{"id":"dd927e87-13e7-4fc3-abbc-d8d287645626","arxiv_id":"2607.21429","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Near one trace bubble, the Euler-Lagrange residual controls the L^p-gradient distance with sharp power max{1,p-1}, and a Struwe-type compactness decomposition holds.","lead":"A new theorem shows that approximate solutions of the critical Sobolev trace equation on a half-space are quantitatively close to a single bubble, with the best possible error rate, plus a global decomposition into bubbles. It supplies the trace-inequality analogue of the known sharp stability result for the classical Sobolev inequality.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1's proof hinges on the unproved linear spectral gap (Prop. 2.2) and compactness lemmas imported from companion preprint [13]; if the gap degenerates or the lemmas do not cover the trace setting, the sharp one-bubble estimate collapses.","rationale":"After reading the full text, I confirm the reader's weakest assumption: the proof of Theorem 1.1 rests on Proposition 2.2, which is merely cited from [13, Corollary 2.6], and on [13, Lemma 4.3] and [13, Theorem 2.3] used in the proof of Proposition 2.3. These are unproved here and are by overlapping authors. A careful reading of Section 3 shows that the three cases depend on the disturbed spectral gap (2.13)/(2.14) to turn a residual bound into ε ≤ C R; without a uniform positive λ_T, the argument collapses. I also note the optimality assertion after Theorem 1.1 is stated without proof. However, I did not find an internal inconsistency: the proofs of Lemma 2.1, the contradiction argument in Proposition 2.3 (conditional on [13]), Lemma 4.1, Lemma 4.2, Lemma 4.3, and Lemma 4.4 are coherent and follow standard Struwe-profile arguments. The global compactness proof in Section 4 is largely self-contained except for the classification theorem [20]. Thus the reader's CONDITIONAL verdict is appropriate; no change is needed.","tokens_in":19609,"tokens_out":13913,"duration_ms":116549,"concrete_test":"Independently re-derive Proposition 2.2: for the bubble U in (1.15), show that the quadratic form Q(φ)=∫_{R^n_+} A_U ∇φ·∇φ dx is positive on the L^2(U^{p∗−2}dy) orthocomplement of T_U M_T with a uniform gap λ_T>0, and verify that the gap is invariant under the group (1.15). If this can be proved without citing [13, Cor. 2.6], the dependency is benign; if the gap degenerates with λ or the amplitude, Theorem 1.1 is unsupported. Alternatively, obtain [13] and check that Cor. 2.6 applies to the trace operator A_U and yields exactly (2.10) with constants independent of v∈M_T^+.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 2.2 — the uniform trace spectral gap on the orthogonal complement of T_v M_T — is imported without proof from [13, Corollary 2.6], and Proposition 2.3 further invokes [13, Lemma 4.3] and [13, Theorem 2.3] for the nonlinear compactness and weighted trace embedding needed to pass to the limit. The proof of Theorem 1.1 (Section 3) then uses the disturbed gap (2.13)–(2.14) as the sole source of coercivity for the Euler–Lagrange remainder H_v[h]. If λ_T were not strictly positive, or depended on the bubble parameters λ, ξ, a in a degrading way, the estimates ε ≤ C R in the three cases of Theorem 1.1 would fail. Because [13] is by overlapping authors and is not verified here, the central claim is conditional: it holds if and only if the imported spectral gap and compactness lemmas are correct. The optimality assertion for the exponent max{1, p−1} is also stated without a construction, so the sharpness claim is not independently established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves two results for the Sobolev trace inequality on R^n_+ for 1<p<n: (i) a sharp local one-bubble critical-point stability estimate (Theorem 1.1): if u is W^{1,p}-close to the normalized trace-bubble manifold N_T, then its Euler-Lagrange residual P_T(u) controls the gradient distance to N_T with power max{1,p−1}; the statement also asserts this power is optimal. (ii) a global Struwe-type compactness theorem (Theorem 1.2) for bounded sequences with vanishing residual and vanishing negative boundary part, giving a finite decomposition into asymptotically orthogonal trace bubbles plus a strongly convergent remainder, with energy quantization. The proof of Theorem 1.1 uses pointwise remainders, an orthogonal modulation step, and a disturbed spectral gap built on a linear trace gap imported from the companion paper [13]. The proof of Theorem 1.2 is a Mercuri–Willem style profile extraction with detailed nonlinear Brézis–Lieb splittings.","tokens_in":19806,"tokens_out":20833,"duration_ms":181514,"significance":"If the result is correct, it gives a natural trace counterpart of the Liu–Zhang local