Pith. sign in

REVIEW 2 major objections 4 minor 52 references

On two-dimensional steady compactly supported Euler flows with constant vorticity

T0 review · 2 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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

Pith's one-line read For steady constant-vorticity flows in compact domains, non-annular vortex shapes exist, yet the circle remains the only rigid shape.

desk verdict Solid bifurcation paper with genuinely new results; Theorem 1.5 has a real proof gap in the H2 verification, but the gap is repairable and the result looks true. read the letter →

arxiv 2602.07407 v4 pith:BKVPATX3 submitted 2026-02-07 math.AP

Changfeng Gui , Jun Wang , Wen Yang , Yong Zhang This is my paper · ORCID
classification math.AP MSC 35B3235N0551M10
keywords steadyEulerflowsconstantvorticityfreeboundaryannulardomainbifurcationshapederivativeoverdeterminedellipticproblemstability
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 studies steady, two-dimensional, incompressible Euler flows with constant vorticity whose fluid region is compact and has one or two free boundaries. It proves that for each of three overdetermined free-boundary problems there are special vorticity values at which non-circular, non-annular solution domains branch off from the standard circular annulus, so the circle is not the only shape a constant-vorticity vortex can occupy. It also proves rigidity: under natural positivity or curvature conditions the only admissible domains are circular, and the bifurcating nontrivial solutions must be sign-changing. Finally, it shows via the implicit function theorem that the circular solutions respond smoothly and locally uniquely to small perturbations of the Neumann data, with explicit linear response coefficients. If correct, closed-streamline configurations with constant vorticity are simultaneously flexible and rigid in a controlled way, opening a path beyond the laminar strip-like water-wave setting.

What carries the argument

The central mechanism is the shape derivative of the stream function with respect to normal boundary perturbations η (and ξ for the inner boundary). Restricted to the trivial annulus, the linearized shape-derivative operator is diagonalized by the Fourier modes cos(kθ); for each mode the spectral condition becomes an explicit scalar dispersion relation (σ_k(γ)=0 for the first problem, μ_k(γ_2)=0 for the two-phase problem, det M_{k,γ}=0 for the fully overdetermined problem). The zeros of these functions give the bifurcation vorticities, and a standard local bifurcation theorem for one-dimensional kernels, with transversality checked explicitly, produces the nontrivial branches. The same inver

What would settle it

Fix λ=1/2, compute γ*_1 from the explicit formula, and solve the free-boundary problem numerically with boundary perturbation η=s cosθ for γ near γ*_1. If no non-annular solution branch appears at that vorticity, or if the leading angular mode is not cosθ, then Theorem 1.1's bifurcation claim fails. Alternatively, check the transversality condition ∂_{γη}G(γ*_1,0)[1,cosθ]≠0 numerically; if it vanishes, the one-dimensional kernel is degenerate and the described curve does not exist.

Watch

Extended reading notes

Core claim

For any inner radius λ∈(0,1), the Bernoulli free-boundary problem admits a C^1 curve of solutions (γ(s),η(s)) branching from the standard annulus at the explicit vorticity value γ*_1 = 4/(λ^2−2λ^2 ln λ−1), with leading boundary perturbation η(s)=s α1 cosθ+o(s); hence non-annular admissible domains exist. The same structure appears for the two-phase problem at γ_2*=γ_1 and for the fully overdetermined problem at two explicit values γ*_1 and γ**_1. The bifurcating solutions are necessarily sign-changing under the rigidity theorems that force positivity to yield radial symmetry. Moreover, the trivial annular flows are locally unique and stable under small Neumann perturbations, with explicit le

Load-bearing premise

The proof leans on the assumption that the map from a boundary perturbation to the normal derivative of the resulting stream function is twice continuously differentiable (C^2) near the trivial annulus; this regularity is imported from external shape-derivative theory and, in the two-phase problem, must survive across an interface where the vorticity jumps.

