{"id":"4805d569-45d0-4fb1-b52e-5ec31d375429","arxiv_id":"2608.07691","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For p>1, the relative p-Faber-Krahn inequality is shown to be equivalent to volume doubling together with a sub-Gaussian upper estimate for Trudinger subsolutions, with improved long-time bounds under uniform Faber-Krahn decay.","lead":"This paper proves that, on any complete non-compact Riemannian manifold, a relative p-Faber-Krahn inequality is exactly equivalent to volume doubling plus a sub-Gaussian estimate for subsolutions of the nonlinear Trudinger equation. This generalizes a known characterization for the heat equation to a whole family of doubly nonlinear diffusion equations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; Theorem 1.4 is coherent and the main cited input [21] is disclosed, with the L^sigma-monotonicity citation in Lemma 4.2 a minor secondary caveat.","rationale":"The reader's verdict ACCEPT is well supported. I examined the two new directions of Theorem 1.4. Proposition 3.1 is a standard iteration with a small omitted justification about the terminal ball factor; Theorem 3.6 correctly combines the eigenfunction construction, the UE_p estimate, and volume doubling to produce the relative Faber–Krahn inequality. The main external input, Theorem 1.1 of [21], is the one-way implication (FK_p)⇒(UE_p); relying on a prior published theorem is normal mathematical practice and is explicitly disclosed. I do not regard that as a load-bearing defect. The reader's secondary concern about Lemma 4.2 is real but only affects the improved long-time estimate: Lemma 4.2 applies Lemma 2.3, stated for solutions, to a subsolution. If the asserted L^σ monotonicity extends to subsolutions, which is plausible and likely follows from the same arguments as in [9], then Theorem 4.3/1.7 is fine; otherwise a short additional proof is needed. Since this does not touch the equivalence in Theorem 1.4, the verdict should remain unchanged.","tokens_in":13622,"tokens_out":24358,"duration_ms":227748,"concrete_test":"Independently verify Theorem 1.1 of [21]: follow its proof that (FK_p) implies the on-diagonal upper estimate ||u(t)||_{L∞} ≤ C||u_0||_{L1}/μ(B(x,t^{1/p})) and then the off-diagonal sub-Gaussian factor. If that implication is sound, the only unverified input to Theorem 1.4 is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection to the central claim. Proposition 3.1 derives volume doubling from the relative Faber–Krahn inequality by a standard iterative argument; the apparent issue that the terminal factor μ(B(x,r/2^m)) vanishes is harmless because on a Riemannian manifold its 1/(1+ν)^m-th power tends to 1, although this justification is omitted. Theorem 3.6 is internally sound: the constructed subsolution v=e^{-λt}φ^{p-1} satisfies the hypotheses of (UE_p), the density-point choice of x_D is standard, and the two-case eigenvalue argument correctly yields (FK_p) with ν=p/N. The other direction (FK_p)⇒(UE_p) is imported from the author's prior Theorem 1.1 in [21]; this is disclosed, standard citation practice, and not a red flag. The only concrete soft spot I found is in the secondary result: Lemma 4.2, used in Theorem 4.3/1.7, invokes Lemma 2.3, which is stated for solutions, although u in Lemma 4.2 is only a subsolution. If the L^1 monotonicity of subsolutions is not available, that proof needs an extra argument. This does not affect Theorem 1.4.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Trudinger equation ∂t u = Δp u^{1/(p-1)} on geodesically complete non-compact Riemannian manifolds and proves an equivalence between a relative p-Faber-Krahn inequality and the conjunction of volume doubling and a sub-Gaussian upper estimate for non-negative bounded weak subsolutions. Theorem 1.4 states (FK_p) ⇔ (VD)+(UE_p). The forward direction (FK_p)⇒(VD) is proved in Proposition 3.1; the implication (FK_p)⇒(UE_p) is cited from the author's earlier Theorem 1.1 in [21]. The reverse direction (VD)+(UE_p)⇒(FK_p) is proved in Theorem 3.6 by constructing an exponentially decaying subsolution from the first p-Laplacian eigenfunction and applying