stability theorem and extends the p=2 trace stability of Zhang–Zhou–Zou to the full range 1<p<n, with the expected power max{1,p−1}. The global compactness theorem and its corollaries are also useful and are largely self-contained: the nonlinear splitting lemmas and the profile extraction are written out in detail. The main reservations are that the local stability theorem is conditional on unsupplied spectral-gap and compactness machinery from the companion preprint [13], and that the claimed optimality of the exponent is not proved in the text. Both issues are load-bearing but appear fixable.","major_comments":[{"comment":"The uniform linear trace spectral gap in Prop. 2.2 is stated as a direct consequence of [13, Cor. 2.6], but neither the statement of that corollary nor a proof is given. This gap is the sole source of coercivity in Theorem 1.1: it supplies the λ_T term in (2.13)–(2.14), which is used in all three cases of §3. Proposition 2.3 further relies on [13, Lemma 4.3] and [13, Theorem 2.3], and Claim 3.2 on [13, Prop. 4.5 and Cor. 4.6]. Since [13] is a companion preprint with overlapping authors and is not verified in this manuscript, the central estimate is conditional: if any of those companion results fails, or if its hypotheses are not met in this trace setting, the proof of Theorem 1.1 collapses. Please include proofs or exact statements of these results, or make the dependence fully explicit.","section":"§2.2, Props. 2.2–2.3 and Claim 3.2"},{"comment":"The assertion that the exponent max{1,p−1} is optimal is not proved anywhere in the paper. Sharpness in the whole-space theorem of Liu–Zhang [12] does not formally transfer to the trace setting, because the trace residual, the bubble family, and the boundary nonlinearity have different scaling and homogeneity. A lower-bound sequence or construction is needed to show that no smaller exponent can replace max{1,p−1}. Without this, the words “sharp” in the abstract and title are unsupported.","section":"Theorem 1.1, after (1.25)"},{"comment":"Lemma 3.1 (orthogonal modulation) is stated without proof; the text only says it is a normalized-manifold version of [13, Prop. 5.3]. This lemma is used to pass from a bubble W attaining the distance to a perturbation h orthogonal to T_v N_T, which is the starting point of the proof of Theorem 1.1. Although a standard implicit-function-theorem argument is likely available, it is not supplied here. The lemma should either be proved or the precise statement and proof of [13, Prop. 5.3] reproduced, since this is a nontrivial step on the trace manifold with amplitude normalization.","section":"Lemma 3.1"}],"minor_comments":[{"comment":"The text says “we prove Theorem 1.2, then deduce Theorem 1.3 and Corollary 1.4”; there is no Theorem 1.3. It should read Corollary 1.3.","section":"§4, opening"},{"comment":"The lower bound for ∫_{E^c} |∇v|^{p−2}|∇φ|^2 is attributed to Hölder's inequality, but in the range p<2 the exponent p−2 is negative. The bound actually follows from the definition E^c: |∇v| ≥ ε|∇φ|. The argument is correct but the wording is misleading and should be revised.","section":"Prop. 2.3, Step 1, p<2 lower bound"},{"comment":"The “standard continuity argument” for the boundary Lévy concentration function Q_k(ρ) should be spelled out or referenced. For a fixed L^1 function, the supremum of integrals over balls of radius ρ is only lower semicontinuous in general; the existence of λ_k with Q_k(λ_k)=δ needs a short justification.","section":"Lemma 4.4, choice of λ_k"},{"comment":"The proof of Corollary 1.4 is compressed into one sentence (“Combining Theorem 1.1 with the case ν=1 of Corollary 1.3”). A standard contradiction argument is presumably intended, but it should be sketched so that the rôle of the qualitative compactness and the quantitative local estimate is clear.","section":"Corollary 1.4"}],"recommendation":"major_revision","confidential_remarks":"The central local theorem cannot be evaluated independently of the companion preprint [13], which supplies the linear trace spectral gap, the nonlinear compactness lemma, the weighted trace embedding, and the disturbed energy gap. The editor may wish to ensure that [13] is either included as an appendix or publicly and verifiably available before acceptance. The optimality claim is also currently unproved. I would not recommend rejection if these points are addressed, since the global compactness part is substantially self-contained and the overall strategy is credible."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid extension of Liu–Zhang's one-bubble stability program to the half-space trace setting, and it deserves a real referee. But the central local-stability theorem is conditional on two lemmas imported wholesale from a companion preprint by overlapping authors, and the 'optimal exponent' claim is asserted without a construction. Treat it as work in progress until those imports are pinned down.