-dimensional, double-phase setting; it is not used as an unverified authority. The only explicit obstacle stated in the paper is the scale degeneration in Theorem 1.3(c): 'We explicitly remark that this works on a scale τR0 such that τ→0 as p↓n, so the newly gained uniform regularity (deduced from Theorem 1.3 point (a)) cannot be extended to a uniform scale.' This is a possible gap in the proof of uniform C^{1,α0} regularity for n=3, and hence Theorem 1.1(ii) may be conditional on completing that argument. But a gap is not circularity: no equation is defined in terms of the quantity it is supposed to predict, no fitted parameter is renamed as a prediction, and the local minimality used in part (c) is obtained from the uniqueness Proposition 3.3/Corollary 3.4 rather than assumed from the conclusion. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper's central claims rest on standard theorems in geometric analysis (Nash embedding, Luckhaus, Toro-Wang, Struwe monotonicity) and on the explicit hypotheses of the theorems (topological non-triviality, entropy condition). No ad hoc free parameters or invented entities are introduced.

assumptions (6)
  • standard math Nash embedding theorem: (N,h) is isometrically embedded into some Euclidean space, so maps into N are treated as vector-valued.
    Used throughout (Section 2.2) and prerequisite for the Euler-Lagrange system (1.5).
  • standard math Luckhaus' regularity theorem (Proposition 2.15) provides partial Hölder regularity for minimizers of p-energies, uniformly as p↘n.
    Used in Theorem 1.3(a) via the blow-up argument; uniformity is argued from Remarks 2.16-2.17.
  • standard math Toro-Wang regularity theory for p-harmonic maps into homogeneous spaces (Theorem 1.3(b)) is valid with constants uniform in p near n.
    Adopted in Section 3.2; the paper adapts the compensated compactness argument but relies on that external result.
  • standard math Struwe's monotonicity trick (Lemma 5.1) yields critical points satisfying the entropy condition (1.8) for the min-max construction.
    Invoked in Section 5 to connect the energy identity to min-max problems.
  • domain assumption The Struwe-type entropy condition (1.8) is assumed for the sequence of critical points in Theorem 1.5.
    It is an explicit hypothesis of the energy identity theorem; without it the energy identity can fail (cf. [35]).
  • domain assumption Topological hypothesis: π_{n+k}(N) ≠ 0 for some k in Theorem 1.1.
    Necessary for the min-max argument to produce non-trivial critical points (Theorem 2.11).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Higher dimensional Sacks-Uhlenbeck-type functionals and applications." pith.science (2026). https://pith.science/paper/YXRCI3Q5

@misc{pith2026250617166,
  author       = {Pith},
  title        = {Pith review of: Higher dimensional Sacks-Uhlenbeck-type functionals and applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YXRCI3Q5}},
  note         = {Machine review of arXiv:2506.17166}
}
abstract

In this work, we generalize Sacks-Uhlenbeck's existence result for harmonic spheres, constructing for $n \ge 2$, regular, non-trivial, $n$-harmonic $n$-spheres into suitable target manifolds. We obtain an infinite family of new null-homotopic such maps. The proof follows a similar perturbative argument, which in high dimensions leads to a degenerate and double-phase-type Euler-Lagrange system, making the uniform regularity needed to formalize the bubbling harder to achieve. Then, we develop a refined neck-analysis leading to an energy identity along the approximation, assuming a suitable Struwe-type entropy bound along a sequence of critical points. Finally, we combine these results to solve quite general min-max problems for the $n$-energy modulo bubbling.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

76 extracted references · 66 canonical work pages

  1. [33]

    T. Lamm. Energy identity for approximations of harmonic maps from surfaces.Trans. Amer. Math. Soc.362(8), (2010)

    work page 2010

  2. [35]

    Y. Li, Y. Wang. A counterexample to the energy identity for sequences of α-harmonic maps. Pac. J. Math. , V ol. 274(1), (2015)

    work page 2015

  3. [1]

    Adams, M

    D.R. Adams, M. Frazier. Composition operators on potential spaces. Proc. Amer. Math. Soc., 114(1), 155–165, (1992)

    work page 1992

  4. [2]

    Brezis, J.M

    H. Brezis, J.M. Coron. Convergence of solutions of H-systems or how to blow bubbles. Arch. Rat. Mech. Anal. , 89, 21-56 (1985)

    work page 1985

  5. [3]

    Bojarski, T

    B. Bojarski, T. Iwaniec. P-harmonic equation and quasiregular mapping. Banach Center Publ. : 19(1), 25-38 (1987)

    work page 1987

  6. [4]

