{"id":"98a6ba0b-b454-4017-b382-dacf9a24eb9a","arxiv_id":"2411.16499","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper establishes the principal eigenvalue theory and bifurcation from zero and infinity for a mixed local-nonlocal elliptic operator under mixed Dirichlet-Neumann boundary conditions.","lead":"This paper studies the first eigenvalue and eigenfunctions of the operator that sums the classical Laplacian and the fractional Laplacian, under mixed Dirichlet and Neumann conditions on different parts of the boundary. It also proves bifurcation results for an associated nonlinear problem and tracks the eigenvalue as the Dirichlet or Neumann portions vanish.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.4(3) simplicity proof uses an L2(U) normalization that the variational construction does not provide; without simplicity, the Crandall-Rabinowitz step in Theorem 2.9 lacks a one-dimensional kernel.","rationale":"The central claim comprises existence and uniqueness of bifurcation points and unbounded continua. Uniqueness from zero requires only principality, but the local Crandall-Rabinowitz branch in the proof of Theorem 2.9 explicitly requires ker T = span{φ1}, i.e., simplicity. The only proof of simplicity is Proposition 3.4(3), and it contains a concrete normalization error: the variational problem fixes the L2(Ω) norm, while the proof abruptly switches to L2(U). Because the eigenfunction on N is slaved to its values on Ω by the nonlocal Neumann condition, the L2(U) norm is not constant on the minimizing manifold, so the equality of integrals that drives the conclusion is unsupported. This is load-bearing because the rest of the bifurcation argument, including the analytic local branch and the exclusion of closed loops, is built on it. The reader identified the same gap as the first fragile premise. I do not make the f(t) > 0 issue primary: under the stated (f2) with f ≥ 0, Lemma 5.10 can be repaired by using '≥' instead of '>', and the logistic example h = s − s^p violates (f2) because f = −t^p is negative, so it is an inconsistency in an application remark rather than a defect in Theorem 2.10 itself. The appropriate verdict remains CONDITIONAL: the main results are plausible and likely repairable, but the manuscript as written does not prove simplicity, and hence does not justify the Crandall-Rabinowitz step in Theorem 2.9.","tokens_in":1063,"tokens_out":7256,"duration_ms":177797,"concrete_test":"Re-derive Assertion 3 with both eigenfunctions normalized by ||·||_{L2(Ω)} = 1, and compute ∫_N(φ1^2 − φ2^2) using the Neumann formula φ_i(x) = (∫_Ω φ_i(y)|x−y|^{−n−2s} dy) / (∫_Ω |x−y|^{−n−2s} dy) for x ∈ N. If ∫_U(φ1^2 − φ2^2) does not vanish, the proof is not repairable as written. Independent numerical check: discretize the one-dimensional problem with Ω = (0,1), N = (1,2), D = R minus (0,2) and verify whether the first eigenspace is exactly one-dimensional; if it is, the gap is fixable, and if not, Theorem 2.9 is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 3.4(3) is the hinge for Theorem 2.9: the proof needs ker(Id − λ1(D)K) = span{φ1}. The simplicity argument normalizes φ1 by ||φ1||_{L2(U)} = 1, citing 'from the proof of assertion 2', but Assertion 2's manifold M and the Rayleigh quotient (2.0.2) impose ∫Ω |u|^2 = 1, not ∫U |u|^2 = 1. Since φ1 satisfies N_sφ1 = 0 in N, its values on N are determined as weighted averages of φ1|Ω; hence ||φ1||_{L2(U)} is not forced to be 1 by the minimization. Consequently the identity (3.0.16), ∫_U(φ1^2 − φ2^2) dx = 1 − 1 = 0, is unjustified, and the dichotomy (3.0.15) does not imply φ1^2 = φ2^2. This is not a cosmetic misprint: without a one-dimensional first eigenspace, the Crandall-Rabinowitz bifurcation step, and the later use of the unique local branch to exclude closed loops in the unboundedness proof, have no valid basis. The f-positivity issue in Lemmas 5.4/5.10 is secondary: under the stated (f2) with f ≥ 0, Lemma 5.10 survives by replacing '>' with '≥', and the logistic example in Remark 5.11 is outside the hypotheses rather than a counterexample to the theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the eigenvalue problem for the mixed