In mathematics, the Neumann–Poincaré operator or Poincaré–Neumann operator, named after Carl Neumann and Henri Poincaré, is a non-self-adjoint compact operator introduced by Poincaré to solve boundary value problems for the Laplacian on bounded domains in Euclidean space. Within the language of potential theory it reduces the partial differential equation to an integral equation on the boundary to which the theory of Fredholm operators can be applied. The theory is particularly simple in two dimensions—the case treated in detail in this article—where it is related to complex function theory, the conjugate Beurling transform or complex Hilbert transform and the Fredholm eigenvalues of bounded planar domains.
Dirichlet and Neumann problems
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. Any such is diffeomorphic to the closed unit disk. By change of variables it is enough to prove the result there. 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. This converts the integral over the disk into the integral over its boundary.
Let Ω be a region bounded by a simple closed curve. Given a smooth function f on the closure of Ω its normal derivative ∂nf at a boundary point is the directional derivative in the direction of the outward pointing normal vector. Applying Green's theorem with A = vx u and B = vy u gives the first of Green's identities:
where the Laplacian Δ is given by
Swapping u and v and subtracting gives the second of Green's identities:
If now u is harmonic in Ω and v = 1, then this identity implies that
so the integral of the normal derivative of a harmonic function on the boundary of a region always vanishes.
A similar argument shows that the average of a harmonic function on the boundary of a disk equals its value at the centre. Translating the disk can be taken to be centred at 0. Green's identity can be applied to an annulus formed of the boundary of the disk and a small circle centred on 0 with v = z2: it follows that the average is independent of the circle. It tends to the value at its value at 0 as the radius of the smaller circle decreases. This result also follows easily using Fourier series and the Poisson integral.
For continuous functions f on the whole plane which are smooth in Ω and the complementary region Ωc, the first derivative can have a jump across the boundary of Ω. The value of the normal derivative at a boundary point can be computed from inside or outside Ω. The interior normal derivative will be denoted by ∂n− and the exterior normal derivative by ∂n+. With this terminology the four basic problems of classical potential theory are as follows:
Interior Dirichlet problem: ∆u = 0 in Ω, u = f on ∂Ω Interior Neumann problem: ∆u = 0 in Ω, ∂n− u = f on ∂Ω Exterior Dirichlet problem: ∆u = 0 in Ωc, u = f on ∂Ω, u continuous at ∞ Exterior Neumann problem: ∆u = 0 in Ωc, ∂n+ u = f on ∂Ω, u continuous at ∞
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. Up to composition with complex conjugation, this is the special case of the Kelvin transform in two dimensions. In this case, since a function is harmonic if and only if it is the real part of a holomorphic function, the statement follows from the fact that the composition of holomorphic functions is holomorphic. The transform v of u is continuous in a small disc |z| ≤ r and harmonic everywhere in the interior except possibly 0. Let w be the harmonic function given by the Poisson integral on |z| ≤ r with the same boundary value g as v on |z| = r. Applying the maximum principle to v − w + ε log |z| on δ ≤ |z| ≤ r, it must be negative for δ small. Hence v(z) ≤ u(z) for z ≠ 0. The same argument applies with v and w swapped, so v = w is harmonic in the disk. Thus the singularity at ∞ is removable.
By the maximum principle the interior and exterior Dirichlet problems have unique solutions. For the interior Neumann problem, if a solution u is harmonic in 0 and its interior normal derivative vanishes, then Green's first identity implies the ux = 0 = uy, so that u is constant. This shows the interior Neumann problem has a unique solution up to adding constants. Applying inversion, the same holds for the external Neumann problem.
For both Neumann problems, a necessary condition for a solution to exist is
For the interior Neumann problem, this follows by setting v = 1 in Green's second identity. For the exterior Neumann problem, the same can be done for the intersection of Ωc and a large disk |z| < R, giving
At ∞ u is the real part of a holomorphic function F with
The interior normal derivative on |z| = R is just the radial derivative ∂r, so that for |z| = R
Hence
so the integral over ∂Ω must vanish.
The fundamental solution of the Laplacian is given by
N(z) = − E(z) is called the Newtonian potential in the plane. Using polar coordinates, it is easy to see that E is in Lp on any closed disk for any finite p ≥ 1. To say that E is a fundamental solution of the Laplacian means that for any smooth function φ of compact support
The standard proof uses Green's second identity on the annulus r ≤ |z| ≤ R where the support of φ is contained in |z| < R. In fact, since E is harmonic away from 0,
As r tends to zero, the first term on the right hand side tends to φ(0) and the second to 0, since r log r tends to 0 and the normal derivatives of φ are uniformly bounded. (That both sides are equal even before taking limits follows from the fact that the average of a harmonic function on the boundary of a disk equals it value at the centre, while the integral of its normal derivative vanishes.)
Neumann–Poincaré kernel The properties of the fundamental solution lead to the following formula for recovering a harmonic function u in Ω from its boundary values:
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. Translating, it can be assumed that z = 0 and the identity becomes
which was proved above. A similar formula holds for functions harmonic in Ωc:
The signs are reversed because of the direction of the normal derivative.
