Pith. sign in

REVIEW 3 major objections 4 minor 32 references

Rigidity for perimeter inequality under spherical symmetrisation

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read The paper proves that equality in the spherical-symmetrisation perimeter inequality forces a single global rotation exactly when the cap-angle function is locally absolutely continuous on one interval.

desk verdict First real candidate for the spherical rigidity characterization, but the (ii)=>(i) direction currently rests on sketched circular-symmetrisation machinery that a referee cannot certify from the written proof. read the letter →

arxiv 1908.04865 v2 pith:ALXBRKQI submitted 2019-08-13 math.AP

Filippo Cagnetti , Matteo Perugini , Dominik Stöger This is my paper · ORCID
classification math.AP MSC 49Q2028A7526B30
keywords sphericalsymmetrisationperimeterinequalityrigidityofequalitycasessetsfinitecircularfunctionsboundedvariationcap-anglefunctionfoliatedSchwarzsymmetry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper gives a complete answer to when the perimeter inequality under spherical symmetrisation is rigid: when are the only equality cases, up to sets of measure zero, rotated copies of the spherical symmetral? It proves these extremals are exactly the rotated copies if and only if the set of radii where the cap angle lies strictly between $0$ and $\pi$ is a single interval, and on that interval the cap-angle function is locally absolutely continuous. This matters because spherical symmetrisation is a standard tool for showing that minimisers and PDE solutions have partial symmetry, and rigidity is what upgrades partial symmetry to a definite axis. The proof combines a detailed study of equality cases for sets of finite perimeter with a codimension-one circular symmetrisation, and it constructs explicit counterexamples whenever the interval or absolute-continuity condition fails.

What carries the argument

The objects carrying the argument are the cap-angle function $\alpha_v$ and its approximate limits $\alpha_v^\wedge$ and $\alpha_v^\vee$; the measure $\lambda_E$ that records the radial contribution of the boundary where the tangential normal vanishes; the decomposition of the reduced boundary normal into radial and tangential parts; and the circular symmetrisation, the codimension-one version of the spherical one obtained by slicing with planes and symmetrising circumference arcs. The proof of the sufficiency direction works through the average direction $d_E(r)$, the normalized barycentre of the spherical slice $E_r$, showing that under condition (ii) it lies in $W^{1,1}_{\mathrm{loc}}$ and its derivative vanishes almost everywhere, so all slices share one common axis. The necessity direction builds counterexamples: rotating the set beyond a radius where $\alpha_v$ reaches $0$ or $\pi$, rotating across a jump of $\alpha_v$, and for a nonzero Cantor part approximating the Cantor function by step functions and taking a limit of perimeters.

What would settle it

Take $n=3$ and let $\alpha_v$ be a scaled Cantor function on an interval, with $v$ determined by (1.2)-(1.3); form the set $E$ from equations (8.20)-(8.28) in which each slice is rotated by $\beta(r)=\lambda(\alpha_v(r)-\alpha_v(a))$. If a direct computation of $P(E)$ by approximating $\alpha_v$ with step functions gives strict inequality $P(E)>P(F_v)$, the perimeter-preservation claim in Proposition 8.4 fails. Conversely, verifying $P(E)=P(F_v)$ for such an explicit Cantor $\alpha_v$ — for instance by computing the limit of the piecewise-rotated approximations — tests the necessity direction of Theorem 1.2.

Watch

Extended reading notes

Core claim

On its own terms, the central result is Theorem 1.2. Let $v$ be a measurable area-distribution function satisfying the volume constraint, with finite-volume finite-perimeter associated symmetral $F_v$, and let $\alpha_v$ be the function whose value at radius $r$ is the aperture of the spherical cap of area $v(r)$. Rigidity $(R)$ — every spherically $v$-distributed set $E$ with $P(E)=P(F_v)$ equals a rotated copy of $F_v$ up to negligible sets — holds if and only if the effective set $\{0<\alpha_v^\wedge\le \alpha_v^\vee<\pi\}$ is a possibly unbounded interval $I$ and $\alpha_v$ belongs to $W^{1,1}_{\mathrm{loc}}$ on the interior of $I$, where $\alpha_v^\wedge$ and $\alpha_v^\vee$ are the representative-independent approximate lower and upper limits. The equivalence is established by showing in one direction that the average direction of the spherical slices of an extremal is locally absolutely continuous and satisfies a first-order ODE that forces it constant, and in the other direction by constructing extremals that piece together rotated copies of $F_v$ when the interval condition fails, when $\alpha_v$ jumps, or when its Cantor part is nonzero.