the upper estimate at a density point. The paper also derives an improved long-time upper estimate under a uniform Faber-Krahn inequality (Theorem 4.3, Theorem 1.7) and a converse statement (Proposition 4.7).","tokens_in":13824,"tokens_out":17467,"duration_ms":145502,"significance":"If correct, Theorem 1.4 is a nonlinear counterpart of Grigor'yan's characterization for the heat equation and is a valuable contribution to nonlinear potential theory on manifolds. The novel direction (FK_p)⇐(VD)+(UE_p) is proved with standard tools and the dependence on previous results is clearly disclosed. The improved upper estimate in Theorem 1.7 and the converse Proposition 4.7 are useful additions. The paper is generally well written and the main proofs are presented in sufficient detail, apart from the specific gaps noted below.","major_comments":[{"comment":"The proof of Lemma 4.2 applies Lemma 2.3, which is stated for solutions of (2.8), to a subsolution u. Specifically, after the Hölder inequality the proof replaces ∫M u by ∫M u0 'using also Lemma 2.3'. Since u is only assumed to be a non-negative bounded subsolution, the monotonicity of the L1 norm is not covered by Lemma 2.3 as stated, nor by the Caccioppoli inequality of Lemma 2.2, which is restricted to σ ≥ p/(p−1). The author should either prove that t ↦ ∥u(·,t)∥_{L1(M)} is non-increasing for subsolutions (for example, by a cut-off argument in the weak formulation (2.11)) or extend Lemma 2.3 to subsolutions. This is necessary for the proof of (4.33) and hence for Theorem 4.3 and the stated result Theorem 1.7.","section":"Section 4, Lemma 4.2"},{"comment":"In the iteration leading to the volume doubling property, the factor μ(B(x,r/2^m))^{(1/(1+ν))^m} is dropped when sending m→∞. Because μ(B(x,r/2^m)) tends to zero as m→∞, the convergence of this factor to 1 is not automatic. On a Riemannian manifold it follows from the local volume comparability μ(B(x,ρ))≍ρ^n for small ρ, which makes the exponent (1/(1+ν))^m tend to zero while the measure decays at most polynomially, but this justification is omitted. Without it the limiting step in the derivation of (VD) is incomplete; please add a short argument or a reference.","section":"Section 3, Proposition 3.1"}],"minor_comments":[{"comment":"The set D={w>0} is not necessarily open for a general w∈W^{1,p}_0(B). Since the proofs in Section 3 work with precompact open sets D, it would be helpful to state explicitly that the relative Faber-Krahn inequality is assumed for all such D, or to add a remark explaining that the general case follows by approximation.","section":"Definition 1.1"},{"comment":"The approximation of the test function ψ by bounded functions and the passage to the limit ε→0 are only sketched; adding a few more details would improve readability and remove any doubt about the validity of the limiting argument in (3.24).","section":"Lemma 3.5"},{"comment":"There are minor typographical and convention issues, including inconsistent capitalization in the title and abstract, and the reuse of symbols c and C without comment in statements such as Theorem 3.6 and Proposition 4.7. These do not affect the mathematics.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The paper's main theorem depends on the author's previous result [21] for the (FK_p)⇒(UE_p) direction; this is disclosed and is standard citation practice, but the editor may wish to confirm that [21] is published (the reference lists Nonlinear Analysis, 2024). The L1-monotonicity issue in Lemma 4.2 is a genuine gap for the secondary result, but I believe it is fixable within the paper's scope and does not affect Theorem 1.4. The volume-doubling step in Proposition 3.1 needs only a short justification. Overall the central claim is sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Philipp Sürig's paper gives the missing converse half of a nonlinear Faber–Krahn characterization for Trudinger's equation: (FK_p) ⇔ (VD)+(UE_p) for all p>1. The genuinely new direction, (VD)+(UE_p) ⇒ (FK_p), is the core of Theorem 3.6, and it is proved cleanly by applying the upper estimate to the eigenfunction-based subsolution v=e^{-λt}φ^{p-1}. The doubling implication (FK_p)⇒(VD) in Proposition 3.1 is also a straightforward and correct adaptation of Grigor'yan's argument. The other half of the equivalence is imported from the author's earlier result [21], and that is disclosed and cited honestly; [21] is published, so I do not see circularity.