local-nonlocal operator L = -Δ + (-Δ)^s under mixed Dirichlet and Neumann conditions, defines a Rayleigh quotient λ1(D), and proves existence, positivity, boundedness, Hölder regularity, and simplicity of the first eigenfunction. It then studies the asymptotic behavior of λ1(D) as the Neumann or Dirichlet sets dissipate, and applies these results to an asymptotically linear problem Qλ, proving the existence of unique bifurcation points from zero and from infinity, each generating an unbounded connected component of positive solutions. The main theorems are Theorem 2.7, Theorem 2.8, Theorem 2.9, and Theorem 2.10.","tokens_in":1786,"tokens_out":1863,"duration_ms":92541,"significance":"If the proofs are completed, the paper makes a useful contribution by extending principal-eigenvalue and bifurcation theory from purely local and purely nonlocal settings to a mixed operator with mixed boundary conditions. The dissipating-set results and the bifurcation-from-infinity treatment are natural extensions of known work by Leonori et al. and by Arcoya-Gámez, and the paper contains several self-contained tools of independent interest: the weak maximum principle, the L∞ bound for first eigenfunctions, the compactness of the solution operator, and the regularity appendix. The variational construction of λ1(D) and the positivity of the first eigenfunction are standard and appear sound. However, the proof of simplicity contains a normalization error that is load-bearing for the local bifurcation argument, and the hypotheses in Section 5 do not match some stated examples; these issues require a major revision.","major_comments":[{"comment":"The proof of simplicity is not valid as written. The variational constraint used in Assertion 2 and in the definition (2.0.2) is ∫Ω |u|² dx = 1, but Assertion 3 normalizes the second eigenfunction by ‖φ2‖_{L²(U)}=1 and claims that ‖φ1‖_{L²(U)}=1 'from the proof of assertion 2'. This claim is unjustified: since N_s φ1 = 0 in N, the values of φ1 on N are determined by its values on Ω through the weighted-average formula in Lemma 3.1, and the L²(U)-norm is not fixed by the minimization. Consequently, identity (3.0.16), ∫_U(φ1² - φ̃2²)dx = 1 - 1 = 0, is not established, and the dichotomy (3.0.15) does not force φ1² = φ̃2². This is load-bearing because Theorem 2.9 later invokes Crandall-Rabinowitz bifurcation from a simple eigenvalue and uses uniqueness of the local branch to exclude closed loops in the unboundedness proof. The argument can likely be repaired by normalizing both eigenfunctions in L²(Ω), deriving φ1² = φ2² a.e. in Ω, and then propagating equality to N using the nonlocal Neumann condition; but the present proof needs correction.","section":"Section 3, Proposition 3.4(3), Eq. (3.0.16)"},{"comment":"The proofs of Lemmas 5.4 and 5.10 use the strict inequality f(t)>0 for all t, but this is not stated in hypothesis (f2), which only gives f: R → R+ with |f(t)| ≤ C. If R+ means nonnegative, then only f(t) ≥ 0 is available; if R+ means positive, then the example h(s)=s-s^p advertised in Remark 5.11 has f(s)=-s^p and is excluded even from nonnegativity. Thus the strict inequality λ1(D) > λθ in Lemma 5.4 and the strict contradiction in Lemma 5.10 are not justified under the stated hypotheses. Please either add an explicit strict-positivity hypothesis to (f2), or weaken the conclusions accordingly and reconcile this with Remark 5.11.","section":"Section 5, Lemmas 5.4 and 5.10, and Remark 5.11"},{"comment":"The Crandall-Rabinowitz step is stated imprecisely at a point where precision matters. After defining λ0 = λ1(D)/a, the proof writes T = ∂_u I(λ1(D),0) = Id - λ1(D)K and immediately invokes bifurcation from (λ1(D)/a,0). The correct linearization is ∂_u I(λ0,0) = Id - λ0 h'(0) K, which equals Id - λ1(D)K only when h'(0)=a; the phrase 'we may assume h'(0)=1' is not a harmless normalization because λ0 depends on a. The transversality condition is also written in a garbled form. Please state h'(0)=a, evaluate the derivative at λ0, and formulate the transversality condition as ∂²_{λu}I(λ0,0)ψ ∉ R(T).","section":"Section 5.1, proof of Theorem 2.9"}],"minor_comments":[{"comment":"In the proof of Theorem 2.10, the