{"id":"9bc1bf2c-1226-4a82-b3a9-3c5d06ce811a","arxiv_id":"2504.16882","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Symmetry reduces the fractional Q-curvature optimal partition problem on the sphere to one dimension, yielding a minimizer made of disjoint spherical shells.","lead":"This paper proves that an optimal partition problem driven by the fractional Q-curvature on the sphere, under a large symmetry group, has a minimizer shaped like nested spherical shells. It also shows that as competition strengthens in a related fractional Yamabe system, solutions separate into exactly those shells.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 7.2's strict inequality rests on an unproved zero-extension/minimizer step and an external UCP whose hypotheses are not verified; the shell topology in Theorem 1.2 depends on it.","rationale":"The reader's CONDITIONAL verdict is appropriate. The paper's main line is coherent: symmetries restore compactness, Proposition 1.5 gives continuity away from the singular set Z, the competitive system phase-separates to an optimal partition, and Theorem 7.2 converts optimality into the layered structure. The Jacobi-polynomial regularity argument for Proposition 1.5 is well supported, and the existence results follow standard Nehari/PS arguments using Lemma 6.2. The delicate point is exactly the strict inequality in Theorem 7.2. I checked the zero-extension claim: it is not true for the fractional Laplacian that a solution in Λ1 extended by zero solves in Λ, because nonlocal interactions across the interface give a strictly negative directional derivative in directions supported in Λ2. The argument can be repaired by noting that, under the assumed equality, the extension is a global minimizer of J_Λ and hence a critical point, giving the same contradiction; but this step is not written, and the UCP citation is used in place of the needed computation. Additionally, the boundedness of least-energy solutions needed to apply [22, Thm 1.4] is not established in the text. These are fixable gaps, so the verdict should remain CONDITIONAL rather than moving to ACCEPT or REJECT. The concrete test above would settle whether the strict inequality survives and whether the external theorem is actually needed.","tokens_in":34590,"tokens_out":26833,"duration_ms":254156,"concrete_test":"For Λ1=eq^{-1}(a,b), Λ2=eq^{-1}(b,c), Λ=eq^{-1}(a,c) as in Theorem 7.2, compute the directional derivative of J_Λ at the zero-extension u of a least-energy solution in Λ1 in the direction of a nonnegative bump φ∈C_c^∞(Λ2). If ⟨u,φ⟩_{D^s} < 0, as direct integration shows, equality c_Λ=c_{Λ1} is impossible and the strict inequality follows without UCP, so the proof can be patched. Also verify boundedness of least-energy solutions in these annuli, e.g., by Moser iteration, so that q=|u|^{2*_s−2}∈L^∞ and [22, Thm 1.4] applies if one keeps the UCP route. If the derivative computation gives zero, or if boundedness fails, the proof of the shell topology in Theorem 1.2 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The crux is Theorem 7.2(i): the strict inequality c_Λ < min{c_{Λ1}, c_{Λ2}} for nested G-annuli is what forces (0,π)\\∪eq(Θ_i) to be exactly ℓ−1 points, giving the layered topology of Theorem 1.2. The proof says that if equality held, taking a least-energy solution in Λ1 and extending it by 0 in Λ\\Λ1 yields a least-energy solution in Λ, contradicting the unique continuation principle [22, Thm 1.4]. For the fractional Laplacian this does not follow from 'solution in Λ1': for 0≤φ∈C_c^∞(Λ2), φ≠0, the zero extension u satisfies ⟨u,φ⟩_{D^s} = −c_{N,s}∫∫ u(x)φ(y)|x−y|^{−N−2s}dxdy < 0, while the right-hand side ∫|u|^{2*_s−2}uφ = 0, so u is not a weak solution in Λ. The assertion only becomes true if equality makes u a global minimizer of J_Λ and one invokes the unstated fact that minimizers are critical points. The paper also does not verify that the least-energy solutions are bounded, so that q=|u|^{2*_s−2}∈L^∞, a hypothesis of the cited UCP. Thus the strict inequality—and with it the exact ℓ−1-point complement and shell topology—rests on a nonlocal extension step that is at best incomplete and on an external theorem whose hypotheses are not checked.