Load-bearing premise

The main equivalence rests on Lemma 1.3, whose proof depends on Theorem 1.4 and Lemma 1.5 about circular symmetrisation; the paper only sketches those proofs by adapting earlier results, so any hidden technical failure there would break the sufficiency direction.

Editorial extensions

If this is right

  • Whenever condition (ii) holds, every extremal of the spherical-symmetrisation perimeter inequality is, up to Lebesgue-negligible sets, the image of the spherical symmetral under one fixed orthogonal transformation.
  • If the set of active radii $\{0<\alpha_v^\wedge\le\alpha_v^\vee<\pi\}$ is disconnected by a radius with $\alpha_v=0$ or $\alpha_v=\pi$, rigidity fails and equality cases can join independently rotated copies of the symmetral on the two sides.
  • If $\alpha_v$ has an approximate jump, rigidity fails: rotating all slices beyond the jump by any sufficiently small angle preserves the perimeter.
  • If $\alpha_v$ has a nonzero Cantor part, rigidity fails: there are extremals whose slice direction rotates by a continuous function built from the Cantor part, so equality cases form a nontrivial family.
  • On an open interval $I$, the condition $\alpha_v\in W^{1,1}_{\mathrm{loc}}(I)$ is equivalent to the tangential part of the boundary of $F_v$ being $\mathcal{H}^{n-1}$-negligible there, linking rigidity to a geometric non-degeneracy of the symmetral's boundary.

Reading between the lines

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

  • A natural testable extension is that the same dichotomy holds for symmetrisations with densities or anisotropic norms: equality cases are rigid exactly when the analogue of the cap-angle is locally absolutely continuous on a single interval, because the average-direction ODE only uses the local geometry of the spheres.
  • For PDE applications, failure of rigidity means spherical symmetrisation alone cannot force a global foliated-Schwarz axis; one would need a second argument to rule out the continuously varying axes that the Cantor-part counterexamples produce.
  • The construction for Cantor parts suggests that whenever the derivative measure of $\alpha_v$ has a singular continuous part, the family of extremals has positive 'dimension', so quantitative stability estimates for the perimeter deficit cannot hold uniformly over all distributions.
  • In dimension $n=2$, spherical and circular symmetrisation coincide, so the theorem gives a complete rigidity criterion for the original two-dimensional setting introduced in 1950; checking the explicit Cantor example numerically would provide a direct verification of the construction.
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

3 major / 4 minor

Summary. The paper studies the perimeter inequality under spherical symmetrisation and gives a necessary and sufficient condition for rigidity, i.e. for the equality cases in (1.4) to consist only of rotated copies of the spherical symmetral F_v. The main result, Theorem 1.2, states that rigidity holds if and only if the set {0 < α_v^∧ ≤ α_v^∨ < π} is an interval I and α_v is locally W^{1,1} on the interior of I. The proof has two parts: the necessity direction (i)⇒(ii) is proved by constructing explicit extremals when the interval condition or W^{1,1} regularity fails, including a construction using jumps and a Cantorian part; the sufficiency direction (ii)⇒(i) is proved through Theorem 1.1, a new Lemma 1.3 on the vanishing of tangential normal components, and a regularity argument for the barycentre direction d_E(r). The paper also proves the perimeter inequality (1.4) in detail and develops tools for the circular symmetrisation introduced by Pólya.

