SWhat 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.
Scientific 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.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. 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.
Contributors
- The named authors in the research record · SCIENCE: Source-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.
Provisional WRS assessment, edition 2026-10-23.3. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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