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a symmetric optimal partition problem on the round sphere for the energy associated with the conformal fractional Laplacian P_g^s, in the range s∈(1/2,1), for the symmetry group G=O(m)×O(n) with m+n=N+1. The main result, Theorem 1.2, asserts the existence of a minimizer among ℓ G-invariant open disjoint sets, with the additional conclusions that the minimizing sets are smooth, connected, cover S^N, and have an explicit shell topology S^{m-1}×B^n, S^{m-1}×S^{n-1}×(0,1), ..., B^m×S^{n-1}. The proof strategy is to pass to R^N by stereographic projection, prove a new Hölder regularity result for G-invariant fractional Sobolev functions (Proposition 1.5), and then analyze the phase-separation limit of least-energy solutions of a competitive fractional system as the coupling tends to −∞ (Theorems 1.4 and 7.1–7.3). The paper also states the existence of infinitely many G-invariant fully nontrivial solutions to the competitive system (Proposition 6.5) and presents a self-contained spectral analysis of the conformal fractional Laplacian in terms of Jacobi polynomials.","tokens_in":34939,"tokens_out":11470,"duration_ms":112864,"significance":"If established, this is a substantial contribution: it gives the first existence and topological rigidity result for fractional Q-curvature optimal partitions under symmetries, in a critical-exponent setting where the usual compactness fails. The self-contained spectral approach via Jacobi polynomials, the one-dimensional regularity mechanism in Proposition 1.5 with its explicit singular set Z, and the detailed phase-separation analysis are genuine strengths. The paper is also honest about the role of the assumption s>1/2 and about the places where it relies on results such as the unique continuation principle of [22] and the variational machinery of [17]. However, two load-bearing points in the proof need substantial repair before the main theorem can be considered established; these are described below.","major_comments":[{"comment":"The proof of the claim c_Λ < min{c_Λ1, c_Λ2} is not valid as written. If u_1 is a least-energy solution in Λ_1 and u is its zero extension to Λ, then u is not a weak solution of (1.11) in Λ. Indeed, for any nonzero 0≤φ∈C_c^∞(Λ_2), one has ⟨u,φ⟩_{D^s} = −c_{N,s} ∫∫ u(x)φ(y)|x−y|^{−N−2s} dx dy < 0, while ∫ |u|^{2*_s−2} u φ = 0. The most that follows from the equality c_Λ = c_Λ1 is that u belongs to M_Λ and achieves c_Λ; to obtain the contradiction one must additionally prove that minimizers of J_Λ on M_Λ are weak solutions (a standard Lagrange-multiplier step, but not stated) and that the least-energy solution u_1 is bounded, so that q=|u|^{2*_s−2}∈L^∞ satisfies the hypotheses of the cited unique continuation principle [22, Thm. 1.4]. Since the exact ℓ−1-point structure of the complement in Theorem 7.2(i), and hence the shell topology in Theorem 1.2, is deduced from this strict inequality, the gap is load-bearing and must be fixed.","section":"Theorem 7.2(i), proof of the strict inequality"},{"comment":"The transition from the limit profiles v_{∞,i} to the open partition is not fully justified. Proposition 1.5 gives continuity only on R^N∖Y, where Y=Y_1∪Y_2 is the singular stratum. The proof later asserts that v_{∞,i} is continuous in all of R^N and that Ω_i={v_{∞,i}>0}, but no argument is supplied for positivity or continuity of v_{∞,1} on Y_1 and of v_{∞,ℓ} on Y_2. This matters because Ω_1 and Ω_ℓ are defined to contain Y_1 and Y_2, respectively, and the identity Ω_i={v_{∞,i}>0} is needed to conclude that v_{∞,i}|Ω_i solves (1.11). Moreover, the sentence \"Since Ω_i is smooth by Theorem 7.2\" invokes Theorem 7.2 before it is known that the sets {Θ_i} or {Ω_i} actually solve the optimal partition problem; the logical order of the argument needs to be reorganized and the missing regularity up to the singular stratum supplied.","section":"Theorem 