Editorial extensions

If this is right

  • For every λ∈(0,1), the partially overdetermined problem has non-annular admissible domains for γ near γ*_1<−4; for λ small, higher-mode bifurcations give additional branches with positive vorticity.
  • Rigidity: any positive solution of the first problem with γ>0 is radially symmetric on an annulus, and the two-phase problem's concentric-disk configuration is the only one when the curvature-type condition ∂_ννψ_2=m holds; hence the new branches are sign-changing.
  • Stability: for γ avoiding the finitely many resonant values, every sufficiently small Neumann perturbation ρ has a unique nearby admissible domain, with explicit leading-order amplitude formulas such as η(ρ)≈Σ τ_k cos(kθ)/(2√Q_γ σ_k).
  • The choice of vorticity as the bifurcation parameter yields branches for every λ∈(0,1), and the method is claimed to extend to higher dimensions and to affine vorticity functions γ(ψ)=βψ.

Reading between the lines

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

  • The explicit dispersion relations can be read as ready-made predictions: pick a λ, compute γ*_1, and run a numerical continuation from the annulus to look for a cosθ-symmetric branch; a clean detection at that value would confirm the mechanism, while its absence would point to the imported regularity assumption failing.
  • Because the bifurcating solutions are sign-changing, the corresponding physical flows likely contain an interior stagnation curve (where ψ changes sign) separating counter-rotating regions—a topological feature that could be probed in experimentally realizable vortex-core flows.
  • The stability amplitudes 1/(2√Q_γ σ_k) imply a resonance-like amplification as γ approaches a bifurcation value; near higher-mode bifurcation points the response to forcing at that mode should blow up, so the stable regime is precisely the complement of the bifurcation set—a duality the paper does not spell out.
  • For the fully overdetermined problem, the matrix M_{k,γ} resembles objects that appear in shape-optimization resonance problems; studying det M_{k,γ} as a function of γ for all k could classify all possible bifurcation vorticities, analogous to a spectral trace formula.
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. The paper studies three steady Euler free-boundary problems with constant vorticity in two dimensions: the partially overdetermined problem (1.2), the two-phase problem (1.3), and the fully overdetermined two-free-boundary problem (1.4). For each problem the authors compute explicit radial trivial solutions, linearize via shape derivatives, identify explicit bifurcation values of the vorticity, apply the Crandall–Rabinowitz theorem to obtain nontrivial branches, and use the implicit function theorem to establish stability under small Neumann perturbations. The main results are Theorems 1.1, 1.3, and 1.5, together with Corollaries 1.2, 1.4, and 1.6.

Significance. If correct, the paper gives the first systematic local bifurcation construction of nontrivial compactly supported constant-vorticity Euler flows with closed streamlines, with explicit spectral computations and checkable nondegeneracy conditions. The shape-derivative approach is a genuine alternative to the Hanzawa-transformation methods used in earlier work. The main weakness is that the proof of Theorem 1.5, one of the three central existence results, contains a concrete error in the verification of the Crandall–Rabinowitz transversality condition.

