Green's theorem for a bounded region Ω in the plane with smooth boundary ∂Ω states that One direct way to prove this is as follows.
By subtraction, it is sufficient to prove the theorem for a region bounded by a simple smooth curve.
Separating the A and B terms, the right hand side can be written as a double integral starting in the x or y direction, to which the fundamental theorem of calculus can be applied.
A similar argument shows that the average of a harmonic function on the boundary of a disk equals its value at the centre.
With this terminology the four basic problems of classical potential theory are as follows:[3] For the exterior problems the inversion map z−1 takes harmonic functions on Ωc into harmonic functions on the image of Ωc under the inversion map.
The properties of the fundamental solution lead to the following formula for recovering a harmonic function u in Ω from its boundary values:[6] where K is the Neumann−Poincaré kernel To prove this identity, Green's second identity can be applied to Ω with a small disk centred on z removed.
This reduces to showing that the identity holds in the limit for a small disk centred on z shrinking in size.
A similar formula holds for functions harmonic in Ωc:[7] The signs are reversed because of the direction of the normal derivative.
Its diagonal values are given by the formula Another expression for k(s,t) is as follows:[9] where z(t) = x(t) + i y(t) is the boundary curve parametrized by arc length.
This follows because, if ζn is the point on ∂Ω with normal containing zn, then The first term the last product uniformly bounded because of the smoothness of the Gauss map n(t).
The second is uniformly bounded because of the approximate version of Pythagoras' theorem: Continuity of f implies that on ∂Ω which gives the jump formulas.
If the moment φ is smooth, the derivatives of the single and double layer potentials on Ω and Ωc extend continuously to their closures.
As a consequence of these relations, successive derivatives can all be expressed in terms of single and double layer potentials of smooth moments on the boundary.
The following properties of T = TK are required to solve the boundary value problem: In fact since a I + T is a Fredholm operator of index 0, it and its adjoint have kernels of equal dimension.
There is another consequence of the laws governing the derivatives, which completes the symmetry of the jump relations, is that normal derivative of the double layer potential has no jump across the boundary, i.e. it has a continuous extension to a tubular neighbourhood of the boundary given by[12] H is called a hypersingular operator.
Hence in Ω Taking the boundary values of both sides and their normal derivative yields 2 equations.
Then The jump formulas for the boundary values and normal derivatives give and It follows that so that ψ and φ are eigenfunctions of T and T* with eigenvalue λ.
Following Schiffer (2011), let φ be an eigenfunction of TK* with real eigenvalue λ satisfying 0 < |λ| < 1/2.
By Green's theorem Adding the two integrals and using the jump relations for the single layer potential, it follows that Thus This shows that the operator S is self-adjoint and non-negative on L2(∂Ω).
The Fredholm determinant is defined by It can be expressed in terms of the Fredholm eigenvalues λn with modulus less than 1/2, counted with multiplicity, as Now define the complex Hilbert transform or conjugate Beurling transform Tc on L2(C) by This is a conjugate-linear isometric involution.
On the other hand it can be checked that TD = 0 by computing directly on powers zn using Stokes theorem to transfer the integral to the boundary.
By the spectral theorem for compact self-adjoint operators, there is an orthonormal basis un of H consisting of eigenvectors of A: where μn is non-negative by the positivity of A.
Thus an orthonormal basis can be chosen on each eigenspace so that: and by conjugate-linearity of T. The Neumann–Poincaré operator is defined on real functions f as where H is the Hilbert transform on ∂Ω.
Writing h = f + ig,[15] so that The imaginary part of the Hilbert transform can be used to establish the symmetry properties of the eigenvalues of TK.
The precise relationship between single and double layer potentials, Fredholm eigenvalues and the complex Hilbert transform is explained in detail in Schiffer (1981).
The space 𝕳 is naturally an inner product space with corresponding norm given by Each element of 𝕳 can be written uniquely as the restriction of the sum of a double layer and single layer potential, provided that the moments are normalized to have 0 integral on ∂Ω.
It can be verified directly that for φ, ψ real[19] In fact for single layer potentials, applying Green's theorem on the domain Ω ∪ Ωc with a small closed disk of radius ε removed around a point w of the domain, it follows that since the mean of a harmonic function over a circle is its value at the centre.
Using the fact that πz−1 is the fundamental solution for ∂w, this can be rewritten as Applying ∂w to both sides gives Similarly for a double layer potential since the mean of the normal derivative of a harmonic function over a circle is zero.
As above, using the fact πz−1 is the fundamental solution for ∂w, this can be rewritten in terms of complex derivatives as Applying ∂w to both sides, Let L2(∂Ω)0 be the closed subspace of L2(∂Ω) orthogonal to the constant functions.
The Cauchy transform of f in H2(∂Ω) defines a holomorphic function F in Ω such that its restrictions to the level curves ∂Ωs in a tubular neighbourhood of ∂Ω have uniformly bounded L2 norms.