Significance. If the result is correct, it settles a natural rigidity question for spherical symmetrisation that was previously open in this generality; the closest earlier result, [25, Theorem 6.2], only sketched the inequality and did not address rigidity. The paper contains a detailed proof of Theorem 1.1, a genuinely new deformation argument for the necessity direction, and an original reduction of the sufficiency direction to the circular symmetrisation. I also want to credit the authors for making the counterexamples for the non-interval, jump, and Cantor cases explicit. However, the sufficiency direction rests on Sections 6 material — Theorem 1.4 and Lemma 1.5 — whose proofs are only sketched, and on a geometric identification in the proof of Lemma 1.3 that is asserted without proof. These are load-bearing gaps rather than cosmetic omissions.

major comments (3)
  1. [Section 6, Theorem 1.4] Theorem 1.4 is the circular analogue of Theorem 1.1, and its proof is dismissed with the sentence 'Using the results shown above, Theorem 1.4 can be proved by following the lines of the proof of Theorem 1.1.' This is not a routine adaptation: the circular symmetrisation depends on the two-dimensional parameter (r,x′) and involves the singular measures |r D_r^s ξ_ℓ| and |D_{x′}^s ℓ| in Corollary 6.10, which have no counterpart in the one-variable spherical proof of Theorem 1.1. Since Theorem 1.4 is used directly in the proof of Lemma 1.5, and Lemma 1.5 is used to prove Lemma 1.3, this gap affects the implication (ii)⇒(i) of Theorem 1.2.
  2. [Section 6, Lemma 1.5] The proof of Lemma 1.5 is given as 'adapting the arguments used in the proof of [12, Proposition 4.2]'. That proposition concerns codimension-one Steiner symmetrisation with a single spatial variable, whereas Lemma 1.5 concerns the full circular symmetrisation of a set E with distribution ℓ(r,x′) and with the normal decomposition into ν_{12‖} and ν_{12⊥}. The proof passes through Proposition 6.12 and Corollary 6.10, both of which are only stated. Because Lemma 1.5 is the essential bridge from equality in the spherical perimeter inequality to vanishing of the tangential-normal contribution in Lemma 1.3, this is a load-bearing incompleteness in the written proof.
  3. [Lemma 1.3, Step 2b] The proof asserts that after applying circular symmetrisations with respect to (e1,e2), …, (e1,en), one obtains E_n = F_v because 'H1-a.e. spherical section of E is a spherical cap'. This geometric identification is not proved, and it is not obvious when the spherical caps are centred at directions d(r) different from e1; the sequential circular symmetrisations in coordinate planes may interact with the centre of the cap in a nontrivial way. This identification is used to justify the chain of equalities P(F_v; Φ(I×S^{n-1})) = P(E_{n-1}; Φ(I×S^{n-1})) = … = P(E; Φ(I×S^{n-1})), which is central to the proof of (ii)⇒(i).
minor comments (4)
  1. [Corollary 6.10] The displayed formula for P(F_ℓ; Φ_{12}(B×S^1)) contains p_E(r,x′) on the right-hand side, but the corollary is about F_ℓ; this should presumably be p_{F_ℓ}(r,x′), as in the analogous spherical formula (5.9).
  2. [Proposition 4.3, Step 5] The inequality 'P(F_v; Φ(Ω×S^{n-1})) ≤ 2P(E; P(F_v; Φ(Ω×S^{n-1})))' appears to be a typographical error; the right-hand side should be 2P(E; Φ(Ω×S^{n-1})).
  3. [Lemmas 1.3 and 1.5] The word 'Viceversa' should be 'Vice versa' in both statements.
  4. [Figure 1.4] The caption contains the misspelling 'rigitidy' and should read 'rigidity'.

Circularity Check

0 steps flagged · score 0.0 of 10

Theorem 1.2's proof is not circular: rigidity and the interval/W^{1,1} condition are not assumed into each other, and cited prior work is used only as a technical template.