major comments (2)
  1. [§5.2 (Eq. (5.8), proof of Theorem 1.5)] The range of L=∂_{(η,ξ)}G(γ~,(0,0)) is misidentified. At γ~=γ1* of (5.11), M_{1,γ1*}=[[0,0],[C,D]] with C,D≠0, so on the k=1 mode the range is {0}×R and its annihilator is the first-component subspace, not span{(η*,ξ*)}. Hence the statement that the range consists of all pairs orthogonal to (η*,ξ*) is false, and the subsequent quadratic-form computation tests the wrong functional. H2 requires ∂_γ L(η*,ξ*)∉R(L); for γ1* this is equivalent to the first component being nonzero, which is true (the kernel condition forces α1=(λ²−1)β1, so α1+β1=λ²β1≠0), but this is not what is shown. The same error occurs at γ1**. Theorem 1.5 is therefore not proved as written, although the gap appears repairable by a direct annihilator computation.
  2. [§3.2, §4.1, §5.1] The bifurcation and stability arguments require the solution map η↦ψ_η (or (η,ξ)↦ψ_{η,ξ}) to be at least C² into the appropriate Hölder spaces. This is asserted by citing [35, Thm 5.3.2] and Schauder theory. For problem (1.3) the map is for a transmission problem with piecewise-constant coefficients across the fixed interface ∂B_λ; the cited reference is formulated for smooth single-phase shape variations, and the transmission case needs a separate justification (e.g. flattening plus the implicit function theorem). Since all three existence theorems and the stability corollaries depend on this differentiability, a precise lemma or an exact reference covering the transmission case should be supplied.
minor comments (4)
  1. [§4 (after Theorem 1.3)] The proof of Corollary 1.4 is omitted with only a reference to Corollary 1.2. If the argument is truly parallel, please state the relevant linearized operator and the nondegeneracy condition explicitly, or mark the corollary as a remark rather than a numbered result.
  2. [Remark 1.3(i)] The remark states that the bifurcation value γ~ in Theorem 1.5 lies in (−∞,−4) for all λ∈(0,1). This is false for γ1**=4/(2λ²lnλ+λ²−1): for λ=1/2 this value is ≈−3.65. The interval in Theorem 1.5 already gives the correct range; please correct the remark.
  3. [§5.1, display after Eq. (5.8)] The vector multiplying M_{k,γ} is written with β_k sin(kθ), but the Fourier expansion in (2.13) uses cos(kθ). Please correct the typo.
  4. [§3.2, Remark 1.1(i)] The assertion that f_de(0.2483,2)≈0 and hence f_de(λ,k)<0 for all λ∈(0,0.2483) and k≥2 is supported only by a numerical check, not by a proof. Please either prove the required monotonicity in λ or state this as a numerical observation rather than a rigorous claim in the remark.

Circularity Check

0 steps flagged · score 1.0 of 10

No material circularity: bifurcation spectra are computed from explicit linearized shape-derivative problems; the only self-citation is contextual.

full rationale

The central existence results are not circular. The bifurcation parameters are obtained by solving explicit linearized boundary value problems: Lemma 3.2 gives the shape-derivative BVP, Propositions 3.3-3.4 compute the dispersion relation (3.4)-(3.5), and the roots sigma_k=0 are the bifurcation values (3.6); analogous computations appear in Section 4 and Section 5 with the 2x2 matrix M_{k,gamma} in (5.8). These quantities are derived, not fitted from the sought conclusion. The trivial branch is explicit (Lemma 3.1), and the transversality checks are direct computations. Rigidity is quoted from external theorems (Reichel [43], Aleksandrov [2], Serrin [49]), and the stability corollaries follow from the implicit function theorem with explicitly inverted linearized operators. The only self-citation, [31], appears in the introduction as motivation ('The second model extends recent work [31]...') and is not used in any proof. The Frechet differentiability of the solution maps is imported from the external shape-derivative reference [35, Thm 5.3.2] and elliptic regularity [28,39]; while terse, this is external support rather than self-referential circularity. The honest caveats - higher-mode simplicity 'observed numerically' in Remark 1.3(i) and the omitted proof of Corollary 1.4 - are completeness gaps, not circular reasoning. A possible technical flaw in the Theorem 1.5 range/transversality verification would be a correctness issue, not a case of a prediction being equivalent to its inputs. Score 1 reflects only the presence of a minor, non-load-bearing self-citation; no step in the derivation reduces to its own assumption.

Assumptions & free parameters 0 free parameters · 10 assumptions · 0 invented entities

The central claims rest on standard elliptic theory, shape-derivative calculus, Crandall–Rabinowitz bifurcation, Reichel's and Aleksandrov's symmetry theorems, and the implicit function theorem. There are no data-fitted constants; the only choices are the Hölder spaces and the smallness/evenness/zero-mean restrictions on perturbations, which are standard for a local bifurcation analysis. No new physical or mathematical entities are introduced.

