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    "title": "Weyl: area controls the leading Dirichlet eigenvalue count",
    "summary": "Weyl linked high Dirichlet eigenvalues of a planar domain to its area: for eigenvalues λ_n of −Δ ordered with multiplicity, n/λ_n tends to area/(4π). His1911note combines comparison inequalities for symmetric integral kernels with the Green-function singularity, reduction to squares and explicit square eigenvalues. It also states the analogous planar variable-coefficient leading term with effective area ∫k/p for the specified self-adjoint elliptic problem.",
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        "title": "Ueber die asymptotische Verteilung der Eigenwerte (1911 original scan)",
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        "description": "The original TheoremX is stated in two dimensions for a bounded planar region, with u=0 boundary condition and its Green-function/boundary calculus setting. The short proof uses boundary normal derivatives and integrability of log distance-to-boundary; this reading does not certify arbitrary fractal boundaries from the word beliebig. For ∂x(p∂xu)+∂y(p∂yu)−qu+λku=0, p,q,k are continuous through the boundary and p,k stay positive; lim4πn/λ_n=∫k/p. No Neumann/Robin, general dimension/manifold, rough measurable-coefficient or sign-changing weight extension is asserted. Leading asymptotic does not determine shape uniquely or supply a second boundary term or uniform remainder.",
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          "citation": "Ueber die asymptotische Verteilung der Eigenwerte (1911 original scan)",
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      "statement": "# Weyl: area controls the leading Dirichlet eigenvalue count\n\nWeyl linked high Dirichlet eigenvalues of a planar domain to its area: for eigenvalues λ_n of −Δ ordered with multiplicity, n/λ_n tends to area/(4π). His1911note combines comparison inequalities for symmetric integral kernels with the Green-function singularity, reduction to squares and explicit square eigenvalues. It also states the analogous planar variable-coefficient leading term with effective area ∫k/p for the specified self-adjoint elliptic problem.\n\n## Scientific scope\n\nThe original TheoremX is stated in two dimensions for a bounded planar region, with u=0 boundary condition and its Green-function/boundary calculus setting. The short proof uses boundary normal derivatives and integrability of log distance-to-boundary; this reading does not certify arbitrary fractal boundaries from the word beliebig. For ∂x(p∂xu)+∂y(p∂yu)−qu+λku=0, p,q,k are continuous through the boundary and p,k stay positive; lim4πn/λ_n=∫k/p. No Neumann/Robin, general dimension/manifold, rough measurable-coefficient or sign-changing weight extension is asserted. Leading asymptotic does not determine shape uniquely or supply a second boundary term or uniform remainder.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nHermann Weyl\n\nSource-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.\n\n\n\n## Original sources\n\n- [Ueber die asymptotische Verteilung der Eigenwerte (1911 original scan)](https://gdz.sub.uni-goettingen.de/download/pdf/PPN252457811_1911/LOG_0023.pdf)\n",
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      "title": "Weyl: area controls the leading Dirichlet eigenvalue count"
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    "title": "Weyl: area controls the leading Dirichlet eigenvalue count",
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    "what_changed": "Weyl linked high Dirichlet eigenvalues of a planar domain to its area: for eigenvalues λ_n of −Δ ordered with multiplicity, n/λ_n tends to area/(4π). His1911note combines comparison inequalities for symmetric integral kernels with the Green-function singularity, reduction to squares and explicit square eigenvalues. It also states the analogous planar variable-coefficient leading term with effective area ∫k/p for the specified self-adjoint elliptic problem.",
    "scope": "Weyl linked high Dirichlet eigenvalues of a planar domain to its area: for eigenvalues λ_n of −Δ ordered with multiplicity, n/λ_n tends to area/(4π). His1911note combines comparison inequalities for symmetric integral kernels with the Green-function singularity, reduction to squares and explicit square eigenvalues. It also states the analogous planar variable-coefficient leading term with effective area ∫k/p for the specified self-adjoint elliptic problem. The original TheoremX is stated in two dimensions for a bounded planar region, with u=0 boundary condition and its Green-function/boundary calculus setting. The short proof uses boundary normal derivatives and integrability of log distance-to-boundary; this reading does not certify arbitrary fractal boundaries from the word beliebig. For ∂x(p∂xu)+∂y(p∂yu)−qu+λku=0, p,q,k are continuous through the boundary and p,k stay positive; lim4πn/λ_n=∫k/p. No Neumann/Robin, general dimension/manifold, rough measurable-coefficient or sign-changing weight extension is asserted. Leading asymptotic does not determine shape uniquely or supply a second boundary term or uniform remainder.",
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      {
        "title": "Ueber die asymptotische Verteilung der Eigenwerte (1911 original scan)",
        "url": "https://gdz.sub.uni-goettingen.de/download/pdf/PPN252457811_1911/LOG_0023.pdf",
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