# Newtonian potential

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{{Short description|Green's function for Laplacian}}
In [mathematics](/source/mathematics), the '''Newtonian potential''', or '''Newton potential''', is an [operator](/source/Operator_(mathematics)) in [vector calculus](/source/vector_calculus) that acts as the inverse to the negative [Laplacian](/source/Laplacian) on functions that are smooth and decay rapidly enough at infinity. As such, it is a fundamental object of study in [potential theory](/source/potential_theory). In its general nature, it is a [singular integral operator](/source/singular_integral_operator), defined by [convolution](/source/convolution) with a function having a [mathematical singularity](/source/mathematical_singularity) at the origin, the Newtonian kernel <math>\Gamma</math> which is the [fundamental solution](/source/fundamental_solution) of the [Laplace equation](/source/Laplace_equation).  It is named for [Isaac Newton](/source/Isaac_Newton), who first discovered it and proved that it was a [harmonic function](/source/harmonic_function) in the [special case of three variables](/source/Green's_function_for_the_three-variable_Laplace_equation), where it served as the fundamental [gravitational potential](/source/gravitational_potential) in [Newton's law of universal gravitation](/source/Newton's_law_of_universal_gravitation).  In modern potential theory, the Newtonian potential is instead thought of as an [electrostatic potential](/source/electrostatic_potential).

The Newtonian potential of a [compactly supported](/source/compact_support) [integrable function](/source/integrable_function) <math>f</math> is defined as the [convolution](/source/convolution)

<math display="block">u(x) = \Gamma * f(x) = \int_{\mathbb{R}^d} \Gamma(x-y)f(y)\,dy</math>

where the Newtonian kernel <math>\Gamma</math> in dimension <math>d</math> is defined by

<math display="block">\Gamma(x) = \begin{cases} 
\frac{1}{2\pi} \log{ | x | }, &  d=2, \\
\frac{1}{d(2-d)\omega_d} | x | ^{2-d}, &  d \neq 2.
\end{cases} </math>

Here <math>\omega_d</math> is the volume of the unit [''d''-ball](/source/N_sphere) (sometimes sign conventions may vary; compare {{harv|Evans|1998}} and {{harv|Gilbarg|Trudinger|1983}}). For example, for <math>d = 3</math> we have <math>\Gamma(x) = -1/(4\pi |x|) </math>.

The Newtonian potential <math>w</math> of <math>f</math> is a solution of the [Poisson equation](/source/Poisson_equation)

<math display="block">\Delta w = f, </math>

which is to say that the operation of taking the Newtonian potential of a function is a partial inverse to the Laplace operator. Then <math>w</math> will be a classical solution, that is twice differentiable, if <math>f</math> is bounded and locally [Hölder continuous](/source/H%C3%B6lder_continuous) as shown by [Otto Hölder](/source/Otto_H%C3%B6lder). It was an open question whether continuity alone is also sufficient. This was shown to be wrong by [Henrik Petrini](/source/Henrik_Petrini) who gave an example of a continuous <math>f</math> for which <math>w</math> is not twice differentiable.
The solution is not unique, since addition of any harmonic function to <math>w</math> will not affect the equation.  This fact can be used to prove existence and uniqueness of solutions to the [Dirichlet problem](/source/Dirichlet_problem) for the Poisson equation in suitably regular domains, and for suitably well-behaved functions <math>f</math>: 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

<math display="block">\Gamma*\mu(x) = \int_{\mathbb{R}^d}\Gamma(x-y) \, d\mu(y)</math>

when <math>\mu</math> is a compactly supported [Radon measure](/source/Radon_measure). It satisfies the Poisson equation

<math display="block">\Delta w = \mu </math>

in the sense of [distributions](/source/distribution_(mathematics)). Moreover, when the measure is [positive](/source/positive_measure), the Newtonian potential is [subharmonic](/source/subharmonic_function) on <math>\mathbb{R}^d</math>.

If <math>f</math> is a compactly supported [continuous function](/source/continuous_function) (or, more generally, a finite measure) that is [rotationally invariant](/source/rotation), then the convolution of <math>f</math> with <math>\Gamma</math> satisfies for <math>x</math> outside the support of <math>f</math>

<math display="block">f*\Gamma(x) =\lambda \Gamma(x),\quad \lambda=\int_{\mathbb{R}^d} f(y)\,dy.</math>

In dimension <math>d=3</math>, 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.

When the measure <math>\mu</math> is associated to a mass distribution on a sufficiently smooth hypersurface <math>S</math> (a [Lyapunov surface](/source/Lyapunov_surface) of [Hölder class](/source/H%C3%B6lder_space) <math>C^{1,\alpha}</math>) that divides <math>\mathbb{R}^d</math> into two regions <math>D_+</math> and <math>D_-</math>, then the Newtonian potential of <math>\mu</math> is referred to as a '''simple layer potential'''.  Simple layer potentials are continuous and solve the [Laplace equation](/source/Laplace_equation) except on <math>S</math>.  They appear naturally in the study of [electrostatics](/source/electrostatics) in the context of the [electrostatic potential](/source/electrostatic_potential) associated to a charge distribution on a closed surface. If <math>\mathrm{d}\mu=f\mathrm{d}H</math> is the product of a continuous function on <math>S</math> with the <math>(d-1)</math>-dimensional [Hausdorff measure](/source/Hausdorff_measure), then at a point <math>y</math> of <math>S</math>, the [normal derivative](/source/normal_derivative) undergoes a jump discontinuity <math>f(y)</math> when crossing the layer. Furthermore, the normal derivative of <math>w</math> is a well-defined continuous function on <math>S</math>.  This makes simple layers particularly suited to the study of the [Neumann problem](/source/Neumann_problem) for the Laplace equation.

==See also==
* [Double layer potential](/source/Double_layer_potential)
* [Green's function](/source/Green's_function)
* [Riesz potential](/source/Riesz_potential)
* [Green's function for the three-variable Laplace equation](/source/Green's_function_for_the_three-variable_Laplace_equation)

==References==

* {{citation|first=L.C.|last=Evans|authorlink=Lawrence C. Evans|title=Partial Differential Equations|publisher=American Mathematical Society|publication-place=Providence|year=1998|isbn=0-8218-0772-2}}.
* {{citation|first1=D.|last=Gilbarg|first2=Neil|last2=Trudinger|authorlink2=Neil Trudinger|title=Elliptic Partial Differential Equations of Second Order|publisher=Springer|publication-place=New York|year=1983|isbn=3-540-41160-7}}.
* {{springer|id=n/n066580|title=Newton potential|first=E.D.|last=Solomentsev}}
* {{springer|id=s/s085260|title=Simple-layer potential|first=E.D.|last=Solomentsev}}
* {{springer|id=s/s091400|title=Surface potential|first=E.D.|last=Solomentsev}}

{{Isaac Newton}}

Category:Harmonic functions
Category:Isaac Newton
Category:Partial differential equations
Category:Potential theory
Category:Singular integrals

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Adapted from the Wikipedia article [Newtonian potential](https://en.wikipedia.org/wiki/Newtonian_potential) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Newtonian_potential?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.