assumptions (10)
  • standard math Standard elliptic regularity and Schauder estimates for Dirichlet and transmission problems
    Invoked throughout to ensure existence, uniqueness, and regularity of ψ_η and Ψ_η (§2.1, §3.2, §4, §5).
  • standard math Shape derivative framework and Hadamard's formula (Henrot–Pierre [35])
    Used to compute the linearized operators in §3.2, §4.1, and §5.1.
  • standard math Crandall–Rabinowitz local bifurcation theorem (Theorem 6.1)
    Used to prove Theorems 1.1, 1.3, and 1.5; stated in the Appendix.
  • standard math Reichel's radial symmetry theorem (Theorem 6.2)
    Used for the rigidity results in Remark 1.1(ii) and Remark 1.3(ii); stated in the Appendix.
  • standard math Aleksandrov's theorem (constant mean curvature hypersurfaces are spheres)
    Used in the proof of Theorem 1.3(3).
  • standard math Implicit function theorems (Theorems 6.4–6.5)
    Used to prove Corollaries 1.2, 1.4, and 1.6.
  • domain assumption The free boundary is a small radial graph over a circle: ∂Ω_η = {x+η(x)ν(x)} with η small in C^{2,α} and zero mean; similarly ∂D_ξ
    Restricts to local perturbations of annular equilibria; defines the Banach spaces X,Y,X̃,Ỹ in §2.1.
  • domain assumption Constant vorticity γ(ψ)=γ and zero gravity for closed streamlines
    The paper restricts to constant vorticity and sets the gravitational term to zero for closed-streamline configurations; this is the physical modeling assumption stated in the introduction.
  • domain assumption Even symmetry and zero-mean normalization of perturbations
    The spaces X,Y,X̃,Ỹ restrict to even functions with zero mean, fixing the trivial branch and the Fourier-mode decomposition.
  • domain assumption Positivity of the stream function ψ∈(0,1) for the rigidity results
    Assumed in Remark 1.1(ii) and Remark 1.3(ii) to apply the maximum principle and Reichel's theorem.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On two-dimensional steady compactly supported Euler flows with constant vorticity." pith.science (2026). https://pith.science/paper/BKVPATX3

@misc{pith2026260207407,
  author       = {Pith},
  title        = {Pith review of: On two-dimensional steady compactly supported Euler flows with constant vorticity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BKVPATX3}},
  note         = {Machine review of arXiv:2602.07407}
}
read the original abstract

In this paper, we study the two-dimensional steady compactly supported incompressible Euler equations with free boundaries. We consider flows with constant vorticity that are perturbations of annular equilibria, in contrast to the laminar flows that predominate in the existing literature on steady water waves. More precisely, we analyze three distinct classes of steady Euler flows with compact support, which correspond, respectively, to partially overdetermined, two-phase overdetermined, and (fully) overdetermined elliptic free-boundary problems. Our main contributions are threefold. For each class, we first prove a flexibility result-the existence of nontrivial admissible domains-by combining shape derivatives with local bifurcation theory. Second, we establish the corresponding rigidity results. Third, we apply the implicit function theorem to show that the standard annular flows are stable under small perturbations of the Neumann boundary condition. These results provide new perspectives on the theory of overdetermined elliptic problems.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

52 extracted references · 3 linked inside Pith

  1. [35]

    Henrot, M

    A. Henrot, M. Pierre, Shape variation and optimization (a geometrical analysis), EMS Tracts in Mathematics, Vol.28, European Mathematical Society (EMS), Z¨urich, (2018). HenryM

    2018

  2. [1]

    Agostiniani, S

    V . Agostiniani, S. Borghini, L. Mazzieri, On the Serrin problem for ring-shaped domains. J. Eur. Math. Soc. (JEMS) 27 (2025), 2705–2749. Aleksandrov

    2025

  3. [2]

    A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. V . Vestnik Leningrad. Univ. 13 (1958), 5-8; translated in Amer. Math. Soc. Tansl.(2) 21 (1962), 412–416. ambook

    1958

  4. [4]