    J. Chen, G. Tian. Compactification of moduli space of harmonic mappings. Comm. Math. Helv. , 74, 201–237 (1999)

    work page 1999

  7. [5]

    Colombo, G

    M. Colombo, G. Mingione. Calder´ on-Zygmund estimates and non-uniformly elliptic operators.J. Functional Anal., 270, 1416-1478 (2016)

    work page 2016

  8. [6]

    Colombo, R

    M. Colombo, R. Tione. Non-classical solutions of the p-laplace equation. JEMS, accepted, (2022)

    2022

Show all 76 references
  1. [7]

    De Filippis, G

    C. De Filippis, G. Mingione. Manifold constrained non-uniformly elliptic problems. J. Geom. Anal, 30 1661-1723 (2020)

    2020

  2. [8]

    De Filippis, G

    C. De Filippis, G. Mingione. A borderline case of Calderon-Zygmund estimates for non-uniformly elliptic problems. St. Petersburg math. j. , 31 455-477 (2020)

    2020

  3. [9]

    De Filippis, G

    C. De Filippis, G. Mingione. Nonuniformly elliptic schauder theory. Inventiones Mathematicae, 234, 1109-1196, (2023)

    2023

  4. [10]

    Di Matteo, T

    G. Di Matteo, T. Lamm. Sacks-Uhlenbeck type regularity for subcritical generalized p-harmonic maps into Homoge- neous targets. (2025)

    2025

  5. [11]

    W.Y. Ding, G. Tian. Energy identity for a class of approximate harmonic maps from surfaces. Comm. Anal. Geom. , 3, 543–554 (1995)

    1995

  6. [12]

    Duzaar, M

    F. Duzaar, M. Fuchs. On removable singularities of p-harmonic maps. Annales de l’I. H. P. , C7(5), 385-405 (1990)

    1990

  7. [13]

    Duzaar, E

    F. Duzaar, E. Kuwert. Minimization of conformally invariant energies in homotopy classes. Calc. Var. PDE, 6, 285–313 (1998)

    1998

  8. [14]

    Duzaar, G

    F. Duzaar, G. Mingione. The p-harmonic approximation and the regularity of p-harmonic maps. Calc. Var. PDE, 20, 235–256 (2004)

    2004

  9. [15]

    Esposito, F

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

    2004

  10. [16]

    Esposito, F

    L. Esposito, F. Leonetti, G. Mingione. Higher integrability for minimizers of integral functionals with ( p,q )-growth. J. Differential Equations, 157, 414-438, (1999)

    1999

  11. [17]

    Esposito, F

    L. Esposito, F. Leonetti, G. Mingione. Regularity results for minimizers of irregular integrals with (p,q )-growth. Forum Mathematicum, 14, 245-272,(2002)

    2002

  12. [18]

    L.C. Evans. Partial Regularity for Stationary Harmonic Maps into Spheres. Archive for Rat. Mech. Anal., 116, 101–113 (1991)

    1991

  13. [19]

    L.C. Evans. Partial Differential Equations: Second Edition. Graduate Studies in Mathematics, AMS , 19, (2010)

    2010

  14. [20]

    Fardoun, R

    A. Fardoun, R. Regbaoui. Regularity and uniqueness of p-harmonic maps with small range. Geom. Dedicata, 164, 259–271 (2013)

    2013

  15. [21]

    J. Frehse. A discontinuous solution of a mildly nonlinear elliptic system. Math. Zeit., 134, 229–230 (1973)

    1973

  16. [22]

    M. Fuchs. P-harmonic obstacle problems. I: Partial regularity theory. Ann. Mat. Pura Appl. , IV. Ser. 156 , 127–158 (1990)

    1990

  17. [23]

    Fusco, J

    N. Fusco, J. Hutchinson. Partial regularity for minimisers of certain functionals having nonquadratic growth. Ann. Mat. Pura Appl. , IV. Ser. 155 , 1–24 (1989)

    1989

  18. [24]

    Giacquinta, G

    M. Giacquinta, G. Modica. Partial regularity of minimizers of quasiconvex integrals. Annales de l’I.H.P., 3(3), 185-208, (1986)

    1986

  19. [25]

    A. Hatcher. Algebraic topology. Cambridge Univ. Press , (2002)

    2002

  20. [26]

    Hardt, F.H

    F. Hardt, F.H. Lin. Mappings Minimizing the Lp Norm of the Gradient. Comm. Pure and Appl. Math. , V ol. 40(5):555–588, 1987

