Green's identities

Green's identities
Name	Green's identities
Field	Partial differential equations
Introduced	19th century
Notable	George Green

Contents

Green's identities are a set of integral relations in potential theory connecting functions and their Laplacians on domains with boundary. They were formulated in the 19th century by George Green and later used by figures such as Augustin-Louis Cauchy, Carl Friedrich Gauss, Siméon Denis Poisson, Lord Kelvin, and Peter Gustav Lejeune Dirichlet. These identities underpin methods developed by Bernhard Riemann, Joseph-Louis Lagrange, Simeon Poisson, Karl Weierstrass, and modern analysts including Jacques Hadamard, Israel Gelfand, Marshall Stone, John von Neumann, and Mark Kac.

Introduction

Green's identities relate integrals over a region in Euclidean space to integrals over the region's boundary and involve the Laplace operator, normal derivatives, and scalar functions. They are foundational in the analysis of the Dirichlet problem, the Neumann problem, the Poisson equation, and connections to the Newtonian potential, the Fundamental theorem of calculus in higher dimensions, and techniques used by Sofia Kovalevskaya, Émile Picard, David Hilbert, Andrey Kolmogorov, and Elias Stein. Green's identities are instrumental in variational formulations exploited by Leonhard Euler, Joseph-Louis Lagrange, Pierre-Simon Laplace, Siméon Denis Poisson, and in the spectral theory of the Laplace–Beltrami operator used by Hermann Weyl, Atle Selberg, Noether, and Richard Courant.

Green's first identity states that for sufficiently smooth scalar functions defined on a bounded region with smooth boundary in Riemannian manifold settings, an integral of a gradient dot product equals a volume integral involving the Laplacian plus a boundary integral of normal derivative times the function. The proof classically reduces to repeated application of the divergence theorem (also called the Gauss–Ostrogradsky theorem) used by Carl Friedrich Gauss, Mikhail Ostrogradsky, Joseph-Louis Lagrange, and generalized by George Green, with regularity arguments refined by Bernhard Riemann, Otto Hölder, Sergei Sobolev, and Laurent Schwartz. Green's second identity follows by subtracting symmetric first identities for two functions, yielding a relation equivalent to an integration by parts formula central to work by David Hilbert, Emmy Noether, John von Neumann, Marshall Stone, and Richard Courant. Green's representation formula gives a solution representation for harmonic functions using boundary integrals involving the fundamental solution (Green's function) of the Laplacian; construction of such Green's functions was advanced by George Green, Lord Kelvin, Pierre-Simon Laplace, Siméon Denis Poisson, Carl Neumann, Franz Neumann, Hermann Hankel, and later systematized by Laurent Schwartz, Israel Gelfand, and Fritz John.

Green's identities are applied to boundary value problems like the Dirichlet problem and Neumann problem for the Poisson equation, central in electrostatics studied by James Clerk Maxwell, Michael Faraday, André-Marie Ampère, and in potential theory investigations by Isaac Newton, Laplace, Poisson, and George Green. They underpin the method of layer potentials used in scattering theory relevant to Lord Rayleigh, John William Strutt, Paul Dirac, and computational techniques like the Boundary element method employed in engineering by researchers at institutions such as Massachusetts Institute of Technology, Stanford University, Imperial College London, and ETH Zurich. Green's identities are crucial in proving uniqueness theorems in Helmholtz equation problems appearing in acoustics by Hermann von Helmholtz and in electromagnetic theory addressed by Oliver Heaviside, Heinrich Hertz, Ludwig Boltzmann, and Hendrik Lorentz. They also enable variational formulations leading to the Rayleigh–Ritz method and spectral estimates used by Weyl, Courant, Hilbert, and Kac.

Generalizations extend Green's identities to the Laplace–Beltrami operator on Riemannian manifolds studied by Bernhard Riemann, Elie Cartan, Atle Selberg, and Shing-Tung Yau, and to elliptic operators of second order considered by André Weil, Lars Hörmander, Louis Nirenberg, and Enrico Bombieri. Analogues include the Green's theorem in the plane linked to Cauchy integral theorem of Augustin-Louis Cauchy and to versions by Bernhard Riemann and Gotthold Eisenstein, as well as Green's formulas in distribution theory due to Laurent Schwartz, Sergei Sobolev, and Hermann Weyl. In harmonic analysis, relations with the Poisson kernel and boundary integral operators connect to work by Norbert Wiener, Salomon Bochner, Elias Stein, Charles Fefferman, Lars Hörmander, and Terence Tao. Stochastic generalizations relate to the Feynman–Kac formula of Richard Feynman and Mark Kac and to potential theory on Brownian motion studied by Norbert Wiener, Andrey Kolmogorov, and Paul Lévy.

Elementary examples include applying Green's first and second identities to balls and rectangles in Euclidean space to derive mean-value properties for harmonic functions used by Augustin-Louis Cauchy, Pierre-Simon Laplace, Siméon Denis Poisson, Sofya Kovalevskaya, and Carl Friedrich Gauss. Concrete computations of Green's functions for the unit ball, half-space, and annulus are classical results obtained by George Green, Lord Kelvin, Siméon Denis Poisson, Hermann Schwarz, and Henri Poincaré; these lead to explicit integrals appearing in problems studied by Joseph Liouville, Émile Picard, Felix Klein, and Hermann Weyl. Numerical examples using the Boundary element method and fast multipole algorithms from Leslie Greengard and Vladimir Rokhlin illustrate efficient use in applied settings at laboratories like Lawrence Berkeley National Laboratory, Los Alamos National Laboratory, and universities including Princeton University and University of Cambridge.