\n\nSection 4 is a second, independent contribution: improved long-time upper estimates under a uniform FK inequality with slowly growing Λ_p, including logarithmic examples such as hyperbolic space. The estimate (1.5) genuinely improves the earlier t^{n/p} bound in that regime, and Proposition 4.7 supplies a reasonable converse. The proofs there are coherent.\n\nSoft spots, in proportion. The only concrete issue I found is that Lemma 4.2 invokes Lemma 2.3, which is stated for solutions, while u in Lemma 4.2 is a subsolution. In the range σ≥p/(p-1), the Caccioppoli inequality (2.12) itself gives the needed L^σ monotonicity, and for σ=1 the L^1 monotonicity follows directly from the weak formulation. So the gap is cosmetic; a remark or a lemma stated for subsolutions would fix it. Second, the main theorem's converse half is not reproved, so the paper is not self-contained; that is normal for this area and the provenance is clear. Third, in Proposition 3.1 the iteration sends m→∞ with a short omitted limiting justification, but the terminal factor indeed tends to one. None of this affects Theorem 1.4.\n\nThe citation pattern looks honest, with appropriate credit to Grigor'yan, Carron, and the author's own prior work. No red flags in the data or presentation.\n\nWho gets value: anyone working on doubly nonlinear parabolic equations on manifolds, especially the bridge between volume growth and solution estimates. It deserves a serious referee; I would send it out with a request to clean up the Lemma 2.3 wording. I would cite it.","headline":"A solid extension of Grigor'yan's FK/heat-kernel equivalence to Trudinger's equation; the new converse direction is cleanly proved, with only minor citation-level caveats in Section 4.","tokens_in":14453,"tokens_out":3274,"would_cite":true,"duration_ms":30091,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K55","58J35","35K92"],"pacs":[],"model":"deepseek-v4-flash","headline":"For the Trudinger equation, a relative p-Faber–Krahn inequality is equivalent to volume doubling plus a sub-Gaussian upper estimate for subsolutions.","keywords":["Trudinger equation","doubly nonlinear parabolic equation","Riemannian manifold","relative Faber–Krahn inequality","volume doubling","sub-Gaussian upper estimate","p-Laplacian","long-time decay"],"falsifier":"A concrete test would be to compute, on a geodesically complete non-compact manifold known to satisfy volume doubling and the sub-Gaussian estimate (for example, a manifold with polynomial volume growth), the first $p$-eigenvalue $\\lambda_{1,p}(D)$ for a nested family of balls and check the lower bound $\\lambda_{1,p}(D) \\ge c R^{-p}(\\mu(B(x_0,R))/\\mu(D))^{p/N}$; a sequence of domains violating this bound would disprove the equivalence in Theorem 1.4.","tokens_in":13347,"feed_emoji":"📐","tokens_out":11752,"duration_ms":96776,"temperature":0.7,"pith_summary":"The paper studies the Trudinger equation $\\partial_t u = \\Delta_p u^{1/(p-1)}$ on geodesically complete, non-compact Riemannian manifolds and asks how the behavior of its subsolutions is tied to the geometry. Its central claim is a characterization: a relative $p$-Faber–Krahn inequality (a quantitative lower bound on Dirichlet energy in terms of the volume of the support) holds if and only if the manifold satisfies volume doubling and every non-negative bounded weak subsolution obeys a sub-Gaussian upper estimate, meaning decay $\\exp(-c(d(x,A)/t^{1/p})^{p/(p-1)})$ away from the initial support. This is the nonlinear, $p>1$ counterpart of the classical $p=2$ heat-equation characterization, and it turns a purely geometric isoperimetric-type inequality into a quantitative diffusion statement. The paper also proves an improved long-time upper bound under a uniform $p$-Faber–Krahn inequality, showing that the shape of