{"id":"534836cd-f6eb-427d-80d1-5c708dbddffd","arxiv_id":"2412.11701","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For supremal functionals with Hessian and lower-order terms, the paper proves existence of minimizers, derives the third-order Aronsson-type PDE, and establishes generalized D-solutions.","lead":"This mathematics paper studies ways to minimize the largest value of a function that depends on a curve's height, slope, and curvature. It proves minimizers exist, derives the corresponding differential equation, and shows the equation has generalized solutions, extending earlier work to include lower-order terms.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.4's proof relies on an invalid subdomain inequality: global Lp approximate minimality does not imply the local estimate used in (3.17), so the 1D absolute-minimiser claim is unsupported as written.","rationale":"The most load-bearing concern is an internal proof error, not a scope limitation. The paper's abstract advertises existence of absolute minimisers when n=1, and that claim rests entirely on Theorem 3.4. The proof's key step (3.17) asserts that global approximate minimality of the Lp functional transfers to a subdomain, but this transfer is false in general, as the explicit counterexample shows. The D-solution section is restrictive due to the structural ansatz H=h(·,·,·,X^T X), but that restriction is honestly stated in Theorem 5.2 and is a matter of scope rather than correctness. The reader's weakest_assumption identified the D-solution ansatz, but the reader's rationale also flagged the Theorem 3.4 gap as the main flaw; hence partial agreement. Since the reader already rendered a CONDITIONAL verdict based on this gap, our stress-test does not change the verdict: the paper's central architecture remains credible, but Theorem 3.4 needs a repaired proof before the abstract's absolute-minimiser claim is fully supported.","tokens_in":20749,"tokens_out":20375,"duration_ms":197479,"concrete_test":"Run the explicit counterexample: Ω=(0,2), O=(0,1), p=2, f=10·1_{(0,1)}+10^6·1_{(1,2)}, g=0·1_{(0,1)}+10^6·1_{(1,2)}. Verify Ep(f,Ω) ≤ 2^{-2}+Ep(g,Ω) but Ep(f,O) > 2^{-2}+Ep(g,O). This settles that the inference in (3.17) is invalid; to salvage Theorem 3.4 one must supply a different subdomain argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 3.4, the first inequality of (3.17) claims Ep(up,(a,b)) ≤ 2^{-p}+Ep(up+φp,(a,b)). This is asserted to follow from the global approximate-minimizer property Ep(up,Ω) ≤ 2^{-p}+Ep(up+φp,Ω) together with the fact that up and up+φp agree outside (a,b). That implication is false. For nonnegative integrands f,g agreeing off (a,b), the global normalized Lp inequality does not control the local Lp norms. For example, take Ω=(0,2), (a,b)=(0,1), p=2, and define f≡10 on (0,1), f≡10^6 on (1,2), g≡0 on (0,1), g≡10^6 on (1,2). Then Ep(f,Ω) ≈ Ep(g,Ω)+7×10^{-5} ≤ Ep(g,Ω)+2^{-2}, so the global inequality holds, while Ep(f,(0,1)) = 10 > 0.25 = 2^{-2}+Ep(g,(0,1)). Thus the subdomain inequality fails even though the global one holds. Since (3.17) is the only step linking the Lp approximants to the absolute-minimiser property of u∞, the proof of Theorem 3.4 is incomplete. The theorem may be true, but the presented argument does not establish it, and the abstract's claim of absolute minimisers in 1D is not supported by the proof as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies second-order L^∞ variational functionals E∞(u,O)=ess sup_O H(·,u,Du,D²u) with lower-order terms. It claims existence of global minimisers under coercivity and level-convexity assumptions (Theorem 3.1), existence of absolute minimisers in one dimension (Theorem 3.4), a formal derivation of the third-order fully nonlinear PDE A²∞u=0 as the analogue of the Euler-Lagrange equation (Derivation 4.1), a rigorous proof that C³ absolute minimisers solve this PDE classically (Theorem 4.3), and existence of generalised D-solutions to the corresponding Dirichlet problem under a structural ansatz on H (Theorem 5.2). The paper generalises the pure-Hessian results of [26] and offers a streamlined treatment of the D-solution existence via the Baire-category method and the Dacorogna–Marcellini theory of implicit PDEs.","tokens_in":21044,"tokens_out":11695,"duration_ms":113507,"significance":"If the