intervals are written with λ0 where λ∞ is meant, and the condition '0 < ‖u‖ ≤ 1/R' should refer to v in the transformed problem; please correct these statements.","section":"Section 5.2, proof of Theorem 2.10"},{"comment":"Equation (5.0.4) concerns the problem Lu = w, but it contains a factor λ multiplying ∫Ω w φ_k dx; this λ should be removed.","section":"Section 5, Lemma 5.3, Eq. (5.0.4)"},{"comment":"Assertion 4 contains a duplicated sentence defining φ2 and the ε-regularized test function, and the notation uε is introduced twice; the passage should be cleaned up.","section":"Section 3, Proposition 3.4(4)"},{"comment":"The convention for R+ is ambiguous: the statement f: R → R+ with |f(t)| ≤ C is not compatible with the extension h(t)=0 for t≤0 unless the extension is specified separately. Please state explicitly whether R+ means (0,∞) or [0,∞).","section":"Throughout, hypotheses (f1)-(f3)"},{"comment":"The assumption 'lim_{k→∞} |D_k ∩ ∂Ω| = 0' is automatically satisfied for open sets D_k ⊂ R^n \\ Ω̄; the intended condition is presumably lim_{k→∞} |D̄_k ∩ ∂Ω| = 0. Please clarify this point.","section":"Theorem 2.8 and Section 4.2"},{"comment":"The displayed 'curves' in Remark 5.11 appear to be placeholders rather than actual figures; please include the plots or remove the visual references.","section":"Remark 5.11"}],"recommendation":"major_revision","confidential_remarks":"The central variational existence theory is standard and likely correct, and the dissipating-set results are plausible. The main obstacle is the simplicity proof in Proposition 3.4(3): it is currently invalid and it is genuinely needed for the Crandall-Rabinowitz step in Theorem 2.9. I did not recommend rejection because a normalization in L²(Ω) plus propagation through the Neumann condition appears to repair the proof. The f-positivity mismatch with Remark 5.11 also needs a clear fix. If the authors can supply the corrected simplicity argument and align the hypotheses with the claims in Section 5, the paper would be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear X,\n\nThe short version: this is a serious paper with a genuinely new setting, but the proof that lambda1(D) is simple contains a normalization error that the bifurcation results lean on. I would send it to a referee, but the authors need to fix Proposition 3.4(3) first.\n\nWhat is new and good: nobody has done the principal eigenvalue for L = -Delta + (-Delta)^s under mixed Dirichlet-Neumann conditions, and the dissipating-set results (Theorems 2.7, 2.8) adapt the known fractional framework to the mixed operator. The variational existence, Holder regularity, and the strong maximum principle are standard but competently handled. The bifurcation section is a careful transplant of Arcoya-Gamez / Rabinowitz machinery and is mostly readable.\n\nThe soft spots, in proportion:\n\n1. Proposition 3.4(3) is the real problem. The proof normalizes phi1 by ||phi1||_{L2(U)} = 1, citing 'from the proof of assertion 2', but the manifold M in Assertion 2 and the Rayleigh quotient constrain integral_Omega |u|^2 = 1, not integral_U. Since eigenfunctions are generally nonzero on N, the L2(U) norm is not fixed. Equation (3.0.16) is therefore unjustified, and the dichotomy does not imply phi1^2 = phi2^2. This matters because Theorem 2.9 uses ker(Id - lambda1(D)K) = span{phi1} to invoke Crandall-Rabinowitz. Without simplicity, the local branch and the uniqueness of lambda0 lack a valid basis. The gap looks fixable—there is a Picone inequality in the paper that may give simplicity directly—but as written it is load-bearing.\n\n2. The f-positivity assumption: Lemmas 5.4 and 5.10 use f(t) > 0, which is not in (f2) (only f >= 0) and fails for h(s) = s - s^p. This one is lower-stakes; the '>=' version would suffice for Lemma 5.10, and the logistic example is outside the stated hypotheses anyway. Worth a clarification, but not a fatal blow.\n\nOverall: the main theorems are plausible and the framework is sound; the missing pieces are mechanical rather than conceptual. I would not cite it in its current form, but I would want to see the revision.