Strict Neumann Polya inequality on simply connected planar domains
A Lean 4 / Mathlib formalization of the strict Neumann Pólya inequality for bounded simply connected Lipschitz domains in the plane. The plane is identified with ℂ; the Neumann eigenvalues μ_0 ≤ μ_1 ≤ ... are defined by the Courant–Fischer min–max formula for the form ∫|∇u|² on H¹(Ω) (weak gradients in L²), indexed from 0 and repeated according to multiplicity, and N_N(E) = #{j ≥ 0 : μ_j |Ω| E / (4π) for every E > 0. The proof development (about 100,000 lines, namespace PolyaNeumann) also identifies the variational eigenvalues with the spectrum of the constructed self-adjoint Neumann…
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