main claims were established, the paper would be a valuable contribution to higher-order L^∞ variational theory: it identifies the correct third-order PDE for supremal functionals with lower-order terms, provides a clean reduction of D-solution existence to known implicit-PDE machinery, and simplifies earlier proofs from [26]. The global-minimiser theorem and the D-solution theorem are substantial and appear largely sound under their stated assumptions. However, the central new variational claim — the existence of absolute minimisers in one dimension — is not proved by the argument given: the key local inequality in Theorem 3.4 does not follow from global approximate minimality. The D-solution existence is also conditional on a quite restrictive structural assumption, which should be transparently highlighted in the abstract and introduction.","major_comments":[{"comment":"The first inequality of (3.17), Ep(up,(a,b)) ≤ 2^{-p}+Ep(up+φp,(a,b)), is not a consequence of the global approximate-minimiser property Ep(up,Ω) ≤ 2^{-p}+Ep(up+φp,Ω) together with the fact that up and up+φp agree outside (a,b). For nonnegative integrands with equal values outside (a,b), the global normalised Lp inequality does not control the local normalised Lp norms. For example, with Ω=(0,2), (a,b)=(0,1), p=2, f≡10 on (0,1), f≡10^6 on (1,2), g≡0 on (0,1), g≡10^6 on (1,2), the global inequality holds while Ep(f,(0,1))=10 > 0.25 = 2^{-2}+Ep(g,(0,1)). Since this local estimate is the only step linking the Lp approximants to the absolute-minimiser property of u∞, the proof of Theorem 3.4 is incomplete and the abstract's assertion that absolute minimisers exist when n=1 is not supported by the argument presented.","section":"§3, Theorem 3.4, Eq. (3.17)"},{"comment":"The proof of Lemma 4.2 starts from the global inequality E∞(u+ϕ,Ω)≥E∞(u,Ω) and concludes that h(t):=max_O H(·,u+tϕ,Du+tDϕ,D²u+tD²ϕ) satisfies h(t)≥h(0) for all t. This implication is false in general: if the maximum of H(J²u) over Ω\\O is larger than max_O H(J²u), the global inequality holds even when h(t)<h(0). The correct argument should use the absolute-minimiser property on O itself, namely E∞(u+ϕ,O)≥E∞(u,O), which is exactly what Definition 1.1 gives for ϕ∈W^{2,∞}_0(O). The repair is immediate, but as written the proof of the lemma is incorrect.","section":"§4, Lemma 4.2"},{"comment":"Corollary 3.3 is a load-bearing ingredient in the proof of Theorem 3.4, since it is used to pass from the Lp approximants to E∞(u∞,(a,b)). Its proof is omitted with only a reference to [19, Lemma 5.1]. The authors should either include a full proof or state precisely how [19, Lemma 5.1] applies to the present second-order functional with lower-order terms; a reader cannot otherwise verify the diagonal lower semicontinuity step.","section":"§3, Corollary 3.3"}],"minor_comments":[{"comment":"The abstract and introduction state that D-solutions are proved to exist, but Theorem 5.2 requires the structural assumption H(x,η,p,X)=h(x,η,p,X^T X) with h strictly increasing along t↦tI and the boundedness condition (5.3). This is substantially more restrictive than the hypotheses of Theorem 3.1 and is not satisfied by a generic C¹ supremand; the statement should be qualified accordingly.","section":"§5, Theorem 5.2"},{"comment":"The summation indices in the display for A∞ include p and q, while p is also used as the gradient argument of H; this makes the formula harder to read. Consider using different letters for the summation indices or explicitly stating that p,q are dummy indices.","section":"§1, Eq. (1.11)"},{"comment":"The inequality chain in (3.17) contains a factor ((b−a)/|Ω|)^{1/p} multiplying E∞, which is not needed and seems inconsistent with the normalisation of Ep. The factor tends to 1 as p→∞, so this does not affect the conclusion, but the line should be corrected for clarity.","section":"§3, proof of Theorem 3.4"},{"comment":"The formal derivation assumes H is C² and u is C⁴, while Theorem 4.3 only assumes H∈C¹ and u∈C³. The relation between the formal calculation and the rigorous theorem would be clearer if the different regularity assumptions were separated explicitly.","section":"§4, Derivation 4.1"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a promising framework and the D-solution existence is a clean reduction, but the proof of