\n\nRecommendation: send it to an expert referee. The novelty justifies the referee time, and the flaws are specific enough to be checked in a single pass.","headline":"A genuinely new setting with a load-bearing normalization error in the simplicity proof; the main results will likely survive once Proposition 3.4(3) is fixed.","tokens_in":33622,"tokens_out":2784,"would_cite":false,"duration_ms":25253,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47A75","35J25","35J20"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single eigenvalue of the mixed operator fixes both the bifurcation from zero and the bifurcation from infinity.","keywords":["mixed local-nonlocal operators","fractional Laplacian","mixed boundary conditions","principal eigenvalue","bifurcation from zero","bifurcation from infinity","asymptotically linear problems","maximum principle"],"falsifier":"Check whether a first eigenfunction on a domain with $\\mathcal{N}$ nonempty has $\\int_{\\mathcal{N}} \\phi_1^2 > 0$: if so, the normalization $\\|\\phi_1\\|_{L^2(U)} = 1$ used in the simplicity proof is incompatible with the variational constraint $\\int_\\Omega |\\phi_1|^2 = 1$, and the proportionality conclusion needs a different argument. A second check: for $h(s) = s - s^p$, $f(t) = -t^p$, so the strict positivity of $f$ used in Lemmas 5.4 and 5.10 is false; testing the claimed bound $\\lambda < \\theta \\lambda_1(\\mathcal{D})$ for this $h$ would show whether the branch estimates still hold.","tokens_in":2534,"feed_emoji":"📈","tokens_out":2776,"duration_ms":76735,"temperature":0.7,"pith_summary":"This paper studies the eigenvalue problem for the mixed local-nonlocal operator $\\mathcal{L} = -\\Delta + (-\\Delta)^s$ under mixed boundary conditions: Dirichlet data on one exterior set $\\mathcal{D}$, a nonlocal Neumann condition on another exterior set $\\mathcal{N}$, and a classical Neumann condition on the part of $\\partial\\Omega$ bordering $\\mathcal{N}$. It establishes that the first eigenvalue $\\lambda_1(\\mathcal{D})$, defined by a Rayleigh quotient on a partially Dirichlet space, is positive, simple, and has a positive H\\\"older-continuous eigenfunction, and that every higher eigenvalue has sign-changing eigenfunctions. Building on this spectral foundation, it proves that the asymptotically linear problem $\\mathcal{L}u = \\lambda h(u)$ has exactly one bifurcation point from zero, $\\lambda_0 = \\lambda_1(\\mathcal{D})/a$, and exactly one from infinity, $\\lambda_\\infty = \\lambda_1(\\mathcal{D})/\\theta$, each generating an unbounded connected branch of positive solutions. The interest is that a single spectral quantity of the mixed operator organizes the full bifurcation picture, and the results also track how $\\lambda_1(\\mathcal{D})$ moves when the Dirichlet or Neumann sets dissipate.","feed_headline":"One eigenvalue sets both bifurcation points of mixed problems","feed_subtitle":"The first eigenvalue is positive and simple; each bifurcation yields an unbounded branch of positive solutions.","key_machinery":"The central object is the Hilbert space $X_{\\mathcal{D}}^{1,2}(U)$ of $H^1$ functions on $\\mathbb{R}^n$ that vanish outside $U = \\Omega \\cup \\mathcal{N} \\cup (\\partial\\Omega \\cap \\overline{\\mathcal{N}})$, equipped with the norm $\\eta(u)^2 = \\int_\\Omega |\\nabla u|^2\\,dx + \\int_Q \\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\\,dx\\,dy$, where $Q$ excludes pairs both outside $\\Omega^c \\times \\Omega^c$; the Rayleigh quotient in this space defines $\\lambda_1(\\mathcal{D})$. The argument is carried by three devices: a strong maximum principle built from Bony's maximum principle and a Hopf lemma, a Picone-type inequality that yields simplicity and sign-changing behavior, and compactness of the inverse operator $K = \\mathcal{L}^{-1}: X \\to X$. For bifurcation, the paper uses the Leray-Schauder degree to show the index of $I_\\lambda$ changes only at $\\lambda_0$, then applies the classical Crandall-Rabinowitz and Rabinowitz global bifurcation theorems; bifurcation from infinity is converted to bifurcation from zero by the inversion $v = u/\\|u\\|^2$.","core_discovery":"On the paper's own terms, the central discovery is that the spectrum of the mixed local-nonlocal