the Faber–Krahn function $\\Lambda_p$ controls the large-time decay profile. If correct, these results give a geometric criterion for sub-Gaussian decay of Trudinger subsolutions and a method for bounding the first $p$-eigenvalue of geodesic balls.","feed_headline":"Faber–Krahn inequality equals doubling plus sub-Gaussian decay","feed_subtitle":"For Trudinger's equation on complete manifolds, the equivalence is proved for every p>1 and yields sharper long-time decay.","key_machinery":"The load-bearing object is the first eigenfunction of the $p$-Laplacian on a precompact domain $D$: a non-negative $\\varphi\\in W^{1,p}_0(D)$ with $-\\Delta_p \\varphi = \\lambda_{1,p}(D) \\varphi^{p-1}$ and $\\int_D \\varphi^p = 1$. Its zero extension gives a separated weak subsolution $v(x,t)=e^{-\\lambda_{1,p}(D)t}\\tilde\\varphi(x)^{p-1}$ of the Trudinger equation, and applying $(\\mathrm{UE}_p)$ to $v$ converts the exponential decay rate into a lower bound on $\\lambda_{1,p}(D)$; comparing this with the volume-doubling ball lower bound $\\mu(B(x_D,r)) \\ge c(r/R)^N \\mu(B(x_0,R))$ yields the required Faber–Krahn inequality. In the long-time half, the engine is the Caccioppoli-type inequality for powers $u^{\\sigma/p}$, together with the monotonicity of the $L^\\sigma$ norms of solutions; these turn the uniform Faber–Krahn inequality into a differential inequality for $\\Phi(t)=\\int_M u^\\sigma$ whose solution is controlled by the integral $\\gamma$ defined by $t = \\int_0^{\\gamma(t)} dv/(\\Lambda_p(v)v)$. The reverse doubling estimate $\\mu(B(x,Ar)) \\ge c A^{N_0} \\mu(B(x,r))$ is used to handle the case where a ball is nearly filled by the support.","core_discovery":"On a geodesically complete non-compact Riemannian manifold, for any $p>1$, the relative $p$-Faber–Krahn inequality is equivalent to volume doubling together with the sub-Gaussian upper estimate for non-negative bounded weak subsolutions of the Trudinger equation. The forward half $(\\mathrm{FK}_p) \\Rightarrow (\\mathrm{VD})$ is proved here by an iteration argument; the half $(\\mathrm{FK}_p) \\Rightarrow (\\mathrm{UE}_p)$ is taken from the author's previous Theorem 1.1 in [21]. The reverse half $(\\mathrm{VD})+(\\mathrm{UE}_p) \\Rightarrow (\\mathrm{FK}_p)$ is proved by feeding the separated solution $v(x,t)=e^{-\\lambda_{1,p}(D)t}\\varphi(x)^{p-1}$ into the sub-Gaussian estimate, where $\\varphi$ is the first $p$-Laplacian eigenfunction on a precompact domain $D$; this forces the lower bound $\\lambda_{1,p}(D) \\ge c R^{-p}(\\mu(B_R)/\\mu(D))^{p/N}$, which is exactly the relative Faber–Krahn inequality. Under a uniform Faber–Krahn inequality with function $\\Lambda_p$, the paper derives a two-sided correspondence between the long-time decay of $L^\\sigma$ norms and the integral $\\gamma(t)=\\int_0^t dv/(\\Lambda_p(v)v)$, yielding Theorem 1.7 as an improvement over the earlier $t^{-n/p}$ bound in cases where $\\Lambda_p$ decays logarithmically.","pith_inferences":["The proof of Theorem 3.6 does not use the full nonlinear structure beyond homogeneity and the eigenvalue equation, so the same equivalence may hold for other doubly nonlinear parabolic equations with a separated exponential solution; testing this on porous-medium-type variants would be a natural extension.","Theorem 4.3 suggests a sharp transition in long-time decay: polynomial $\\Lambda_p$ gives $t^{-n/p}$, while logarithmic $\\Lambda_p$ degrades the time factor to $t^{-n/(p\\sigma)}$ with an extra stretched-exponential in $t^{1/(\\alpha+1)}$; the paper does not prove matching lower bounds, so whether these rates are sharp remains open.","Corollary 1.5 could be used as a bootstrap in proving $L^1$–$L^\\infty$ smoothing: on a doubling manifold, establishing the sub-Gaussian estimate at a single exponent automatically upgrades it to all larger exponents, which may simplify regularity arguments for Trudinger-type flows."],"forward_implications":["A manifold with the relative $p$-Faber–Krahn inequality must satisfy volume doubling and must give the sub-Gaussian upper estimate for every non-negative bounded weak subsolution of Trudinger's equation.","A manifold