    Amick, L

    C. Amick, L. Fraenkel, J. Toland, On the Stokes conjecture for the wave of extreme form, Acta Math. 148 (1982), 193–214. AS

    1982

  5. [5]

    Athanasiadis, I.G

    C. Athanasiadis, I.G. Stratis. On some elliptic transmission problems. Annales Polonici Mathe- matici 63.2 (1996), 137–154. AT

    1996

  6. [6]

    Atkinson, W

    K. Atkinson, W. Han, Spherical Harmonics and Approximations on the Unit Sphere: An Introduc- tion, Lecture Notes in Mathematics vol 2044, Springer (2012). Cavallina

  7. [7]

    Cavallina, Local analysis of a two-phase free boundary problem concerning mean curvature, Indiana Univ

    L. Cavallina, Local analysis of a two-phase free boundary problem concerning mean curvature, Indiana Univ. Math. J. 71 (2022), 1411–1435. CavallinaY

    2022

  8. [8]

    Cavallina, Stability analysis of the two-phase torsional rigidity near a radial configuration, Appl

    L. Cavallina, Stability analysis of the two-phase torsional rigidity near a radial configuration, Appl. Anal. 98 (2019), 1889–1900. CFM

    2019

Show all 52 references
  1. [9]

    Cerezo, I

    A. Cerezo, I. Fern ´andez, P . Mira, On the classification of Serrin planar domains, arXiv:2601.09649. ConstantinS

  2. [10]

    Constantin, W

    A. Constantin, W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math. 57 (2004) 481–527. ConstantinSV

    2004

  3. [11]

    Constantin, W

    A. Constantin, W. Strauss, E. V ˘arv˘aruc˘a, Global bifurcation of steady gravity water waves with critical layers, Acta Math. 217 (2016) 195–262 ConstantinV

    2016

  4. [12]

    Constantin, E

    A. Constantin, E. V ˘arv˘aruc˘a, Steady periodic water waves with constant vorticity: regularity and local bifurcation, Arch. Ration. Mech. Anal. 199 (2011), 33–67. CrandallR

    2011

  5. [13]

    Crandall, P .H

    M.G. Crandall, P .H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321–340. DaiFZ

    1971

  6. [14]

    G. Dai, T. Feng, Y. Zhang, The existence and geometric structure of periodic solutions to rotational electrohydrodynamic waves problem, J. Geom. Anal. 35 (2025), 23pp. DaiSZ

    2025

  7. [15]

    G. Dai, P . Sicbaldi, Y. Zhang, Overdetermined elliptic problems in exterior domains of the hyper- bolic space, arXiv:2405.04348v1. DaiXZ

    arXiv

  8. [16]

    G. Dai, F. Xu, Y. Zhang, The dynamics of periodic traveling interfacial electrohydrodynamic waves: bifurcation and secondary bifurcation, J. Nonlinear Sci. 34 (2024), 31pp. DaiZC

    2024

  9. [17]

    G. Dai, Y. Zhang, Sign-changing solution for an overdetermined elliptic problem on unbounded domain, J. Reine Angew. Math. 803 (2023), 267–293. DaiZ

    2023

  10. [18]

    G. Dai, Y. Zhang, Global bifurcation structure and some properties of steady periodic water waves with vorticity, J. Differential Equations, 349 (2023), 125–137. dpmw

    2023

  11. [19]

    D ´avila, M

    J. D ´avila, M. del Pino, M. Musso, M. H. Wheeler, Overhanging solitary vater waves, (2026) to appear in Invent. Math. DelPW

    2026

  12. [20]

    Del Pino, F

    M. Del Pino, F. Pacard, J. Wei, J. Serrin’s overdetermined problem and constant mean curvature surfaces, Duke Math. J. 164 (2015), 2643–2722. DEP

    2015

  13. [21]

    Dom ´ınguez-V´azquez, A

    M. Dom ´ınguez-V´azquez, A. Enciso, D. Peralta-Salas, Piecewise smooth stationary Euler flows with compact support via overdetermined boundary problems, Arch. Ration. Mech. Anal. 239 (2021), 1327–1347. Eh

