{"id":"708e2c91-2a8e-4494-a4df-761ed0d00e64","arxiv_id":"2509.03468","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A new Sobolev-space theory for nonlocal gradients with space-dependent horizons is developed, including trace, Poincare, and existence results for nonlocal hyperelasticity with local boundary conditions.","lead":"This paper introduces heterogeneous nonlocal gradients with interaction ranges that vanish at the boundary, creating a seamless bridge between nonlocal models and classical boundary conditions. It develops the full function-space theory and proves existence of minimizers for hyperelasticity-type variational problems under local Dirichlet, Neumann, or mixed boundary conditions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed existence and Poincaré results depend on an unquantified smallness condition ('mildly varying', (4.13)) that is never verified for any concrete horizon and is absent from the abstract's advertised conclusions.","rationale":"The reader's weakest_assumption is exactly the unquantified 'mildly varying' condition, and my independent read identifies the same condition as the single most load-bearing assumption in the paper. Without it, Theorem 4.17 only yields a finite-dimensional kernel; Corollary 4.19 and Corollary 5.3, which are among the central advertised achievements, do not follow. The trace theorem and translation mechanism do not require smallness and seem correctly proved; in particular, the step flagged by the reader in Proposition 4.2 appears repairable as stated, because CΩρ(·)∈LBH+Ω provides the needed mapping Hλ−μ,p(Ω)→Lp(Ω) for any μ∈(0,λ). Thus the decisive unresolved point is not a local proof gap but the global applicability of a hidden smallness hypothesis. Since the authors explicitly introduce the condition and do not claim to verify it for examples, the correct verdict remains CONDITIONAL rather than ACCEPT or REJECT: the mathematical framework is credible and conditional, but the practical reach of the main existence results is not demonstrated. My recommendation is therefore UNCHANGED relative to the reader's verdict, with the mild-variation condition identified as the load-bearing concern that should be addressed by an explicit threshold or a verifiable criterion.","tokens_in":38417,"tokens_out":21305,"duration_ms":184876,"concrete_test":"Choose a concrete kernel satisfying (H0)-(H4), e.g., the truncated fractional kernel in Example 2.1 with s=1/2 on the unit ball. Track the constants in Lemma 3.3 and in the Neumann-series argument of Proposition 2.4 to produce an explicit numerical threshold ¯δ0 such that δ is mildly varying whenever max δ < ¯δ0. Then evaluate the model horizon δ(x)=e^{1/(|x|^2−1)} from Example 2.6: if max δ < ¯δ0, the condition is checkable and satisfied for that example; if not, the main existence theorem does not apply to the paper's own illustrative horizon. A complementary numerical check is to discretize Qρ on L2(B1) for this δ and compute the smallest singular value; if it stays bounded away from zero while the analytical threshold is impossible to extract, the unquantified nature of the assumption is confirmed as the decisive obstruction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The kernel characterization (Theorem 4.17), all Poincaré inequalities (Corollary 4.19), and the minimizer existence results (Corollary 5.3, Remark 5.4) require the horizon function to be 'mildly varying', meaning max δ < an unspecified threshold ¯δ0 from Proposition 2.4. The threshold depends on the kernel ρ1, the normalized horizon ηδ, and on p, but no formula, bound, or verification procedure is provided. Hence, for any concrete horizon function — including the admissible example δ(x)=e^{1/(|x|^2−1)} in Example 2.6 — one cannot decide whether the main theorems apply. Moreover, ¯δ0 is defined through an invertibility statement for Op(aε), i.e., essentially through injectivity of the same operator Qρ whose kernel is the object of study; checking the condition is therefore as difficult as the theorem itself. Remark 4.18 concedes that the proof does not cover non-mildly-varying horizons and cites only numerical simulations as evidence that the condition may be unnecessary. This is not an internal contradiction, but it creates a real gap between the paper's central claim of existence under local boundary conditions and the verified hypothesis set: the advertised result is conditional on a hidden, unquantified smallness parameter. The trace theory and translation mechanism are unconditional and appear sound; the bottleneck is specifically the passage from finite-dimensional kernel of Dρ to the identification with constants, which is required for coercivity and the direct method.