7.3 and Proposition 1.5"}],"minor_comments":[{"comment":"In the statement of Theorem 7.3, the coupling parameter is written as λ_{ij}=η_{ij,k}; this should be η_{ij}=η_{ij,k}.","section":"Theorem 7.3 statement"},{"comment":"In the proof of Theorem 7.3, the notation c_{Θ_i} is used before it is shown that the sets Θ_i are admissible open sets and before c_{Ω_i} is defined; the chain of inequalities involving c_{Θ_i} appears to be intended for c_{Ω_i} after taking interiors.","section":"Proof of Theorem 7.3"},{"comment":"The proof of Lemma 2.7 relies on \"standard arguments with Lyapunov functions\" and cites [24, Lemma 3.2] without giving the details; since the two-sided spectral bound is used in Corollary 2.8 and in the regularity estimates of Section 4, the advertised self-contained derivation is not fully substantiated, although the statement itself is standard via the known formula (2.7).","section":"Lemma 2.7"},{"comment":"There are several minor typographical issues, including the spacing in the notation ∥·∥_{H^s_g(S^N)} and the phrase \"an interior product\" in Section 2.1, which should read \"an inner product.\"","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is honest about its reliance on [22] and [17], and I see no circularity or parameter fitting. The main obstacles are the proof of the strict inequality in Theorem 7.2(i) and the treatment of the singular stratum in Theorem 7.3; both are repairable with additional arguments, so I do not recommend rejection. The paper should be sent back for a revision in which the zero-extension step is either replaced by a correct argument or explicitly supported by a minimizer-regularity lemma, and in which continuity/positivity at the singular stratum is proved or cited."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Juan,\n\nThe paper proves existence and exact layered structure for an optimal partition problem driven by the fractional Q-curvature on the sphere, under an O(m)×O(n) symmetry. The genuinely new piece is Proposition 1.5, a Hölder regularity result for symmetric functions in the fractional Sobolev space, proved via Jacobi polynomials. That part looks correct and is the key technical enabler. The phase-separation machinery is adapted from earlier work, but the adaptation is careful and the limit-profile argument is coherent.\n\nThe main soft spot is the proof of Theorem 7.2(i), the strict inequality c_Λ < min{c_{Λ1}, c_{Λ2}} for nested annuli. The proof says a least-energy solution in Λ1, extended by zero to Λ, would be a least-energy solution in Λ, contradicting unique continuation. That extension is not a solution in Λ: pairing it with a test function in Λ2 gives a negative inner product from the nonlocal cross-term while the right-hand side is zero. So the contradiction as written doesn't hold. The inequality itself is true; you can prove it directly by noting that if equality held, the Λ1-minimizer would be a minimizer in Λ, hence a critical point, and the same cross-term computation contradicts the Euler-Lagrange equation. So it's a repairable gap, not a fatal flaw. The paper also never verifies that least-energy solutions are bounded, which would matter for the cited UCP; the direct argument avoids that need.\n\nOther issues are minor: a few steps lean on cited results (spectral asymptotics of φ_{N,s}, the compact embedding under symmetries), and the shell topology in Theorems 1.2 and 7.1 rests on the strict inequality. But once the gap is fixed, the main arguments should work.