the 1D absolute-minimiser theorem has a genuine gap at Eq. (3.17). This is a load-bearing point for one of the main abstract claims. If the authors can supply a valid local approximate-minimality argument, or alternatively weaken the theorem accordingly, the paper would likely be publishable after a careful revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, the core architecture is sound: the global-minimiser existence theorem (Thm 3.1) is a careful extension of the Lp-approximation method to lower-order terms, and the D-solution existence for A∞(J²u, D³u)=0 (Thm 5.2) is a clean reduction to Katzourakis's machinery [20]. Second, the proof of the 1D absolute-minimiser theorem (Thm 3.4) has a real gap that the authors need to fix before the abstract's claim is supported.\n\nThe novel material is real: the PDE (1.3) was noted in passing in [9], but here it is systematically derived, verified for C³ absolute minimisers, and given a D-solution theory with lower-order terms. Theorem 4.3's proof is a genuinely simpler route to the Aronsson equation than [26], and the D-solution section is honest about the structural ansatz H(x,η,p,X)=h(x,η,p,X^T X). That ansatz is restrictive, but it is clearly stated and the result is still useful.\n\nThe soft spot is in Theorem 3.4. The first inequality in (3.17) claims that the global approximate-minimiser property on Ω implies the local Lp estimate on (a,b). That does not follow. Because the Lp norm is averaged over each set, the contribution outside (a,b) can dominate the global norms while the local norms misbehave. A simple counterexample (take F=10 on (0,1), 10⁶ on (1,2); G=0 on (0,1), 10⁶ on (1,2)) shows the global inequality can hold while the local inequality fails by a large margin. Since (3.17) is the only step linking the Lp approximants to absolute minimality, the proof of Theorem 3.4 is incomplete. The theorem may be true, but this argument does not establish it. There is also a small technical issue in Lemma 4.2, where a maximum is taken over an open set; this is fixable by working on compact subdomains or invoking the compact support of the test functions. The D-solution section holds up, though the last inequality in (3.17) has a strange factor of ((b-a)/|Ω|)^{1/p}; it tends to 1 as p→∞, so it is not load-bearing, but it should be corrected.\n\nThis paper deserves a serious referee. The intended audience is specialists in calculus of variations in L∞, and for them the global-minimiser and D-solution parts will be valuable. The 1D absolute-minimiser claim needs a repaired proof; if the gap is closed, the paper is a solid contribution.\n\nRecommendation: send it to peer review, but with explicit instructions to the authors to fix Theorem 3.4 and to double-check the inequalities in (3.17).","headline":"A genuine extension of the second-order L-infinity programme with a solid D-solution section, but the 1D absolute-minimiser proof as written does not work.","tokens_in":21637,"tokens_out":6524,"would_cite":false,"duration_ms":58428,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J47","35J60","35D30","35A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For second-order supremal functionals, the Euler-Lagrange analogue is a third-order fully nonlinear PDE, and the Dirichlet problem admits generalised D-solutions.","keywords":["calculus of variations in L∞","supremal functionals","absolute minimisers","variational PDEs","fully nonlinear PDE","D-solutions","Young measures","Hessian"],"falsifier":"Set n=1, take H(X)=X², and choose first-order boundary data g with g″ non-constant. The unique absolute minimiser for ‖u″‖∞ is piecewise quadratic with one jump in u″; compute its diffuse third derivative from difference quotients and check the reduced-support condition sup_{Z∈supp*(D³u(x))} |A∞(J²u(x),Z)|=0. Any point where the support of D³u(x) contains a third-derivative value with A∞(J²u(x),Z)≠0 would make Theorem 5.2 false for this H; if the condition holds, the D-solution notion is compatible with the known non-smooth minimiser.","tokens_in":20504,"feed_emoji":"📐","tokens_out":9994,"duration_ms":89203,"temperature":0.7,"pith_summary":"The paper seeks to complete the second-order L∞ calculus of variations for supremands depending on the full second-order jet: given E∞(u,O)=ess sup_O H(·,u,Du,D²u), it establishes