operator with mixed boundary conditions has a principal eigenvalue $\\lambda_1(\\mathcal{D}) > 0$ with the properties expected of a cooperative linear operator: simplicity, sign-definite eigenfunctions, and H\\\"older continuity, with all higher eigenvalues having sign-changing eigenfunctions. This eigenvalue alone determines where the nonlinear problem $(Q_\\lambda)$ can bifurcate: the paper proves that $\\lambda_0 = \\lambda_1(\\mathcal{D})/a$ is the unique bifurcation point from the trivial solution and $\\lambda_\\infty = \\lambda_1(\\mathcal{D})/\\theta$ the unique bifurcation point from infinity, and that each bifurcation produces an unbounded connected set $\\Gamma_0$, $\\Gamma_\\infty$ of positive solutions. It further proves qualitative behavior of $\\lambda_1(\\mathcal{D})$ under geometric limits: $\\lambda_1(\\mathcal{D}_k) \\to \\lambda_1(\\mathbb{R}^n \\setminus \\Omega)$ as the Neumann sets dissipate, and $\\lambda_1(\\mathcal{D}_k) \\to 0$ as the Dirichlet sets dissipate when $0 < s < 1/2$, with partial extensions for $s \\ge 1/2$.","pith_inferences":["A natural testable extension: the same uniqueness-and-unbounded-branch structure should hold for the $p$-Laplacian version $-\\Delta_p + (-\\Delta_p)^s$ under the same mixed boundary data, provided the analogue of $\\lambda_1(\\mathcal{D})$ is simple; the paper cites the $p$-eigenvalue literature but does not prove this case.","The bifurcation-from-infinity analysis via $v = u/\\|u\\|^2$ suggests that rescaling any solution family in $L^\\infty$ produces a compact perturbation problem; this inversion could be used to extract refined asymptotics of $\\Gamma_\\infty$ near $\\lambda_\\infty$, which the paper does not compute.","If simplicity of $\\lambda_1(\\mathcal{D})$ fails in some geometry, the bifurcation picture could be richer, with multiple local branches; a numerical check on a domain with a nonempty Neumann set could discriminate.","The condition $f(t) > 0$ used in the bound $\\lambda < \\theta \\lambda_1(\\mathcal{D})$ is not guaranteed by the hypotheses and fails for the logistic-type example $h(s) = s - s^p$; if it is dropped, the paper's claimed branch range for such $h$ may require an alternative argument."],"forward_implications":["For any asymptotically linear $h$ satisfying (f1)-(f3), the only way positive solutions can emerge from the zero solution is at $\\lambda_0 = \\lambda_1(\\mathcal{D})/a$, and the branch that emerges is unbounded.","The only possible bifurcation from infinity occurs at $\\lambda_\\infty = \\lambda_1(\\mathcal{D})/\\theta$, and it also yields an unbounded connected set of positive solutions.","If the Neumann region shrinks to nothing, the first eigenvalue of the mixed problem converges to that of the pure Dirichlet problem for $\\mathcal{L}$ on $\\mathbb{R}^n \\setminus \\Omega$.","If the Dirichlet region disperses and $0 < s < 1/2$, the first eigenvalue collapses to zero, matching the pure Neumann case; a partial result holds for $s \\ge 1/2$ under additional geometric conditions.","Every eigenvalue above $\\lambda_1(\\mathcal{D})$ has sign-changing eigenfunctions, so $\\lambda_1(\\mathcal{D})$ is the only eigenvalue with a one-signed eigenfunction."],"supporting_citations":[{"why":"Defines the nonlocal normal derivative $\\mathcal{N}_s$ used in the mixed boundary condition.","marker":"[29]"},{"why":"Supplies regularity and maximum principles for mixed local-nonlocal operators, used in Lemma 3.1 and Remark 5.2.","marker":"[13]"},{"why":"Provides Bony's maximum principle, a key ingredient in the strong maximum principle of Lemma 3.1.","marker":"[33]"},{"why":"Supplies the Hopf lemma and gradient regularity used to control the boundary behavior in Lemma 3.1.","marker":"[5]"},{"why":"Gives the Picone-type inequality used to prove simplicity and sign-changing properties of eigenfunctions.","marker":"[43]"},{"why":"Provides the model for principal eigenvalues under mixed boundary conditions for the fractional Laplacian, including the dissipating-set arguments.","marker":"[39]"},{"why":"Rabinowitz's global bifurcation theorem is the basis for the unbounded continua in Theorems 2.9 and 2.10.","marker":"[47]"},{"why":"Supplies