    2021

  14. [22]

    Ehrnstr ¨om, J

    M. Ehrnstr ¨om, J. Escher, E. Wahl ´en, Steady water waves with multiple critical layers, SIAM J. Math. Anal. 43 (2011), 1436–1456. EncisoARS

    2011

  15. [23]

    Enciso, A.J

    A. Enciso, A.J. Fernandez, D. Ruiz, P . Sicbaldi, A Schiffer-type problem for annuli with applica- tions to stationary planar Euler flows, Duke Math. J. 174 (2025), 1151–1208. FallMW

    2025

  16. [24]

    Fall, I.A

    M.M. Fall, I.A. Minlend, T. Weth, Unbounded periodic solutions to Serrin’s overdetermined boundary value problem, Arch. Ration. Mech. Anal. 223 (2017), 737–759. FallMW1

    2017

  17. [25]

    Fall, I.A

    M.M. Fall, I.A. Minlend, T. Weth, The Schiffer problem on the cylinder and on the 2-sphere, to appear in J. Eur. Math. Soc. (JEMS) (2025). FarinaV

    2025

  18. [26]

    Farina, E

    A. Farina, E. Valdinoci, On partially and globally overdetermined problems of elliptic type, Amer. J. Math., 135 (2013), 1699–1726. FragalG

    2013

  19. [27]

    Fragal `a, F

    I. Fragal `a, F. Gazzola, Partially overdetermined elliptic boundary value problems. J. Differential Equations 245, 1299–1322 (2008) Gilbarg

    2008

  20. [29]

    Gilsbach, M

    A. Gilsbach, M. Onodera, Linear stability estimates for Serrin’s problem via a modified implicit function theorem, Calc. Var. Partial Differential Equations, 60 (2021), 19pp. Gomez

    2021

  21. [30]

    G ´omez-Serrano, J

    J. G ´omez-Serrano, J. Park, Jaemin, J. Shi, Existence of non-trivial non-concentrated compactly sup- ported stationary solutions of the 2D Euler equation with finite energy, Mem. Amer. Math. Soc. 311 (2025), no. 1577, v+82 pp. GuiWYZ

    2025

  22. [31]

    C. Gui, J. Wang, W. Yang, Y. Zhang, Bifurcation analysis of Stokes waves with piecewise smooth vorticity in deep water, arXiv:2511.03973. HamelN

    arXiv

  23. [32]

    Hamel, N

    F. Hamel, N. Nadirashvili, Circular flows for the Euler equations in two-dimensional annular domains, and related free boundary problems. J. Eur. Math. Soc. (JEMS) 25, 323–368 (2023) Haziot

    2023

  24. [33]

    Haziot, Stratified large-amplitude steady periodic water waves with critical layers, Comm

    S.V . Haziot, Stratified large-amplitude steady periodic water waves with critical layers, Comm. Math. Phys., 381 (2021), 765–797. HenrotO

    2021

  25. [34]

    Henrot, M

    A. Henrot, M. Onodera, Hyperbolic solutions to Bernoulli’s free boundary problem, Arch. Ration. Mech. Anal. 240 (2021), 761–784. HenrotP

    2021

  26. [36]

    Henry, B.V

    D. Henry, B.V . Matioc, On the existence of steady periodic capillary-gravity stratified water waves, Ann. Sc. Norm. Super. Pisa CI. Sci., 12 (2013), 955–974. KamburovS

    2013

  27. [37]

    Kamburov, L

    N. Kamburov, L. Sciaraffia, Nontrivial solutions to Serrin’s problem in annular domains. Ann. Inst. H. Poincar´e C Anal. Non Lin´eaire 38, 1–22 (2021) KeadyN

    2021

  28. [38]

    Keady, J

    G. Keady, J. Norbury, On the existence theory for irrotational water waves, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 1, 137–157 Lady

    1978

  29. [40]

    Levi-Civita, Determinazione rigorosa delle onde irrotazionali periodiche in acqua profonda, Rend