full rationale

The central claim is an equivalence between rigidity (R) and condition (ii). Neither side is presupposed by the other: the implication (i)=> (ii) constructs explicit counterexamples when the interval or W^{1,1} condition fails (Propositions 8.1, 8.3, 8.4), while (ii)=> (i) proves that any extremal has a locally constant direction map d_E (Section 7), relying on Lemma 1.3. Lemma 1.3 is established through the circular symmetrisation results Theorem 1.4 and Lemma 1.5. The paper explicitly says 'We will only sketch the proofs' for these auxiliary results, and Lemma 1.5 is said to follow by adapting [12, Proposition 4.2]; this is a completeness gap in the written proof, not a circular reduction, since [12] is an external result on Steiner symmetrisation and the circular machinery does not presuppose the spherical rigidity theorem. Similarly, the proof of Theorem 1.4 states that it 'can be proved by following the lines of the proof of Theorem 1.1', and Theorem 1.1 itself is proved in detail while adapting the argument of [3, Theorem 1.1]; these are methodological analogies, not logical inputs that already contain the target conclusion. No parameter is fitted and no known result is merely renamed: the spherical rigidity theorem requires a genuinely new construction of non-rigid examples and a new analysis of the circular symmetrisation. The only self-citations are contextual and non-load-bearing. Hence no circular step can be exhibited, and the appropriate score is 0.

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

The central claim rests on standard geometric measure theory (coarea formula, Vol'pert slicing, sphere isoperimetric inequality) and on the finite-perimeter/finite-volume framework for v-distributed sets. The sketched circular symmetrisation results are an additional, less standard dependency.

assumptions (4)
  • standard math Federer's theorem, De Giorgi structure theorem, coarea formula, and Vol'pert slicing theorem for sets of finite perimeter and BV functions
    Invoked in Theorem 3.7, Propositions 3.6 and 6.1, and throughout the perimeter computations.
  • standard math Isoperimetric inequality on the sphere: if a set has measure equal to a geodesic ball, its boundary measure is at least that ball's (see (3.17))
    Critical in Theorem 1.1 and Proposition 4.3 to compare slice perimeters.
  • domain assumption The domain assumption that v satisfies 0 ≤ v(r) ≤ nω_n r^{n-1} and that F_v has finite perimeter and finite volume (or its complement does)
    This is the stated framework of Theorem 1.2; the finite-volume reduction to complements is explained in the introduction.
  • ad hoc to paper The sketched Theorem 1.4 and Lemma 1.5 on circular symmetrisation, which the paper says can be proved by adapting earlier arguments
    These results underpin Lemma 1.3 and hence the (ii) => (i) direction; their proofs are not fully written out in the submitted text.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Rigidity for perimeter inequality under spherical symmetrisation." pith.science (2026). https://pith.science/paper/ALXBRKQI

@misc{pith2026190804865,
  author       = {Pith},
  title        = {Pith review of: Rigidity for perimeter inequality under spherical symmetrisation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ALXBRKQI}},
  note         = {Machine review of arXiv:1908.04865}
}
read the original abstract

Necessary and sufficient conditions for rigidity of the perimeter inequality under spherical symmetrisation are given. That is, a characterisation for the uniqueness (up to orthogonal transformations) of the extremals is provided. This is obtained through a careful analysis of the equality cases, and studying fine properties of the circular symmetrisation, which was firstly introduced by P\'olya in 1950.

Figures

Figures reproduced from arXiv: 1908.04865 by the authors.

Figure 1.1
Figure 1.1. A pictorial idea of the spherical symmetral Fv of a v￾distributed set E, in the case n = 3. Moreover, if A ⊂ R n is any Borel set, we define the perimeter of E relative to A as the extended real number given by P(E; A) := Hn−1 (∂ eE ∩ A), and we set P(E) := P(E; R n ). When E is a set with smooth boundary, it turns out that ∂ eE = ∂E, and the perimeter of E agrees with the usual notion of (n − 1)-dimensional surface… view at source ↗
Figure 1
Figure 1. shows a set [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 1.2
Figure 1.2. shows a set E ∈ N (v) that cannot be obtained by applying a single orthogonal transformation to Fv. This is due to the fact that the set {0 < αv < π} is disconnected r˜ x1 x2 E x1 x2 r˜ Fv [PITH_FULL_IMAGE:figures/full_fig_p005_1_2.png] view at source ↗
Figures from the paper (3 more)
Figure 1.3
Figure 1.3. Figure 1.3: The set E above cannot be obtained by applying an orthog￾onal transformation around the origin to the set Fv shown in the right, therefore rigidity (R) fails. This happens because the set {0 < αv < π} is disconnected by a point ˆr ∈ (0, ∞) such that αv(ˆr) = π. One p…
Figure 1.4
Figure 1.4. Figure 1.4: Modifying the function αv given in [PITH_FULL_IMAGE:figures/full_fig_p006_1_4.png]
Figure 1.5
Figure 1.5. Figure 1.5: An example in which rigidity fails. In this case, the tangential part of ∂ ∗Fv gives a non trivial contribution to P(Fv). This allows to slide a proper subset of Fv around the origin, without modifying the perimeter. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_1_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

