Results on Neumann eigenfunctions, conformal coordinates, boundary kernels and reciprocal matrices

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• The eigenvalue problems — Proposition 3.1. Assume (SOB) and (ER). The first Dirichlet eigenvalue and the first positive Neumann eigenvalue are attained; the Neumann eigenspace is finite-dimensional, consists of functions smooth up to the boundary, and contains no nonzero function vanishing on an open set.
• Filonov's gap — Proposition 4.1. Assume (SOB) and (ER). For a bounded simply connected planar domain with smooth boundary (the proof uses only connectedness), the first positive Neumann eigenvalue is strictly smaller than the first Dirichlet eigenvalue.
• Strict hot spots — Theorem 5.1. Assume (SOB) and (ER). On a bounded simply connected planar domain with smooth boundary, no eigenfunction of the first positive Neumann eigenvalue has an interior critical point; its maximum and minimum are attained only on the boundary.
• Hopf's boundary point lemma — Lemma 1.2. A subharmonic function on a disc, continuously differentiable up to a boundary point where it attains a strict maximum, has positive outward normal derivative there.
• Conformal maps smooth up to the boundary — Proposition 2.1. A bounded simply connected planar domain with smooth boundary is the image of the unit disc under a conformal map that extends to a diffeomorphism of the closures with nonvanishing derivative.
• A positive boundary weight — Lemma 3.1. Assume (SOB) and (ER). A smooth positive weight on the circle annihilates the pulled-back boundary values of every first Neumann eigenfunction; these boundary values are nonzero and take both signs.
• Planar separation — Lemma 4.2. Two paths in the closed unit disc, with interiors in the open disc, joining alternating pairs of points of the circle, intersect.
• Connected sign sets — Lemma 5.1. Assume (SOB) and (ER). The positive and negative sets of a first Neumann eigenfunction are connected, and its boundary values show no four alternating strict signs.
• Multiplicity at most two — Proposition 5.2. Assume (SOB) and (ER). The first positive Neumann eigenvalue of a bounded simply connected planar domain with smooth boundary has multiplicity at most two.
• Rohleder's variational principle — Proposition 1.1. Assume (SOB) and (ER). For tangent H^1 vector fields the divergence-curl energy is at least mu times the squared L^2 norm, with equality exactly for gradients of first Neumann eigenfunctions.
• The curvature identity — Lemma 2.1. For tangent vector fields of class C^2 up to the boundary, the divergence-curl energy equals the Dirichlet energy plus the boundary integral of the curvature times |X|^2.
• Green representation in the disc — Lemma 1.2. The Green integral of an L^2 function on the unit disc is defined and continuous at every point and is the weak solution of the Dirichlet problem with zero boundary values.
• The Green operator of a subcritical potential — Lemma 2.2. For a bounded nonnegative potential q with a Dirichlet inequality of constant d < 1, the Green operator is self-adjoint on L^2(q) with norm at most d.
• Boundary kernels — Lemma 4.1. The kernels K and R built from the Poisson kernel and the Green operator are finite off the diagonal, R is symmetric with total integral at most q(D)/(1-d), and K represents the solution of the Dirichlet problem with the potential.
• Douglas's identity — Lemma 5.1. The Dirichlet energy of the Poisson extension of a Lipschitz function h on the circle is half the double integral of |h(s)-h(t)|^2/(pi |s-t|^2).
• The multiplier identity — Proposition 6.2. Assume (SOB) and (ER). For the boundary gradient g of a first Neumann eigenfunction and every real Lipschitz b on the circle, E(bg) is half the double integral of N(s,t)(b(s)-b(t))^2 g(s).g(t).
• Matrices with one positive eigenvalue — Lemma 1.1. Having at most one positive eigenvalue is preserved by congruence, by subtracting positive semidefinite matrices, by principal submatrices and by limits; a rank-one matrix minus a positive semidefinite one has it.
• Conditional negativity — Theorem 2.1. For a bounded nonnegative subcritical potential on the disc and every interior point p, the kernel K(p,s)K(p,t)/N(s,t) on the circle is conditionally negative semidefinite.
• Regularized Green kernels — Lemma 3.1. There are bounded continuous kernels C_eps increasing to the Green function of the disc whose finite entrywise-reciprocal matrices have at most one positive eigenvalue.
• Rank-one updates — Lemma 4.1. If the entrywise reciprocal of a positive symmetric matrix M has at most one positive eigenvalue, so does the entrywise reciprocal of M + c M_iz M_zj for c >= 0.
• A Hilbert-space embedding of the circle — Lemma 1.1. For every interior point p of the disc there is an injective Lipschitz map b from the circle into a separable Hilbert space with |b(s)-b(t)|^2 = K(p,s)K(p,t)/N(s,t).

Reading order

Six lessons, to be read in order. The proof assumes the named hypotheses (SOB), standard facts on Sobolev spaces of smooth bounded planar domains, and (ER), elliptic regularity for the Dirichlet and Neumann problems; apart from the Dirichlet cases noted in the first lesson, these are not proved in the programme.