    T. Levi-Civita, Determinazione rigorosa delle onde irrotazionali periodiche in acqua profonda, Rend. Accad. Lincei 33 (1924), 141–150. Nekrasov

    1924

  30. [41]

    A. I. Nekrasov, On steady waves, Izv. Ivanovo-Voznesensk. Politekhn. In-ta 3 (1921). Onodera

    1921

  31. [42]

    Onodera, Linear stability analysis of overdetermined problems with non-constant data, Math

    M. Onodera, Linear stability analysis of overdetermined problems with non-constant data, Math. Eng. 5 (2023), 18 pp. Reichel

    2023

  32. [43]

    W. Reichel, Radial symmetry by moving planes for semilinear elliptic BVPs on annuli and other non-convex domains, Elliptic and parabolic problems (Pont- `a-Mousson, 1994), 164–182, Pitman Res. Notes Math. Ser., 325, Longman Sci. Tech., Harlow,

    1994

  33. [44]

    A. Ros, D. Ruiz, P . Sicbaldi, Solutions to overdetermined elliptic problems in nontriavial exterior domains, J. Eur. Math. Soc. 22 (2020), 1223–1252. Ruiz

    2020

  34. [45]

    Ruiz, Symmetry results for compactly supported steady solutions of the 2D Euler equations, Arch

    D. Ruiz, Symmetry results for compactly supported steady solutions of the 2D Euler equations, Arch. Ration. Mech. Anal. 247 (2023), 25 pp. Ruiz1

    2023

  35. [47]

    D. Ruiz, P . Sicbaldi, J. Wu, Overdetermined elliptic problems in nontrivial contractible domains of the sphere, J. Math. Pures Appl. (9) 180 (2023), 151–187. SchlenkS

    2023

  36. [48]

    Schlenk, P

    F. Schlenk, P . Sicbaldi, Bifurcating extremal domains for the first eigenvalue of the Laplacian, Adv. Math. 229 (2012), 602–632. Serrin

    2012

  37. [49]

    Serrin, A symmetry problem in potential theory, Arch

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

    1971

  38. [50]

    Sicbaldi, New extremal domains for the first eigenvalue of the Laplacian in flat tori, Calc

    P . Sicbaldi, New extremal domains for the first eigenvalue of the Laplacian in flat tori, Calc. Var. Partial Differential Equations 37 (2010), 329–344. Stokes

    2010

  39. [52]

    Varholm, Global bifurcation of waves with multiple critical layers, SIAM J

    K. Varholm, Global bifurcation of waves with multiple critical layers, SIAM J. Math. Anal., 52 (2020), 5066–5089. Wahlen

    2020

  40. [53]

    Wahl ´en, Steady water waves with a critical layer, J

    E. Wahl ´en, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468–

    2009

  41. [54]

    Wahl ´en, J

    E. Wahl ´en, J. Weber, Global bifurcation of capillary-gravity water waves with overhanging profiles and arbitrary vorticity, Int. Math. Res. Not., (2023), 17377–17410. WahlenW2

    2023

  42. [55]

    Wahl ´en, J

    E. Wahl ´en, J. Weber, Large-amplitude steady gravity water waves with general vorticity and crit- ical layers, Duke Math. J., 173 (2024), 2197–2258. WangXZ

    2024

  43. [56]

    J. Wang, F. Xu, Y. Zhang, The existence of stratified linearly steady two-mode water waves with stagnation points, J. Math. Fluid Mech., 27 (2025), 21 pp. Wheeler

    2025

  44. [57]

    M.H. Wheeler, Non-symmetric solutions to an overdetermined problem for the Helmholtz equa- tion in the plane, arXiv:2509.00455v1 STEADY COMPACTLY SUPPORTED EULER FLOWS WITH CONSTANT VORTICITY 33 CHANGFENGGUI, DEPARTMENT OFMATHEMATICS, FACULTY OFSCIENCE ANDTECHNOLOGY, UNIVERSIT...

    arXiv

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.