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces heterogeneous nonlocal gradients with a spatially varying horizon that vanishes at the boundary of a bounded Lipschitz domain. It defines associated Sobolev spaces, embeds these spaces between W^{1,p} and Bessel potential spaces, establishes a translation mechanism relating the heterogeneous nonlocal gradient to the classical gradient up to lower-order terms, proves a trace theorem, density results, extension operators, regularity properties, and a characterization of the kernel of the nonlocal gradient. Under an additional 'mildly varying' smallness hypothesis on the horizon, it derives Poincaré inequalities and, for quasiconvex or polyconvex integrands, proves existence of minimizers for nonlocal hyperelasticity-type functionals under local Dirichlet, Neumann, and mixed boundary conditions, together with Euler-Lagrange equations for the associated boundary value problems.","tokens_in":38826,"tokens_out":18000,"duration_ms":158272,"significance":"If the main theorems hold, this is a substantial contribution to the nonlocal calculus of variations and to local-to-nonlocal coupling: it provides a systematic Sobolev-space toolbox for heterogeneous nonlocal gradients, including a natural trace theory that matches classical local boundary values. The use of pseudo-differential techniques to handle the space-dependent horizon is elegant and goes substantially beyond the constant-horizon theory. The trace theorem, density results, extension operator, finite-dimensionality of the kernel, and the quasiconvexity characterization are valuable and appear technically sound. The practical reach of the existence and Poincaré results, however, is currently gated by an unquantified smallness condition that is not verified for any concrete admissible horizon.","major_comments":[{"comment":"The 'mildly varying' hypothesis is the gatekeeper for the kernel identification with constants, for all three Poincaré inequalities, and for the existence theorems, but it is only an existential smallness statement. The threshold δ̄0 in (4.13) comes from Proposition 2.4, which asserts only that some ε0 exists; no formula, lower bound, or verification procedure is given. Consequently, for a concrete admissible horizon such as Example 2.6, the paper does not enable the reader to decide whether Theorem 4.17(ii), Corollary 4.19, or Corollary 5.3 apply. Remark 4.18 itself concedes that the non-mildly-varying case is open and cites only numerical simulations. Since the abstract advertises existence of minimizers under local Dirichlet, Neumann, and mixed boundary conditions without this caveat, the advertised statement is conditional on an invisible constant. Please add a quantitative or otherwise checkable sufficient condition, or exhibit a nontrivial family of admissible horizons that is provably mildly varying, and qualify the abstract and introduction accordingly.","section":"§4.5, Eq. (4.13); Theorem 4.17; Corollary 4.19; Corollary 5.3"},{"comment":"The proof of the embedding H_{ρ(·),p}(Ω) → H^{λ,p}(Ω) contains a bootstrap step that is not justified as written. The first estimate bounds ||u||_{H^{λ−μ,p}(Ω)} by applying P_{ρ(·),Ω} to Q^Ω_{ρ(·)}u and therefore requires Q^Ω_{ρ(·)}u ∈ H^{1−μ,p}(Ω), but at that stage only Q^Ω_{ρ(·)}u ∈ L^p(Ω) is known from (3.15). Using the norm equivalence (2.10) to infer membership in H^{1−μ,p}(Ω) before that membership is established is circular as printed. The argument can be repaired by first showing that ∇(Q^Ω_{ρ(·)}u) = D_{ρ(·)}u + C^Ω_{ρ(·)}u lies in H^{−μ,p}(Ω), which together with Q^Ω_{ρ(·)}u ∈ L^p(Ω) ⊂ H^{−μ,p}(Ω) yields Q^Ω_{ρ(·)}u ∈ H^{1−μ,p}(Ω) via (2.10); please rewrite this step explicitly. Since the embedding is used in Lemma 4.3 and hence throughout Section 4, this proof must be clearly valid.","section":"§4, Proposition 4.2, Eq. (4.2)"}],"minor_comments":[{"comment":"The asserted identification H_{ρ(·),p}(Ω)|_U = H^{s,p}(U) does not follow from Corollary 4.14. That corollary gives only H^{s,p}_0(U)|_U + W^{1,p}(Ω)|_U ⊂ H_{ρ(·),p}(Ω)|_U ⊂ H^{s,p}(U), and H^{s,p}_0(U)|_U is in general a proper subspace of H^{s,p}(U). Please either prove the equality or state only the inclusions.","section":"§4.4, Example 4.15(b)"},{"comment":"The proof of uniqueness of the trace operator refers to Theorem 4.7(i), which appears only later in the text; the forward reference is acceptable, but the logical order would be cleaner if the density theorem were stated and proved before the trace theorem.","section":"§4.1, Theorem 4.5"},{"comment":"The derivation of the natural boundary condition uses the formal