32 extracted references · 28 canonical work pages

  1. [1]

    Ambrosio, V

    L. Ambrosio, V. Caselles, S. Masnou, and J. Morel , Connected components of sets of finite perimeter and applications to image processing , J. Eur. Math. Soc. (JEMS), 3 (2001), pp. 39–92

    work page 2001

  2. [2]

    Ambrosio, N

    L. Ambrosio, N. Fusco, and D. Pallara , Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000

    work page 2000

  3. [3]

    Barchiesi, F

    M. Barchiesi, F. Cagnetti, and N. Fusco , Stability of the Steiner symmetrization of convex sets , J. Eur. Math. Soc., 15 (2013), pp. 1245–1278

    work page 2013

  4. [4]

    Bartsch, T

    T. Bartsch, T. Weth, M. Willem , Partial symmetry of least energy nodal solutions to some vari- ational problems, J. Anal. Math., 96 (2005), pp. 1–18

    work page 2005

  5. [5]

    Baernstein, A unified approach to symmetrization , Symposia Matematica, 35 (1995), pp

    A. Baernstein, A unified approach to symmetrization , Symposia Matematica, 35 (1995), pp. 47–91

    work page 1995

  6. [6]

    I , Birindelli, F Leoni, F Pacella, Symmetry and spectral properties for viscosity solutions of fully nonlinear equations, J. Math. Pures Appl. 107(9), (2017), pp. 409–428

    work page 2017

  7. [7]

    B ¨ogelein, F

    V. B ¨ogelein, F. Duzaar, N. Fusco , A quantitative isoperimetric inequality on the sphere , Adv. Calc. Var. (3), 10 (2017), pp. 223–265

    work page 2017

  8. [8]

    J. E. Brothers and W. P. Ziemer , Minimal rearrangements of Sobolev functions , J. Reine Angew. Math., 384 (1988), pp. 153–179

    1988

Show all 32 references
  1. [9]

    Cagnetti, M

    F. Cagnetti, M. Colombo, G. De Philippis, and F. Maggi, Rigidity of equality cases in Steiner’s perimeter inequality, Anal. PDE, 7 (2014), pp. 1535–1593

    2014

  2. [10]

    , Essential connectedness and the rigidity problem for Gaussian symmetrization , J. Eur. Math. Soc. (JEMS), 19 (2017), pp. 395–439

    2017

  3. [11]

    G. M. Capriani, The Steiner rearrangement in any codimension, Calc. Var. Partial Differential Equa- tions, 49 (2014), pp. 517–548. 47

    2014

  4. [12]

    Chleb´ık, A

    M. Chleb´ık, A. Cianchi, and N. Fusco , The perimeter inequality under Steiner symmetrization: cases of equality, Ann. of Math. (2), 162 (2005), pp. 525–555

    2005

  5. [13]

    Cianchi and N

    A. Cianchi and N. Fusco, Functions of bounded variation and rearrangements, Arch. Ration. Mech. Anal., 165 (2002), pp. 1–40

    2002

  6. [14]

    Math., 203 (2006), pp

    , Steiner symmetric extremals in P´ olya-Szeg¨ o type inequalities, Adv. Math., 203 (2006), pp. 673– 728

    2006

  7. [15]

    Cianchi, N

    A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli , On the isoperimetric deficit in Gauss space , Amer. J. Math., 133 (2011), pp. 131–186

    2011

  8. [16]

    De Giorgi, Sulla propriet` a isoperimetrica dell’ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita , Atti Accad

    E. De Giorgi, Sulla propriet` a isoperimetrica dell’ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita , Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I (8), 5 (1958), pp. 33–44