    1987

  21. [27]

    H´ elein, J.C

    F. H´ elein, J.C. Wood. Harmonic maps.Handbook of global analysis , 1213, 417-491

  22. [28]

    H´ elein

    F. H´ elein. Regularity of weakly harmonic maps from a surface into a manifold with symmetries. Manuscripta Math., 70, 203–218, (1991)

    1991

  23. [29]

    Iwaniec, A

    T. Iwaniec, A. Lutoborski. Integral estimates for null Lagrangians. Arch. Rational Mech. Anal., 125, 25–79, (1993)

    1993

  24. [30]

    J. Jost. Two-dimensional geometric variational problems. John Wiley and Sons, Chichester , (1991)

    1991

  25. [31]

    M. K¨ uhn. On viscosity solutions and the normalized p-Laplace operator. Doctoral Thesis, Universit¨ at zu K¨ oln, (2017)

    2017

  26. [32]

    Ladyzhenskaya, N.N

    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

  27. [34]

    Y. Li, Y. Wang. A weak energy identity and the length of necks for a sequence of Sacks–Uhlenbeck α-harmonic maps. Adv. in Math. , V ol. 225(3), 1134-1184 (2010)

    2010

  28. [36]

    J. Li, X. Zhu. Energy identity and necklessness for a sequence of Sacks-Uhlenbeck maps to a sphere. Ann. I. H. Poincar´ e, 36, 103–118 (2019)

    2019

  29. [37]

    F.H. Lin, T. Riviere. Energy quantization for harmonic maps. Duke Math. J. 111(1), 177-193 (2002). 46 GIANMICHELE DI MATTEO, TOBIAS LAMM

    2002

  30. [38]

    Luckhaus

    S. Luckhaus. Partial H¨ older continuity for minima of certain energies among maps into Riemannian manifold.Indiana Univ. Math. J. , 37, 349–367 (1988)

    1988

  31. [39]

    Manfredi, A

    J.J. Manfredi, A. Weitsman. On the fatou theorem for p-harmonic function. Commun. PDE, 13(6), 651–668, (1988)

    1988

  32. [40]

    Marcellini

    P. Marcellini. On the definition and the lower semicontinuity of certain quasiconvex integrals. Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 3, 391-409,(1986)

    1986

  33. [41]

    Marcellini

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

    1989

  34. [42]

    Marcellini

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

    1991

  35. [43]

    Marcellini

    P. Marcellini. Regularity for elliptic equations with general growth conditions. J. Differ. Equ. , 105, 296-333, (1993)

    1993

  36. [44]

    Martino, A

    D. Martino, A. Schikorra. Regularizing properties of n-Laplace systems with antisymmetric potentials in Lorentz spaces. Math. Ann., accepted (2023)

    2023

  37. [45]

    Maugeri, D.K

    A. Maugeri, D.K. Palagachev, L.G. Softova. Elliptic and Parabolic Equations with Discontinuous Coefficients. Math. Research, 109, (2002)

    2002

  38. [46]

    Mingione, V

    G. Mingione, V. Radulescu. Recent developments in problems with nonstandard growth and nonuniform ellipticity. J. math. anal. app. , 501, (2021)

    2021

  39. [47]

    Mingione

    G. Mingione. Short tales from nonlinear calder´ on-zygmund theory.Nonlocal and nonlinear diffusions and interactions: new methods and applications , 159–204, springer verlag, (2017)

    2017

  40. [48]

    Mi´ skiewicz, B

    M. Mi´ skiewicz, B. Petraszczuk, P. Strzelecki. Regularity for solutions of H-systems and n-harmonic maps with n/2 square integrable derivatives. Nonlinear Anal., V ol. 232, 2023

    2023

  41. [49]

    J.D. Moore. Energy growth in minimal surface bubbles. Preprint, (2007)

    2007

  42. [50]

    L. Mou, P. Yang. Regularity for n-harmonic maps. J. Geom. Anal., V ol. 6(1):91–112, 1996

    1996

  43. [51]

    L. Mou, C. Wang. Bubbling phenomena of Palais-Smale-like sequences of m-harmonic type systems. Calc. Var. PDE, 4, 341–367 (1996)

    1996

  44. [52]

    Naber, D

    A. Naber, D. Valtorta. Energy identity for stationary harmonic maps. arXiv:2401.02242v1, (2024)

    2024 arXiv

  45. [53]

    Nakauchi, S

    N. Nakauchi, S. Takakuwa. A remark on p-harmonic maps. Nonlinear Anal., 25(2), 169-185 (1995)

    1995

  46. [54]