statement that D_{ρ(·)} = ∇ on ∂Ω. Since the existence results in Corollary 5.3 do not rely on this derivation, the point is not load-bearing, but the formal passage should be flagged as heuristic or justified more carefully.","section":"§5.3"}],"recommendation":"major_revision","confidential_remarks":"This is a strong paper with substantial unconditional results and a coherent pseudo-differential framework. The main obstacle to accepting the advertised scope is the unquantified mild-variation condition; if the authors can provide a checkable criterion or a provably mildly varying family of horizons, and repair the bootstrap step in Proposition 4.2, I would be willing to support acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know: this is a serious, carefully built paper, and I do not share the reader's suspicion that Proposition 4.2 has a load-bearing gap. The genuinely new content is the heterogeneous nonlocal gradient, defined as a pseudo-differential operator with symbol q(x,ξ)=Q-hat(δ(x)ξ), and the associated Sobolev spaces. The trace theorem (Theorem 4.5) — a unique bounded trace that agrees with the classical one on W^{1,p} and has a right inverse — is the cleanest result in the paper. The translation lemma (Lemma 4.3), with commutators and parametrices that switch between D_{ρ(·)} and ∇ up to lower-order errors, is a powerful and well-executed mechanism. Density, extension, regularity, and the finite-dimensional kernel result all hang together coherently.\n\nOn the alleged gap: the step in Proposition 4.2 where the proof bounds ||Q u||_{H^{1−μ,p}} is not circular. It uses that Q u ∈ L^p ⊂ H^{−μ,p} and ∇Q u = D u + C u ∈ H^{−μ,p}; by the norm equivalence (2.10), or rather the underlying characterization of Bessel potential spaces, this forces Q u ∈ H^{1−μ,p}. The second embedding step then legitimately uses the already-proved weaker embedding to control C u in L^p. I read the chain as sound.\n\nThe real soft spot is the 'mildly varying' assumption (4.13). The threshold δ̄0 comes from the Neumann series in Proposition 2.4, but no quantitative bound or verification procedure is given. For a concrete horizon, including their own Example 2.6, you cannot decide whether the main existence theorems apply. The authors are honest about this in the body — Remark 4.18 admits the proof does not cover non-mildly-varying horizons and mentions numerics only as informal evidence — but the abstract advertises existence under local boundary conditions without mentioning the smallness parameter. That is a presentation gap, not a mathematical falsehood. I would ask the authors to either state a sufficient bound on δ̄0 in terms of the kernel and the normalized horizon, or explicitly reformulate the main theorems as 'for sufficiently small horizon amplitude.'\n\nWho gets value: anyone working on peridynamics, nonlocal calculus of variations, or local-to-nonlocal coupling. The trace theory and translation mechanism are worth citing on their own, independent of the mildly varying condition. This deserves a serious referee and likely publication after the smallness condition is clarified. Send it to peer review.","headline":"A substantial and convincing functional-analytic toolbox for nonlocal gradients with boundary-vanishing heterogeneous horizons; the trace theory is the gem, the mildly varying smallness condition is the soft spot the abstract glosses over.","tokens_in":39224,"tokens_out":11223,"would_cite":true,"duration_ms":97644,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R11","46E35","49J45","47G30","74A70","74G65"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper builds a Sobolev-space theory for nonlocal gradients whose interaction range vanishes at the boundary, giving classical boundary traces and minimizers in nonlocal hyperelasticity.","keywords":["nonlocal gradients","fractional gradients","nonlocal Sobolev spaces","pseudo-differential operators","trace operators","Poincaré inequality","nonlocal variational problems","heterogeneous horizons"],"falsifier":"Numerically discretize $Q^\\Omega_{\\rho(\\cdot)}$ on a fixed Lipschitz domain and check whether a nonconstant function with $D_{\\rho(\\cdot)}u=0$ appears for some horizon below the stated threshold; finding one would disprove Theorem 4.17, and with it the Poincaré inequalities and existence results that depend on it.","tokens_in":38186,"feed_emoji":"📐","tokens_out":8747,"duration_ms":73107,"temperature":0.7,"pith_summary":"Nonlocal hyperelasticity models replace the classical deformation gradient by a nonlocal gradient averaged over a finite interaction range, or horizon. This paper studies what happens when the horizon varies in space