\n\nNet: this is a serious contribution. The regularity result is new, the existence theorem meaningful, and the paper deserves a serious referee—with the request to focus on Theorem 7.2(i) and provide a correct proof of the strict inequality. I'd cite the regularity result if I worked on symmetric nonlocal problems. For reading group, maybe after the revision.","headline":"Main theorem likely correct; the new regularity result is solid, but the proof of the strict inequality behind the shell topology has a fixable gap.","tokens_in":35462,"tokens_out":7722,"would_cite":true,"duration_ms":68731,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B33","35R01","35R11","58J40","58J70","58J90","35S15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a symmetric optimal partition problem driven by fractional Q-curvature always has a minimizer on the sphere, and the minimizer consists of ℓ smooth nested shell-shaped regions with sphere-product interfaces.","keywords":["conformal fractional Laplacian","fractional Q-curvature","fractional Yamabe problem","optimal partition problem","phase separation","Jacobi polynomials","cohomogeneity one action","critical Sobolev exponent"],"falsifier":"A concrete check is to compute the least energies $c_\\Lambda$ and $c_{\\Lambda_1},c_{\\Lambda_2}$ for nested ring-shaped regions $\\Lambda_1\\subset\\Lambda\\subset\\mathbb{R}^N$ in the one-dimensional Jacobi-coordinate model, for instance with $N=3$, $m=n=2$, and $s=3/4$; equality $c_\\Lambda=\\min\\{c_{\\Lambda_1},c_{\\Lambda_2}\\}$ would contradict Theorem 7.2 and with it the discrete-layer topology of Theorem 1.2. Another check: exhibit a function in $H^s_g(S^N)^G$, $s\\in(1/2,1)$, with an essential discontinuity outside $Z=S^{m-1}\\times\\{0\\}\\cup\\{0\\}\\times S^{n-1}$, which would disprove Proposition 1.5.","tokens_in":34420,"feed_emoji":"🌐","tokens_out":10287,"duration_ms":91246,"temperature":0.7,"pith_summary":"This paper tackles a variational division of the round sphere $S^N$ into $\\ell$ regions, with cost equal to the least energy of the critical fractional Yamabe equation (the conformal fractional Laplacian of order $2s$) in each region. It proves that, once $O(m)\\times O(n)$ symmetry is imposed with $m,n\\ge2$ and $m+n=N+1$, this optimal partition problem always attains its infimum, and the minimizer has a fully explicit shape: $\\ell$ smooth connected regions stacked like nested tubes, with interfaces diffeomorphic to $S^{m-1}\\times S^{n-1}$. The interest is that without symmetries, critical nonlocal problems fail to be compact, so existence is not automatic. The proof obtains the partition as the phase-separation limit of least-energy solutions of a weakly coupled competitive system as the coupling strength tends to $-\\infty$. A new Hölder regularity theorem for symmetric functions in the fractional Sobolev space on the sphere, proved through Jacobi polynomials and a one-dimensional isometry, is the key step that makes the limit profiles continuous and their supports open.","feed_headline":"Fractional Q-curvature minimizers are nested shells on the sphere","feed_subtitle":"With O(m)×O(n) symmetry, the variational infimum is attained and every interface is a product of spheres.","key_machinery":"The carrying object is the isometric isomorphism between the symmetric fractional Sobolev space and a one-dimensional weighted Sobolev space. Every $G$-invariant function on $S^N$ has the form $u=w\\circ f$ with $f(x,y)=|x|^2-|y|^2$, so the $G$-orbits are level sets of $f$ parametrized by $[-1,1]$. The Jacobi polynomials $P_i^{(\\alpha,\\beta)}$ with $\\alpha=m/2-1$ and $\\beta=n/2-1$, composed with $f$, form an orthogonal eigenbasis for the conformal fractional Laplacian $P^s_g$, with eigenvalues $\\varphi_{N,s}(2i(N-1+2i))$; this yields an isomorphism $j:H^s_g(S^N)^G\\to H^s_h([-1,1])$ with weight $h(t)=(1-t)^\\alpha(1+t)^\\beta$. In this one-dimensional model, elements of $H^s_h$ are Hölder continuous for $s>1/2$, and quantitative Jacobi-polynomial estimates give a Hölder bound away from the endpoints, which translates back to continuity of symmetric functions away from the singular set $Z=S^{m-1}\\times\\{0\\}\\cup\\{0\\}\\times S^{n-1}$. The second piece of machinery is the variational phase-separation argument: least-energy fully nontrivial solutions of the competitive system are shown to converge, as $\\eta_{ij}\\to-\\infty$, to segregated limit profiles whose supports form the optimal partition.","core_discovery":"The paper's central claim is Theorem 1.2. For $N\\ge3$, $s\\in(1/2,1)$, $\\ell\\in\\mathbb{N}$, and $G=O(m)\\times O(n)$ with $m,n\\ge2$ and $m+n=N+1$, the $(G,\\ell)$-optimal partition problem (1.10) admits a solution $\\{U_1,\\ldots,U_\\ell\\}$ consisting of smooth, connected, $G$-invariant open sets that cover $S^N$; after relabeling, $U_1\\cong S^{m-1}\\times B^n$, intermediate $U_i\\cong S^{m-1}\\times S^{n-1}\\times(0,1)$, and $U_\\ell\\cong B^m\\times S^{n-1}$, with $U_i\\cap U_{i+1}\\cong S^{m-1}\\times S^{n-1}$ and $U_i\\cap U_j=\\emptyset$ for $|j-i|\\ge2$. The route is Theorem 1.4: positive least-energy fully nontrivial $G$-invariant solutions of the critical competitive system (1.14), as the competition parameters tend to $-\\infty$, converge strongly in $H^s_g$ to limit profiles whose positivity sets form the optimal partition. Along the way the paper proves the existence of infinitely many $G$-invariant fully nontrivial solutions of that system (Proposition 6.5) and, as the main new analytic tool, Proposition 1.5: every function in $H^s_g(S^N)^G$, $s\\in(1/2,1)$, has a continuous representative on the complement of the singular set $Z=S^{m-1}\\times\\{0\\}\\cup\\{0\\}\\times S^{n-1}$. Via stereographic projection, the same result yields an optimal partition of $\\mathbb{R}^N$ into $\\ell-1$ bounded shell regions plus one unbounded region (Theorem 1.3).","pith_inferences":["Because the proof's only use of $s>1/2$ is the Hölder regularity of limit profiles, a different regularity argument could plausibly extend the same optimal-partition theorem to $0<s\\le1/2$; the paper itself leaves this open.","The one-dimensional Jacobi-coordinate model suggests that the optimal partition problem is essentially a question about $\\ell-1$ ordered transition points in $[-1,1]$, which could be tested numerically by computing least-energy levels of symmetric annular regions and observing where their supports separate.","If unique continuation were unavailable for a broader class of nonlocal operators, the same phase-separation construction could produce minimizers with thick interfaces, so the strict nested-annulus inequality is the place where nonlocality shapes the topology.","The same strategy may apply to any cohomogeneity-one group action on a closed manifold for which an analogous Jacobi-polynomial basis diagonalizes the operator; the topological type of the partition would be dictated by the orbit space."],"forward_implications":["The infimum in the $(G,\\ell)$-optimal partition problem (1.10) is attained for every $\\ell\\in\\mathbb{N}$, so the least-energy cost functional is well posed under this symmetry.","The minimizer has the explicit layered topology described in Theorem 1.2: two end pieces are diffeomorphic to $S^{m-1}\\times B^n$ and $B^m\\times S^{n-1}$, the intermediate pieces are $S^{m-1}\\times S^{n-1}\\times(0,1)$, and interfaces are $S^{m-1}\\times S^{n-1}$.","Via stereographic projection the same result gives an optimal partition of $\\mathbb{R}^N$ into $\\ell-1$ bounded shell-like regions and one unbounded region (Theorem 1.3).","The competitive system (1.14) admits infinitely many fully nontrivial $G$-invariant solutions, and its least-energy positive solutions exhibit complete phase separation in the strong limit as coupling tends to $-\\infty$ (Theorem 1.4).","The regularity theorem Proposition 1.5 shows that $G$-invariant fractional Sobolev functions on the sphere are intrinsically one-dimensional in their regularity: for $s>1/2$ they are locally Hölder continuous away from the two singular orbits."],"supporting_citations":[{"why":"Supplies the unique continuation property for the fractional Laplacian used to prove the strict inequality $c_\\Lambda < \\min\\{c_{\\Lambda_1},c_{\\Lambda_2}\\}$ for nested annular regions.","marker":"[22]"},{"why":"Provides the abstract variational construction for weakly coupled competitive systems, including the Nehari-type projection and genus argument.","marker":"[17]"},{"why":"Establishes the local ($s=1$) phase-separation and optimal-partition method that this paper adapts to the fractional setting.","marker":"[16]"},{"why":"Resolves the same optimal partition problem for