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
∫ ∂ Ω A d x + B d y = ∬ Ω ( B x − A y ) d x d y . {\displaystyle \displaystyle {\int _{\partial \Omega }A\,dx+B\,dy=\iint _{\Omega }(B_{x}-A_{y})\,dx\,dy.}}
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:
∫ ∂ Ω u ∂ n v = ∬ Ω u x v x + u y v y − u Δ v , {\displaystyle \displaystyle {\int _{\partial \Omega }u\,\partial _{n}v=\iint _{\Omega }u_{x}v_{x}+u_{y}v_{y}-u\,\Delta v,}}
where the Laplacian Δ is given by
Δ = − ∂ x 2 − ∂ y 2 . {\displaystyle \displaystyle \Delta =-\partial _{x}^{2}-\partial _{y}^{2}.}
Swapping u and v and subtracting gives the second of Green's identities:
∫ ∂ Ω u ∂ n v − ∂ n u v = ∬ Ω Δ u v − u Δ v . {\displaystyle \displaystyle {\int _{\partial \Omega }u\,\partial _{n}v-\partial _{n}u\,v=\iint _{\Omega }\,\Delta u\,v-u\,\Delta v.}}
If now u is harmonic in Ω and v = 1, then this identity implies that
∫ ∂ Ω ∂ n u = 0 , {\displaystyle \displaystyle {\int _{\partial \Omega }\partial _{n}u=0,}}
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:
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