existence of global minimisers under Dirichlet data, of absolute minimisers in one dimension, and derives the third-order fully nonlinear PDE H_X(J²u):D(H(J²u))⊗D(H(J²u))=0 as the Euler-Lagrange analogue. It proves that C³ absolute minimisers satisfy this PDE classically, and that under a structural factorisation H=h(·,·,·,XᵀX) with monotonicity along the identity direction, the Dirichlet problem for the expanded third-order equation has generalised solutions in the D-sense. If correct, the theory provides a PDE that canonically belongs to the supremal functional and a rigorous generalised-solution framework where classical, weak, and viscosity notions do not apply.","feed_headline":"Third-order PDE found for Hessian L∞ minimisers","feed_subtitle":"Supremal variational problems with lower-order terms now have an Euler-Lagrange equation and generalised solutions.","key_machinery":"The load-bearing object is the third-order fully nonlinear operator A²∞u, the contraction of the Hessian-derivative of the supremand with the outer product of the gradient of the composed supremand H(J²u). Its expanded form A∞(J²u,D³u)=0 is the equation for which generalised solutions are defined. The proof rests on three mechanisms: shifted L^p approximations E_p=(∫(M+H)^p)^{1/p}−M, which select a preferred minimiser while handling sign-changing H; a variational lemma that uses a max-differentiation theorem on the one-sided derivatives of the supremal functional to obtain inequalities at points where H(J²u) is maximal or minimal; and the D-solution framework, in which the third derivative is a Young-measure-valued diffuse derivative and the equation is required to hold on the reduced support of that measure. Feeding that framework is the category-method solvability of the implicit equation H(J²u)=C, obtained by prescribing the eigenvalues of D²uᵀD²u and using the identity-direction monotonicity of h.","core_discovery":"For the supremal functional (1.1), the correct Euler-Lagrange analogue is A²∞u=0 in (1.3), and the paper proves this in two directions. Global minimisers exist by passing to the limit p→∞ in shifted L^p functionals (3.1), under coercivity and level-convexity; in dimension one the L^p-selected minimiser is absolute. If an absolute minimiser is C³, it solves (1.3) classically, via a new argument using radial test functions and one-sided max/min inequalities; the derivation of (1.3) from the L^p Euler-Lagrange equations is formal but defines the candidate. The PDE is then studied independently of the variational problem: for H=h(·,·,·,XᵀX) with h strictly increasing along t↦tI and bounded at δ₀I, the implicit equation H(J²u)=C has infinitely many $W^{{2,∞}}$ solutions matching g and Dg on ∂Ω, and each is a D-solution of the expanded third-order equation A∞(J²u,D³u)=0. This extends the earlier pure-Hessian second-order theory to lower-order terms with shorter proofs.","pith_inferences":["The paper leaves open whether the L^p-selected minimiser is absolute in dimension two or higher; the one-dimensional proof relies on cubic Hermite interpolation on intervals, so higher-dimensional absolute minimality is the natural next target.","The structural factorisation H=h(·,·,·,XᵀX) suggests reading D²uᵀD²u as a Hessian metric and the identity-direction monotonicity as a trace-type growth condition; this could be tested numerically on radial solutions where the eigenvalue problem reduces to an ODE.","Because the shifted approximation replaces H by M+H without changing minimisers, the existence theorem is effectively invariant under adding a constant to the supremand, which simplifies numerical realisation for sign-changing H."],"forward_implications":["Any C³ absolute minimiser of E∞ solves the third-order equation (1.3) classically, so regularity of the supremand immediately implies a pointwise PDE without differentiability of the functional.","The L^p-approximation route yields a global minimiser that in one dimension is automatically an absolute minimiser, so the selected object is locally optimal, not merely globally optimal.","When H factors through XᵀX and is monotone along identity multiples, the Dirichlet problem for A∞(J²u,D³u)=0 has infinitely many W^{2,∞} D-solutions for arbitrary first-order Dirichlet data.","The generalised-solution result subsumes and shortens the