What is recorded here?
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.
Source-linked catalogue note. A detailed explanatory guide has not yet been prepared for this entry.
Scientific scope
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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People and systems
Hermann Weyl
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.
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English catalogue narrative
Weyl: area controls the leading Dirichlet eigenvalue count
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
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.
Reading status
This 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.
People and systems
Hermann Weyl
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.