integer $s$ and supplies the strategy for critical polyharmonic systems.","marker":"[12]"},{"why":"Provides the compact embedding of symmetric fractional Sobolev spaces used to overcome critical-exponent lack of compactness.","marker":"[2]"},{"why":"Characterizes eigenvalues and eigenfunctions of the Laplacian invariant under $O(m)\\times O(n)$, used for the Jacobi-polynomial basis.","marker":"[29]"},{"why":"Supplies the Jacobi polynomial norm, derivative, and interior asymptotic estimates behind the Hölder regularity of symmetric functions.","marker":"[43]"},{"why":"Gives the principle of symmetric criticality that identifies symmetric critical points with solutions of the original equations.","marker":"[38]"}],"fun_headline_variants":["Nested shells are optimal sphere partitions","Sphere partition minimizers form nested shells","Symmetry forces nested shell minimizers","Fractional Q-curvature yields shell minimizers"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that combining two adjacent ring-shaped regions always strictly lowers the least energy of the fractional problem; this strictness is proved with a unique continuation property of the fractional Laplacian, and if it failed the partition interfaces could have positive thickness instead of the sphere-product layers.","fun_headline_variants_meta":{"raw":{"variants":["Nested shells are optimal sphere partitions","Sphere partition minimizers form nested shells","Symmetry forces nested shell minimizers","Fractional Q-curvature yields shell minimizers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000376,"raw_usage":{"total_tokens":2073,"prompt_tokens":1083,"completion_tokens":990,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":699,"completion_tokens_details":{"reasoning_tokens":936}},"tokens_in":699,"tokens_out":990,"duration_ms":10231,"temperature":1.0,"reasoning_tokens":936,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:54:16.848460+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check is to compute the least energies $c_\\Lambda$ and $c_{\\Lambda_1},c_{\\Lambda_2}$ for nested ring-shaped regions $\\Lambda_1\\subset\\Lambda\\subset\\mathbb{R}^N$ in the one-dimensional Jacobi-coordinate model, for instance with $N=3$, $m=n=2$, and $s=3/4$; equality $c_\\Lambda=\\min\\{c_{\\Lambda_1},c_{\\Lambda_2}\\}$ would contradict Theorem 7.2 and with it the discrete-layer topology of Theorem 1.2. Another check: exhibit a function in $H^s_g(S^N)^G$, $s\\in(1/2,1)$, with an essential discontinuity outside $Z=S^{m-1}\\times\\{0\\}\\cup\\{0\\}\\times S^{n-1}$, which would disprove Proposition 1.5.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the unique continuation property for the fractional Laplacian used to prove the strict inequality $c_\\Lambda < \\min\\{c_{\\Lambda_1},c_{\\Lambda_2}\\}$ for nested annular regions."},{"cited_title":"Clapp and A","cited_arxiv_id":null,"evidence_quote":"Provides the abstract variational construction for weakly coupled competitive systems, including the Nehari-type projection and genus argument."},{"cited_title":"Clapp, J","cited_arxiv_id":null,"evidence_quote":"Resolves the same optimal partition problem for integer $s$ and supplies the strategy for critical polyharmonic systems."},{"cited_title":"Abreu, E","cited_arxiv_id":null,"evidence_quote":"Provides the compact embedding of symmetric fractional Sobolev spaces used to overcome critical-exponent lack of compactness."},{"cited_title":"Henry and J","cited_arxiv_id":null,"evidence_quote":"Characterizes eigenvalues and eigenfunctions of the Laplacian invariant under $O(m)\\times O(n)$, used for the Jacobi-polynomial basis."},{"cited_title":"Szego, Orthogonal Polynomials, vol","cited_arxiv_id":null,"evidence_quote":"Supplies the Jacobi polynomial norm, derivative, and interior asymptotic estimates behind the Hölder regularity of symmetric functions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the principle of symmetric criticality that identifies symmetric critical points with solutions of the original equations."}],"review_version":1}