and shrinks to zero at the domain's boundary, so that the nonlocal operator becomes local near the boundary. It establishes that the resulting heterogeneous nonlocal Sobolev spaces behave like classical Sobolev spaces at the boundary: functions in them have well-defined boundary traces matching the classical trace, and Poincaré inequalities hold. This makes it possible to prove existence of minimizers for nonlocal hyperelastic energies with quasiconvex or polyconvex stored-energy densities under local Dirichlet, Neumann, and mixed boundary conditions.","feed_headline":"Nonlocal Sobolev spaces admit classical boundary traces","feed_subtitle":"With a horizon that vanishes at the boundary, nonlocal hyperelasticity models satisfy local Dirichlet, Neumann and mixed data.","key_machinery":"The load-bearing machinery is the identification of the heterogeneous nonlocal gradient as a restricted pseudo-differential operator whose symbol is $q_{\\rho(\\cdot)}(x,\\xi)=\\widehat{Q}_{\\rho_1}(\\delta(x)\\xi)$, sitting in the symbol class $S^0_{1,\\mu}$. This symbol is the Fourier multiplier of the homogeneous nonlocal gradient, rescaled by the space-dependent horizon. From it the paper builds a parametrix (an almost-inverse), commutators measuring the mismatch between $D_{\\rho(\\cdot)}$ and $\\nabla Q^\\Omega_{\\rho(\\cdot)}$, and a bidirectional translation between the nonlocal space $H_{\\rho(\\cdot),p}(\\Omega)$ and the classical Sobolev space $W^{1,p}(\\Omega)$ up to lower-order operators. This translation is what carries classical facts about traces, density, compactness, and quasiconvex lower semicontinuity over to the nonlocal setting.","core_discovery":"At the center of the paper is the heterogeneous nonlocal Sobolev space $H_{\\rho(\\cdot),p}(\\Omega)$, defined by requiring both $u$ and its nonlocal gradient $D_{\\rho(\\cdot)}u$ to be $p$-integrable, where the kernel is rescaled by a smooth horizon $\\delta(x)$ that vanishes near $\\partial\\Omega$. The authors show this space sits between the classical Sobolev space $W^{1,p}(\\Omega)$ and the Bessel potential space $H^{\\lambda,p}(\\Omega)$, and that smooth functions are dense. The key structural discovery is a translation mechanism: a bounded operator $Q^\\Omega_{\\rho(\\cdot)}$ maps $H_{\\rho(\\cdot),p}(\\Omega)$ into $W^{1,p}(\\Omega)$, with $D_{\\rho(\\cdot)}$ equal to $\\nabla Q^\\Omega_{\\rho(\\cdot)}$ up to lower-order operators, and a parametrix $P_{\\rho(\\cdot),\\Omega}$ acts in the reverse direction. From this they derive a unique bounded trace operator $T_{\\rho(\\cdot)}:H_{\\rho(\\cdot),p}(\\Omega)\\to W^{1-1/p,p}(\\partial\\Omega)$ extending the classical trace, show that the kernel of $D_{\\rho(\\cdot)}$ consists only of constants when the horizon is mildly varying, and prove Poincaré inequalities for functions with zero mean, vanishing trace on a boundary portion, or vanishing on a positive-measure set. These tools yield existence of minimizers for functionals $\\int_\\Omega f(x,D_{\\rho(\\cdot)}u)\\,dx$ with quasiconvex or polyconvex integrands under Dirichlet, Neumann, and mixed local boundary data.","pith_inferences":["The hidden threshold $\\bar\\delta_0$ is an invitation to quantify it: for concrete kernels and domains one could estimate the largest horizon for which the Poincaré and existence theorems hold, turning the mild-variation condition into a checkable design criterion.","The pseudo-differential translation mechanism suggests the same toolbox could treat interfaces inside a domain, not only boundaries, by letting the horizon vanish across a hypersurface.","One testable consequence is that if the horizon is allowed to exceed the threshold, the kernel of the nonlocal gradient may cease to be trivial, so the Poincaré inequality and existence results should fail; numerical experiments with discretized $Q^\\Omega_{\\rho(\\cdot)}$ could locate this transition."],"forward_implications":["Boundary data for nonlocal models can be imposed as classical Sobolev traces, so local and nonlocal regions can be coupled seamlessly at interfaces.","The Poincaré inequalities give coercivity for energy minimization, so existence of minimizers follows by the direct method for quasiconvex and polyconvex stored-energy densities.","Because the spaces are sandwiched between $W^{1,p}(\\Omega)$ and $H^{\\lambda,p}(\\Omega)$, solutions are never more regular than classical Sobolev functions away from the boundary, but they can be less regular.","Minimizers solve nonlocal Euler–Lagrange systems with classical boundary conditions, including a nonlocal Laplace equation with mixed Dirichlet–Neumann data as a special scalar case.","The zero-gradient kernel being