In two dimensions the Neumann–Poincaré kernel K(z,w) has the remarkable property that it restricts to a smooth function on ∂Ω × ∂Ω. It is a priori only defined as a smooth function off the diagonal but it admit a (unique) smooth extension to the diagonal. Using vector notation v(t) = (x(t), y(t)) to parametrize the boundary curve by arc length, the following classical formulas hold:
Thus the unit tangent vector t(t) at t is the velocity vector
so the oriented unit normal n(t) is
The constant relating the acceleration vector to the normal vector is the curvature of the curve:
Thus the curvature is given by
There are two further formulas of Frenet:
The Neumann–Poincaré kernel is given by the formula
For s ≠ t, set
The function
is smooth and nowhere vanishing with a(s,s) = L2 if the length of the curve is 2L.
Similarly the function
is smooth. In fact writing s = t + h,
so that
On the diagonal b(t,t) = κ L2 / 2. Since k is proportional to b / a, it is also smooth. Its diagonal values are given by the formula
Another expression for k(s,t) is as follows:
where z(t) = x(t) + i y(t) is the boundary curve parametrized by arc length. This follows from the identity
and the Cauchy–Riemann equations which can be used to express the normal derivative in terms of the tangential derivative, and
so in the direction normal to the boundary curve K is discontinuous at the boundary.
Double layer potentials The double layer potential with moment φ in C(∂Ω) is defined on the complement of ∂Ω as
It is a continuous function on the complement. Since the restriction of K extends to a smooth function on ∂Ω × ∂Ω, D(φ) can also be defined on ∂Ω. However like the Neumann–Poincaré kernel it will have discontinuities at the boundary. These are jump discontinuities. If φ is real, then the double layer potential is just the real part of a Cauchy integral:
The simplest case is when φ is identically 1 on ∂Ω. In this case D(1) equals
1 on Ω, by the vanishing of the integral of the normal derivative on the boundary region bounded by ∂Ω and a small disk centred on z; so the integral over the ∂Ω equals the average of the function 1 on the boundary of small disk and hence equals 1. (This integral and the one for Ωc can also be calculated using Cauchy's integral theorem.) 0 on Ωc, because it is the integral of a normal derivative of a harmonic function. 1/2 on ∂Ω, since
By definition the Neumann–Poincaré operator TK is the operator on L2(∂Ω) given by the kernel K(z,w). It is a Hilbert–Schmidt operator since the kernel is continuous. It takes values in C∞(∂Ω) since the kernel is smooth. The third computation above is equivalent to the statement that the constant function 1 is an eigenfunction of TK with eigenvalue 1/2.
To establish jump formulas for more general functions, it is necessary to check that the integrals for D(1) are uniformly absolutely convergent, i.e. that there is a uniform finite bound C such that
for all z not in the boundary. It is enough to check this for points in a tubular neighbourhood of the boundary. Any such point u lies on a normal through a unique point, v(0) say, on the curve and it is enough to look at the contribution to the integral from points v(t) with t in a small interval around 0. Writing
it follows that
So for t sufficiently small
for some constant C1. (The first inequality gives an approximate version of Pythagoras' theorem in the tubular neighbourhood.) Hence
Uniform boundedness follows because the first term has a finite integral independent of λ:
The bound above can be used to prove that if the moment φ vanishes at a boundary point z then its double layer potential D(φ) is continuous at z. More generally if φn tends uniformly to φ, then D(φn)(zn) converges to D(φ)(z). In fact suppose that |φ(w)| ≤ ε if |w - z| ≤ δ. Taking zn tending to z
The first integrand tends uniformly to 0 so the integral tends to 0. The second integral is bounded above by 2εC. The third integral is bounded by C times the supremum norm of φn − φ. Hence D(φ)(zn) tends to D(φ)(z).
JUMP FORMULAS. If φ is a continuous function on ∂Ω, the restrictions of its double layer potential u = Dφ to Ω and Ωc extend uniquely to continuous functions on their closures. Let u− and u+ be the resulting continuous functions on ∂Ω. Then
In particular
In fact the expressions for u± are continuous, so it is enough to show that the if zn tends to a boundary point z with zn in Ω or Ωc then u(zn) tends to the expression for u±(z). If zn lie in Ω or Ωc then
where ψ(w) = φ(w) − φ(z). The right hand side tends to zero since ψ vanishes at z.
Single layer potentials The single layer potential with moment φ in C(∂Ω) is defined on C as
where N is the Newtonian potential
The single layer potential is harmonic off ∂Ω. Since
and the first integrand tends uniformly to 0 as |z| tends to infinity, the single layer potential is harmonic at infinity if and only if ∫ φ = 0.
The single layer potential is continuous on C. In fact continuity off ∂Ω is clear. If zn tends to z with z in ∂Ω, then
The first integrand tends uniformly to 0 on |w - z| ≥ ε. For n sufficiently large the last integral is bounded by
which tends to 0 as ε tends to 0, by the Cauchy–Schwarz inequality since the integrand is square integrable.
The same argument shows that S = TN defines a bounded operator on C(∂Ω):
for φ in C(∂Ω).