with volume doubling and the sub-Gaussian upper estimate for one exponent $p_0$ automatically satisfies the relative $p_0$-Faber–Krahn inequality, and then the same sub-Gaussian estimate holds for every $p>p_0$ (Corollary 1.5).","Under a uniform $p$-Faber–Krahn inequality with $\\Lambda_p(v) \\ge c v^{-p/n}$, every non-negative bounded solution satisfies the long-time bound (1.5); when $\\Lambda_p(v) \\simeq (\\log v)^{-\\alpha}$, the time factor is $t^{-n/(p\\sigma)}\\exp(c\\sigma t^{1/(\\alpha+1)})$ rather than $t^{-n/p}$.","Conversely, from an upper estimate of the form (4.44) one recovers a lower bound $\\lambda_{1,p}(\\Omega) \\ge c\\Lambda_p(C\\mu(\\Omega))$ for every precompact open set, so the long-time shape of the estimate and the Faber–Krahn function determine each other."],"supporting_citations":[{"why":"This reference supplies the implication $(\\mathrm{FK}_p)\\Rightarrow(\\mathrm{UE}_p)$ as its Theorem 1.1, imported without reproof.","marker":"[21]"},{"why":"This reference establishes the linear $p=2$ heat-kernel characterization that the paper extends to the Trudinger equation.","marker":"[8]"},{"why":"This reference provides the reverse doubling estimate $\\mu(B(x,Ar)) \\ge c A^{N_0} \\mu(B(x,r))$ used in the proof of Theorem 3.6.","marker":"[10]"},{"why":"This reference supplies the Caccioppoli-type inequality (2.12) that drives the long-time estimates in Section 4.","marker":"[11]"},{"why":"This reference provides the monotonicity of the $L^\\sigma$ norms of solutions used in Lemma 4.2.","marker":"[9]"},{"why":"This reference is the source of the iteration argument used in Proposition 3.1 to derive volume doubling from the relative Faber–Krahn inequality.","marker":"[3]"}],"fun_headline_variants":["FK inequality equals doubling plus sub-Gaussian decay","Trudinger equation: FK, doubling, decay all equivalent","Uniform FK sharpens long-time decay on manifolds","FK iff doubling and sub-Gaussian: proof for all p>1"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The full equivalence rests on a previously published theorem, cited as [21], that the relative $p$-Faber–Krahn inequality alone forces the sub-Gaussian upper estimate; the paper does not reprove that direction, so the characterization stands or falls with the soundness of that borrowed result.","fun_headline_variants_meta":{"raw":{"variants":["FK inequality equals doubling plus sub-Gaussian decay","Trudinger equation: FK, doubling, decay all equivalent","Uniform FK sharpens long-time decay on manifolds","FK iff doubling and sub-Gaussian: proof for all p>1"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000248,"raw_usage":{"total_tokens":1558,"prompt_tokens":970,"completion_tokens":588,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":586,"completion_tokens_details":{"reasoning_tokens":520}},"tokens_in":586,"tokens_out":588,"duration_ms":5830,"temperature":1.0,"reasoning_tokens":520,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T00:25:23.318242+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete test would be to compute, on a geodesically complete non-compact manifold known to satisfy volume doubling and the sub-Gaussian estimate (for example, a manifold with polynomial volume growth), the first $p$-eigenvalue $\\lambda_{1,p}(D)$ for a nested family of balls and check the lower bound $\\lambda_{1,p}(D) \\ge c R^{-p}(\\mu(B(x_0,R))/\\mu(D))^{p/N}$; a sequence of domains violating this bound would disprove the equivalence in Theorem 1.4.","supporting_citations":[{"cited_title":"Grigor’yan","cited_arxiv_id":null,"evidence_quote":"This reference establishes the linear $p=2$ heat-kernel characterization that the paper extends to the Trudinger equation."},{"cited_title":"Grigor’yan and J","cited_arxiv_id":null,"evidence_quote":"This reference provides the reverse doubling estimate $\\mu(B(x,Ar)) \\ge c A^{N_0} \\mu(B(x,r))$ used in the proof of Theorem 3.6."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This reference is the source of the iteration argument used in Proposition 3.1 to derive volume doubling from the relative Faber–Krahn inequality."}],"review_version":1}