pure-Hessian second-order case, because the proof no longer has to be carried out from first principles."],"supporting_citations":[{"why":"The earlier pure-Hessian second-order L∞ theory that this paper generalises, and the benchmark showing solutions need not be C³.","marker":"[26]"},{"why":"Supplies the L^p-approximation scheme, the extended Jensen inequality, and the approximate-minimiser selection used in Theorem 3.1.","marker":"[10]"},{"why":"Category-method solvability of implicit equations with prescribed singular values, used in Lemma 5.1 for H(J²u)=C.","marker":"[15]"},{"why":"Defines D-solutions and supplies the differentiation theorem used to pass from H(J²u)=C to A∞(J²u,D³u)=0 in the D-sense.","marker":"[20]"},{"why":"The max-differentiation theorem used in Lemma 4.2 to produce the one-sided inequalities that force the PDE at maxima and minima.","marker":"[16]"},{"why":"Earlier formal derivation of the same third-order equation with running costs, cited as precedent for the PDE form.","marker":"[9]"}],"fun_headline_variants":["Hessian L∞ minimisers satisfy a third-order PDE","Existence and PDE for L∞ Hessian variational problems","Lower-order terms in L∞ Hessian calculus of variations","Solving L∞ second-order variational problems with lower-order terms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"For the generalised-solution existence theorem, the load-bearing premise is the structural assumption H(x,η,p,X)=h(x,η,p,XᵀX) with h strictly increasing along t↦tI; if a smooth supremand does not factor through XᵀX or is not monotone along identity multiples, the category-method construction of solutions to H(J²u)=C is not available and the D-solution conclusion is not established.","fun_headline_variants_meta":{"raw":{"variants":["Hessian L∞ minimisers satisfy a third-order PDE","Existence and PDE for L∞ Hessian variational problems","Lower-order terms in L∞ Hessian calculus of variations","Solving L∞ second-order variational problems with lower-order terms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000221,"raw_usage":{"total_tokens":1566,"prompt_tokens":1180,"completion_tokens":386,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":796,"completion_tokens_details":{"reasoning_tokens":318}},"tokens_in":796,"tokens_out":386,"duration_ms":3758,"temperature":1.0,"reasoning_tokens":318,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:40:52.393234+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Set n=1, take H(X)=X², and choose first-order boundary data g with g″ non-constant. The unique absolute minimiser for ‖u″‖∞ is piecewise quadratic with one jump in u″; compute its diffuse third derivative from difference quotients and check the reduced-support condition sup_{Z∈supp*(D³u(x))} |A∞(J²u(x),Z)|=0. Any point where the support of D³u(x) contains a third-derivative value with A∞(J²u(x),Z)≠0 would make Theorem 5.2 false for this H; if the condition holds, the D-solution notion is compatible with the known non-smooth minimiser.","supporting_citations":[{"cited_title":"Katzourakis and T","cited_arxiv_id":null,"evidence_quote":"The earlier pure-Hessian second-order L∞ theory that this paper generalises, and the benchmark showing solutions need not be C³."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the L^p-approximation scheme, the extended Jensen inequality, and the approximate-minimiser selection used in Theorem 3.1."},{"cited_title":"Dacorogna and P","cited_arxiv_id":null,"evidence_quote":"Category-method solvability of implicit equations with prescribed singular values, used in Lemma 5.1 for H(J²u)=C."},{"cited_title":"Katzourakis, Generalised Solutions for Fully Nonlinear PDE Systems and E xistence-Uniqueness The- orems, Journal of Differential Equations, 263 (1), 641–686 (2017)","cited_arxiv_id":null,"evidence_quote":"Defines D-solutions and supplies the differentiation theorem used to pass from H(J²u)=C to A∞(J²u,D³u)=0 in the D-sense."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The max-differentiation theorem used in Lemma 4.2 to produce the one-sided inequalities that force the PDE at maxima and minima."},{"cited_title":"Aronsson, E","cited_arxiv_id":null,"evidence_quote":"Earlier formal derivation of the same third-order equation with running costs, cited as precedent for the PDE form."}],"review_version":1}