the analytic global bifurcation framework and the Crandall-Rabinowitz theorem used for the local branch at $\\lambda_0$.","marker":"[18]"},{"why":"Establishes $\\lambda_1(\\emptyset) = 0$ for the pure Neumann case, used in Theorem 4.1 and Proposition 4.1.","marker":"[42]"}],"fun_headline_variants":["First eigenvalue controls both bifurcations in mixed operator","One principal eigenvalue governs all mixed-problem bifurcations","Single eigenvalue sets zero and infinity bifurcation points","Mixed operator's first eigenvalue fixes both solution branches","Eigenvalue one: key to both bifurcation points in mixed problems"],"cache_read_input_tokens":35584,"weakest_assumption_plain":"The proof that $\\lambda_1(\\mathcal{D})$ is simple assumes the eigenfunction $\\phi_1$ can be normalized in $L^2(U)$ even though the variational problem that produces $\\phi_1$ fixes only $\\int_\\Omega |\\phi_1|^2 = 1$, and since eigenfunctions need not vanish on $\\mathcal{N}$, this normalization is not justified; a second fragile assumption is the strict positivity $f(t) > 0$ used in Lemmas 5.4 and 5.10, which is not part of hypotheses (f1)-(f3) and fails for $h(s) = s - s^p$.","fun_headline_variants_meta":{"raw":{"variants":["First eigenvalue controls both bifurcations in mixed operator","One principal eigenvalue governs all mixed-problem bifurcations","Single eigenvalue sets zero and infinity bifurcation points","Mixed operator's first eigenvalue fixes both solution branches","Eigenvalue one: key to both bifurcation points in mixed problems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000632,"raw_usage":{"total_tokens":3041,"prompt_tokens":1194,"completion_tokens":1847,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":810,"completion_tokens_details":{"reasoning_tokens":1768}},"tokens_in":810,"tokens_out":1847,"duration_ms":11962,"temperature":1.0,"reasoning_tokens":1768,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:04:20.840339+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check whether a first eigenfunction on a domain with $\\mathcal{N}$ nonempty has $\\int_{\\mathcal{N}} \\phi_1^2 > 0$: if so, the normalization $\\|\\phi_1\\|_{L^2(U)} = 1$ used in the simplicity proof is incompatible with the variational constraint $\\int_\\Omega |\\phi_1|^2 = 1$, and the proportionality conclusion needs a different argument. A second check: for $h(s) = s - s^p$, $f(t) = -t^p$, so the strict positivity of $f$ used in Lemmas 5.4 and 5.10 is false; testing the claimed bound $\\lambda < \\theta \\lambda_1(\\mathcal{D})$ for this $h$ would show whether the branch estimates still hold.","supporting_citations":[{"cited_title":"Dipierro, X","cited_arxiv_id":null,"evidence_quote":"Defines the nonlocal normal derivative $\\mathcal{N}_s$ used in the mixed boundary condition."},{"cited_title":"Biagi, S","cited_arxiv_id":null,"evidence_quote":"Supplies regularity and maximum principles for mixed local-nonlocal operators, used in Lemma 3.1 and Remark 5.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides Bony's maximum principle, a key ingredient in the strong maximum principle of Lemma 3.1."},{"cited_title":"On elliptic problems with mixed operators and Dirichlet-Neumann boundary conditions","cited_arxiv_id":"2311.02567","evidence_quote":"Gives the Picone-type inequality used to prove simplicity and sign-changing properties of eigenfunctions."},{"cited_title":"Leonori, M","cited_arxiv_id":null,"evidence_quote":"Provides the model for principal eigenvalues under mixed boundary conditions for the fractional Laplacian, including the dissipating-set arguments."},{"cited_title":"Rabinowitz","cited_arxiv_id":null,"evidence_quote":"Rabinowitz's global bifurcation theorem is the basis for the unbounded continua in Theorems 2.9 and 2.10."},{"cited_title":"Buffoni and J","cited_arxiv_id":null,"evidence_quote":"Supplies the analytic global bifurcation framework and the Crandall-Rabinowitz theorem used for the local branch at $\\lambda_0$."},{"cited_title":"Mugnai and E","cited_arxiv_id":null,"evidence_quote":"Establishes $\\lambda_1(\\emptyset) = 0$ for the pure Neumann case, used in Theorem 4.1 and Proposition 4.1."}],"review_version":1}