In mathematics, the Newtonian potential, or Newton potential, is an operator in vector calculus that acts as the inverse to the negative Laplacian on functions that are smooth and decay rapidly enough at infinity. As such, it is a fundamental object of study in potential theory. In its general nature, it is a singular integral operator, defined by convolution with a function having a mathematical singularity which is the fundamental solution of the Laplace equation. It is named for Isaac Newton, who first discovered it and proved that it was a harmonic function in the special case of three variables, where it served as the fundamental gravitational potential in Newton's law of universal gravitation. In modern potential theory, the Newtonian potential is instead thought of as an electrostatic potential.

The Newtonian potential of a compactly supported integrable function is defined as the convolution

is defined by

is the volume of the unit d-ball (sometimes sign conventions may vary; compare and .

is a solution of the Poisson equation

is bounded and locally Hölder continuous as shown by Otto Hölder. It was an open question whether continuity alone is also sufficient. This was shown to be wrong by Henrik Petrini will not affect the equation. This fact can be used to prove existence and uniqueness of solutions to the Dirichlet problem: one first applies a Newtonian potential to obtain a solution, and then adjusts by adding a harmonic function to get the correct boundary data.

The Newtonian potential is defined more broadly as the convolution

is a compactly supported Radon measure. It satisfies the Poisson equation

in the sense of distributions. Moreover, when the measure is positive, the Newtonian potential is subharmonic.

is a compactly supported continuous function (or, more generally, a finite measure) that is rotationally invariant

, this reduces to Newton's theorem that the potential energy of a small mass outside a much larger spherically symmetric mass distribution is the same as if all of the mass of the larger object were concentrated at its center.

(a Lyapunov surface of Hölder class is referred to as a simple layer potential. Simple layer potentials are continuous and solve the Laplace equation. They appear naturally in the study of electrostatics in the context of the electrostatic potential-dimensional Hausdorff measure, the normal derivative. This makes simple layers particularly suited to the study of the Neumann problem for the Laplace equation.

See also

• Double layer potential

• Green's function

• Riesz potential

• Green's function for the three-variable Laplace equation

References

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