    J. Nash. The imbedding problem for Riemannian manifolds. Annals of Math. , 63(1), 20–63 (1956)

    1956

  47. [55]

    T.H. Parker. Bubble tree convergence for harmonic maps. J. Diff. Geom. , (1996)

    1996

  48. [56]

    D.C. Ravenel. Complex Cobordism and Stable Homotopy Groups of Spheres. AMS Chelsea Publishing , (2003)

    2003

  49. [57]

    Riviere, P

    T. Riviere, P. Strzelecki. Sharp Nonlinear Gagliardo Nirenberg Type Estimate and Applications to the Regularity of Elliptic Systems. Comm. PDE, 30(4), (2005)

    2005

  50. [58]

    Sacks, K

    J. Sacks, K. Uhlenbeck. The Existence of Minimal Immersions of 2-Spheres. Annals of Math., V ol. 113(1), 1-24 (1981)

    1981

  51. [59]

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

    1976

  52. [60]

    M. Struwe. The existence of surfaces of constant mean curvature with free boundaries. Acta Math, 160, 19–64, (1988)

    1988

  53. [61]

    M. Struwe. Non uniqueness in the Plateau problem for surfaces of constant mean curvature. Arch. Rat. Mech. Anal. , 93, (1986)

    1986

  54. [62]

    M. Struwe. Partial regularity for biharmonic maps, revisited. Calc. Var. PDE, 33, 249–262, (2008)

    2008

  55. [63]

    Strzelecki

    P. Strzelecki. Stationary p-harmonic maps into spheres. Banach Center Publ. , 33(1), 383-393 (1996)

    1996

  56. [64]

    Strzelecki

    P. Strzelecki. Regularity of p-harmonic maps from the p-dimensional ball into a sphere. Manuscripta Math. , 82(3- 4):407–415, 1994

    1994

  57. [65]

    Takeuchi

    H. Takeuchi. Some conformal properties of p-harmonic maps and a regularity for sphere valued p-harmonic maps. J. Math. Soc. Japan, V ol. 46(2), 1994

    1994

  58. [66]

    T. Toro, C. Wang. Compactness Properties of Weakly p-Harmonic Maps into homogeneous spaces. Indiana Univ. Math. J., 44(1), 87-113 (1995)

    1995

  59. [67]

    K. Tsukada. Totally geodesic submanifolds of Riemannian manifolds and curvature-invariant subspaces. Kodai Math. J., 19(3), 395-437, (1996)

    1996

  60. [68]

    Uhlenbeck

    K. Uhlenbeck. Regularity for a class of nonlinear elliptic systems. Acta Math., 138, 219–240 (1977)

    1977

  61. [69]

    H. Urakawa. Calculus of variations and harmonic maps. Transl. Math. Mono. of AMS , 132, Providence, RI, (1993)

    1993

  62. [70]

    Ural’tseva, A.B

    N.N. Ural’tseva, A.B. Urdaletova. The boundedness of the gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations. Vestnik Leningrad Univ. Math. , 19 (1983) (russian) english. tran. 16, 263-270, (1984)

    1983

  63. [71]

    C. Wang. Bubble phenomena of certain Palais-Smale sequences from surfaces to general targets. Houston J. Math. , 22, 559–590 (1996)

    1996

  64. [72]

    Wang, S.W

    C. Wang, S.W. Wei. Energy identity for m-harmonic maps. Diff. Int. Eq. , 15(12),1519–1532 (2002)

    2002

  65. [73]

    S.W. Wei. Representing Homotopy Groups and Spaces of Maps by p-harmonic maps. Indiana Univ. Math. J. , V ol. 47(2), 625-670 (1998)

    1998

  66. [74]

    V.V. Zhikov. Problems of convergence, duality, and averaging for a class of functionals of the calculus of variations. Dokl. Akad. Nauk SSSR , 267, 524-528, (1982)

    1982

  67. [75]

    V.V. Zhikov. Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk SSSR Ser. Mat., 50, 675-710, (1986)

    1986

  68. [76]

    V.V. Zhikov. On Lavrentiev’s phenomenon. Russian J. Math. Phys. 3, 249-269, (1995) Gianmichele Di Matteo, Scuola Superiore Meridionale, Largo San Marcellino 10, 80138 Napoli, Italy Tobias Lamm, Karlsruhe Institute of Technology (KIT), Englerstrasse 2, 76131 Karlsruhe, Germany

    1995

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.