\n\nWhat's new: Theorem 1.1 is the sharp one-bubble residual-vs-distance estimate for the Sobolev trace inequality for all 1<p<n, and Theorem 1.2 is the Struwe/Mercuri–Willem global compactness decomposition on the half-space. That is a natural capstone for the Bianchi–Egnell-type program on trace inequalities. The authors do real work: they adapt the nonlinear splitting and concentration-compactness machinery to the boundary, prove the Brézis–Lieb-type derivative splittings carefully, and handle the three ranges p<2, 2<p<p*, and p≥2. The proof of Proposition 2.3 is detailed, and the use of the exact remainder Mp is a nice simplification of [12]. The global compactness extraction (Lemmas 4.1–4.6) is mostly self-contained.\n\nSoft spots, in order of seriousness:\n\n1. Load-bearing imports. Proposition 2.2 (the linear trace spectral gap) is not proved; it's 'a direct consequence of [13, Corollary 2.6]'. Proposition 2.3 also invokes [13, Lemma 4.3] and [13, Theorem 2.3] for nonlinear compactness and weighted trace embedding. The companion paper is by overlapping authors, and without those inputs the disturbed spectral gap in (2.13)–(2.14)—and hence Theorem 1.1—has no coercivity. This is not a demonstrated error, but it is a genuine condition: if the gap degenerates with parameters, or [13] has a flaw, the whole local estimate collapses. The paper should either prove Proposition 2.2 or state the dependence more prominently.\n\n2. Optimality. Theorem 1.1 claims the exponent max{1,p−1} is optimal, but no construction or example is given. That is an assertion without support. It may be true, but it needs a proof or a reference.\n\n3. Delegation. Corollary 1.3 and parts of Theorem 1.2 are dispatched with 'arguing as in [15]' rather than proved. For a paper whose contribution is precisely those theorems, that's thin in places. The profile extraction uses a 'standard continuity argument' for the Lévy concentration function and a covering argument; those are standard, so this is a minor point.\n\nBottom line: if Proposition 2.2 and [13]'s lemmas are correct, the paper is correct and a genuine extension. The structure is sound and the PDE argument is careful. It should go to a referee who can check the companion preprint. I wouldn't desk-reject it; I'd send it out with a request to verify the imports and to fill or clearly flag the optimality claim.","headline":"Conditional but genuine extension of the one-bubble stability program to the trace setting; send to a referee who can verify the imported spectral gap.","tokens_in":20371,"tokens_out":2459,"would_cite":true,"duration_ms":22416,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B33","35J92","46E35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Near the normalized trace-bubble family, the Euler-Lagrange residual controls the gradient distance with the optimal exponent max{1,p−1} for all 1<p<n.","keywords":["Sobolev trace inequality","critical-point stability","trace bubble","global compactness","p-Laplacian","Euler-Lagrange residual","sharp quantitative estimates","half-space"],"falsifier":"Compute the second variation at the standard bubble W: if for some n,p there exist admissible perturbations orthogonal to the tangent space whose Rayleigh quotient (∫ A_W∇φ·∇φ)/(∫ W^{p*−2}φ^2) approaches (p*−1)S_T^p from below, then the uniform spectral gap λ_T fails and the sharp local one-bubble estimate cannot hold for those parameters.","tokens_in":19406,"feed_emoji":"📐","tokens_out":5480,"duration_ms":49739,"temperature":0.7,"pith_summary":"The paper proves that the sharp one-bubble stability phenomenon for critical points of the Sobolev inequality transfers to the Sobolev trace inequality on the half-space. If a function is gradient-close to a normalized positive trace bubble, the norm of its Euler-Lagrange residual controls the gradient distance to the bubble family with the optimal power max{1,p−1}. It also establishes a global compactness theorem: bounded sequences with vanishing residual and vanishing negative boundary part split into finitely many asymptotically orthogonal normalized trace bubbles plus a strongly convergent remainder. Combining the local estimate with this compactness yields a sharp global one-bubble stability theorem for nonnegative functions in the one-bubble energy window. This extends the critical-point stability program to the trace setting for the full range 1<p<n.","feed_headline":"Sharp bubble stability proven for trace inequality","feed_subtitle":"Near trace bubbles, the residual controls distance with optimal exponent for all 1<p<n.","key_machinery":"The central