    1958

  9. [17]

    [Author name on title page: Ennio Giorgi], Edited by Luigi Ambrosio, Gianni Dal Maso, Marco Forti, Mario Miranda and Sergio Spagnolo, Reprint of the 2006 edition

    , Selected papers, Springer Collected Works in Mathematics, Springer, Heidelberg, 2013. [Author name on title page: Ennio Giorgi], Edited by Luigi Ambrosio, Gianni Dal Maso, Marco Forti, Mario Miranda and Sergio Spagnolo, Reprint of the 2006 edition

    2013

  10. [18]

    Giaquinta, G

    M. Giaquinta, G. Modica, and J. Soucek , Cartesian currents in the Calculus of Variations. I. Cartesian currents, vol. 37 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Springer-Verlag, Berlin, 1998

    1998

  11. [19]

    Gidas, W.M

    B. Gidas, W.M. Ni, L. Nirenberg , Symmetry and related properties via the maximum principle , Comm. Math. Phys., 68 (1979), 209–243

    1979

  12. [20]

    Kawohl, Rearrangements and convexity of level sets in PDE , vol

    B. Kawohl, Rearrangements and convexity of level sets in PDE , vol. 1150 of Lecture Notes in Math- ematics, Springer-Verlag, Berlin, 1985

    1985

  13. [21]

    Rational Mech

    , On the isoperimetric nature of a rearrangement inequality and its consequences for some vari- ational problems, Arch. Rational Mech. Anal., 94 (1986), pp. 227–243

    1986

  14. [22]

    Maggi, Sets of finite perimeter and geometric variational problems, vol

    F. Maggi, Sets of finite perimeter and geometric variational problems, vol. 135 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2012. An introduction to Geometric Measure Theory

    2012

  15. [23]

    McGillivray, An isoperimetric inequality in the plane with a log-convex density , Ric

    I. McGillivray, An isoperimetric inequality in the plane with a log-convex density , Ric. Mat., 67 (2018), No. 2, pp. 817–874

    2018

  16. [24]

    Morgan, S

    F. Morgan, S. Howe, N. Harman , Steiner and Schwarz symmetrization in warped products and fiber bundles with density , Rev. Mat. Iberoam. 27 (2011), no. 3, 909–918

    2011

  17. [25]

    Morgan, A

    F. Morgan, A. Pratelli , Existence of isoperimetric regions in Rn with density Ann. Global Anal. Geom. 43 (2013), no. 4, 331–365

    2013

  18. [26]

    Perugini, Rigidity of Steiner’s inequality for the anisotropic perimeter (preprint)

    M. Perugini, Rigidity of Steiner’s inequality for the anisotropic perimeter (preprint)

  19. [27]

    P´olya, Sur la sym´ etrisation circulaire

    G. P´olya, Sur la sym´ etrisation circulaire. (French) C. R. Acad. Sci. Paris 230, (1950), pp 25–27

    1950

  20. [28]

    E Schmidt, Beweis der isoperimetrischen Eigenschaft der Kugel im hyperbolischen und sph¨ arischen Raum jeder Dimensionszahl, Math. Z. 49 (1943/44), pp 1–109

    1943

  21. [29]

    Serrin, A symmetry property in potential theory , Arch

    J. Serrin, A symmetry property in potential theory , Arch. Rational Mech. Anal., 43 (1971), pp 304– 318

    1971

  22. [30]

    L. Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, 3, Centre for Mathematical Analysis, Canberra, 1983

    1983

  23. [31]

    Smets, M

    D. Smets, M. Willem, Partial symmetry and asymptotic behavior for some elliptic variational prob- lems, Calc. Var. Partial Differ. Equ., 18 (2003), pp 57–75

    2003

  24. [32]

    A. I. Vol′ pert, Spaces BV and quasilinear equations, Mat. Sb. (N.S.), 73 (115) (1967), pp. 255–302. Department of Mathematics, University of Sussex, Pevensey 2, BN1 9QH, Brighton, UK E-mail address: f.cagnetti@sussex.ac.uk Department of Mathematics, University of Sussex, Peve...

    1967

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.