exactly the constants mirrors local elasticity, so rigid-body motions are the only zero-energy deformations under mild variation."],"supporting_citations":[{"why":"Supplies the kernel assumptions (H0)–(H4) and the Fourier multiplier bounds for $\\widehat{Q}_{\\rho_1}$ used throughout.","marker":"[10]"},{"why":"Provides the pseudo-differential calculus (composition, parametrices, adjoints) that the translation mechanism relies on.","marker":"[33]"},{"why":"Gives the invertibility criterion for slowly varying symbols that produces the mild-variation horizon threshold.","marker":"[37]"},{"why":"Provides the universal extension operator used to build extensions and the parametrix on bounded domains.","marker":"[39]"},{"why":"Introduces nonlocal gradients on bounded domains and the fundamental theorem of calculus that motivates the heterogeneous construction.","marker":"[9]"},{"why":"Shows homogeneous finite-horizon fractional gradients can have infinite-dimensional kernels, against which the finite-dimensional and constant kernel results here are contrasted.","marker":"[32]"},{"why":"Supplies the variational theory for finite-horizon fractional gradients, including weak lower semicontinuity tools adapted in Theorem 5.1.","marker":"[16]"},{"why":"Provides the direct-method framework and quasiconvexity and polyconvexity criteria used for the existence theorems.","marker":"[18]"}],"fun_headline_variants":["Nonlocal gradients with vanishing horizon give classical traces","Heterogeneous nonlocal Sobolev spaces admit local boundary values","Boundary vanishing horizon links nonlocal and local PDEs","Trace and Poincaré for nonlocal gradients at the boundary","Existence of minimizers in nonlocal hyperelasticity with local BCs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the horizon function's maximum value lies below a threshold whose size is never computed; all the main theorems—constant zero-gradient kernel, Poincaré inequalities, and existence of minimizers—depend on this smallness condition.","fun_headline_variants_meta":{"raw":{"variants":["Nonlocal gradients with vanishing horizon give classical traces","Heterogeneous nonlocal Sobolev spaces admit local boundary values","Boundary vanishing horizon links nonlocal and local PDEs","Trace and Poincaré for nonlocal gradients at the boundary","Existence of minimizers in nonlocal hyperelasticity with local BCs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000694,"raw_usage":{"total_tokens":3203,"prompt_tokens":1073,"completion_tokens":2130,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":689,"completion_tokens_details":{"reasoning_tokens":2044}},"tokens_in":689,"tokens_out":2130,"duration_ms":14388,"temperature":1.0,"reasoning_tokens":2044,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:31:32.770159+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically discretize $Q^\\Omega_{\\rho(\\cdot)}$ on a fixed Lipschitz domain and check whether a nonconstant function with $D_{\\rho(\\cdot)}u=0$ appears for some horizon below the stated threshold; finding one would disprove Theorem 4.17, and with it the Poincaré inequalities and existence results that depend on it.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the kernel assumptions (H0)–(H4) and the Fourier multiplier bounds for $\\widehat{Q}_{\\rho_1}$ used throughout."},{"cited_title":"Kumano-go","cited_arxiv_id":null,"evidence_quote":"Provides the pseudo-differential calculus (composition, parametrices, adjoints) that the translation mechanism relies on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the invertibility criterion for slowly varying symbols that produces the mild-variation horizon threshold."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the universal extension operator used to build extensions and the parametrix on bounded domains."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces nonlocal gradients on bounded domains and the fundamental theorem of calculus that motivates the heterogeneous construction."},{"cited_title":"Kreisbeck and H","cited_arxiv_id":null,"evidence_quote":"Shows homogeneous finite-horizon fractional gradients can have infinite-dimensional kernels, against which the finite-dimensional and constant kernel results here are contrasted."},{"cited_title":"Cueto, C","cited_arxiv_id":null,"evidence_quote":"Supplies the variational theory for finite-horizon fractional gradients, including weak lower semicontinuity tools adapted in Theorem 5.1."},{"cited_title":"Dacorogna","cited_arxiv_id":null,"evidence_quote":"Provides the direct-method framework and quasiconvexity and polyconvexity criteria used for the existence theorems."}],"review_version":2}