machinery is the normalized trace-bubble manifold N_T and the Euler-Lagrange residual P_T(u). The key estimate is the 'disturbed spectral gap' proposition: for perturbations orthogonal to the bubble tangent directions, the exact monotonicity remainder is bounded below by a weighted boundary integral with a uniform positive spectral gap λ_T. This is proved by contradiction, using pointwise remainder estimates, a compactness lemma with weighted trace convergence, and a linear spectral gap of the linearized trace operator imported from a companion paper. The global compactness theorem is assembled from nonlinear splittings of the p-Laplacian and the trace nonlinearity, a concentrati","core_discovery":"The paper's central claim is a sharp local quantitative stability estimate for the Sobolev trace inequality on the half-space. Given any function sufficiently close in gradient norm to the family of normalized positive trace bubbles, the Euler-Lagrange residual of the trace functional controls the gradient distance to that family, raised to the power max{1,p−1}. In addition, the paper proves a global compactness theorem of the p-Laplacian type: bounded sequences whose residual tends to zero and whose negative boundary part tends to zero decompose, after a subsequence, into finitely many asymptotically orthogonal normalized trace bubbles plus a remainder that converges strongly. These two res","pith_inferences":["The exact-remainder method used here may extend to a linear multi-bubble stability estimate on the trace side, likely subject to dimension restrictions analogous to the Euclidean case.","The global compactness theorem supplies the qualitative foundation for quantitative multi-bubble estimates beyond one bubble, because it guarantees that limiting bubble configurations exist and are separated.","If the companion-paper spectral gap were proved directly in a self-contained way, the constants could plausibly be made explicit and the method adapted to fractional trace inequalities.","The optimal exponent max{1,p−1} mirrors the whole-space p-Laplacian case, suggesting that further parallels in critical-point stability—such as multi-bubble rates—are likely to hold in the trace setting."],"forward_implications":["Any function sufficiently close in gradient norm to a normalized trace bubble has a residual at least a constant times the bubble-distance to the power max{1,p−1}, upgrading qualitative closeness to a sharp quantitative bound.","Every bounded sequence with residual tending to zero and negative boundary part tending to zero admits a finite bubble decomposition, showing that no energy loss occurs beyond asymptotically orthogonal bubbles on the half-space.","A sequence with gradient energy in the window [(ν−1/2)A_T, (ν+1/2)A_T] and vanishing residual is strongly approximated by a sum of ν normalized trace bubbles.","For nonnegative functions in the one-bubble energy window, the residual controls the distance to a single normalized trace bubble globally, with no a priori one-bubble closeness assumption."],"fun_headline_variants":["Sharp one-bubble stability for trace inequality","Trace Sobolev inequality: sharp stability near bubbles","Optimal exponent for trace bubble stability","Global compactness and sharp stability for trace","Critical-point stability for Sobolev trace inequality"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the linearized trace operator at every normalized bubble has a uniform positive spectral gap λ_T on the orthogonal complement of the tangent space; this gap is imported from a companion paper, and if it degenerates the main estimates collapse.","fun_headline_variants_meta":{"raw":{"variants":["Sharp one-bubble stability for trace inequality","Trace Sobolev inequality: sharp stability near bubbles","Optimal exponent for trace bubble stability","Global compactness and sharp stability for trace","Critical-point stability for Sobolev trace inequality"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000258,"raw_usage":{"total_tokens":1362,"prompt_tokens":632,"completion_tokens":730,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":376,"completion_tokens_details":{"reasoning_tokens":662}},"tokens_in":376,"tokens_out":730,"duration_ms":7088,"temperature":1.0,"reasoning_tokens":662,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T07:27:16.239173+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the second variation at the standard bubble W: if for some n,p there exist admissible perturbations orthogonal to the tangent space whose Rayleigh quotient (∫ A_W∇φ·∇φ)/(∫ W^{p*−2}φ^2) approaches (p*−1)S_T^p from below, then the uniform spectral gap λ_T fails and the sharp local one-bubble estimate cannot hold for those parameters.","supporting_citations":[],"review_version":1}