Inmathematics, theNewtonian potential, orNewton potential, is anoperator invector calculus that acts as the inverse to the negativeLaplacian on functions that are smooth and decay rapidly enough at infinity. As such, it is a fundamental object of study inpotential theory. In its general nature, it is asingular integral operator, defined byconvolution with a function having amathematical singularity at the origin, the Newtonian kernel${\displaystyle \mathrm{\Gamma }}$ which is thefundamental solution of theLaplace equation. It is named forIsaac Newton, who first discovered it and proved that it was aharmonic function in thespecial case of three variables, where it served as the fundamentalgravitational potential inNewton's law of universal gravitation. In modern potential theory, the Newtonian potential is instead thought of as anelectrostatic potential.

The Newtonian potential of acompactly supportedintegrable function${\displaystyle f}$ is defined as theconvolution

$${\displaystyle u(x)=\mathrm{\Gamma }\ast f(x)={\int }_{{\mathbb{R}}^d}\mathrm{\Gamma }(x-y)f(y)dy}$$

where the Newtonian kernel${\displaystyle \mathrm{\Gamma }}$ in dimension${\displaystyle d}$ is defined by

Here${\displaystyle {\omega }_d}$ is the volume of the unitd-ball (sometimes sign conventions may vary; compare (Evans 1998) and (Gilbarg & Trudinger 1983)). For example, for${\displaystyle d=3}$ we have${\displaystyle \mathrm{\Gamma }(x)=-1/(4\pi |x|)}$.

The Newtonian potential${\displaystyle w}$ of${\displaystyle f}$ is a solution of thePoisson equation

$${\displaystyle \mathrm{\Delta }w=f,}$$

which is to say that the operation of taking the Newtonian potential of a function is a partial inverse to the Laplace operator. Then${\displaystyle w}$ will be a classical solution, that is twice differentiable, if${\displaystyle f}$ is bounded and locallyHölder continuous as shown byOtto Hölder. It was an open question whether continuity alone is also sufficient. This was shown to be wrong byHenrik Petrini who gave an example of a continuous${\displaystyle f}$ for which${\displaystyle w}$ is not twice differentiable.The solution is not unique, since addition of any harmonic function to${\displaystyle w}$ will not affect the equation. This fact can be used to prove existence and uniqueness of solutions to theDirichlet problem for the Poisson equation in suitably regular domains, and for suitably well-behaved functions${\displaystyle f}$: 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

$${\displaystyle \mathrm{\Gamma }\ast \mu (x)={\int }_{{\mathbb{R}}^d}\mathrm{\Gamma }(x-y)d\mu (y)}$$

when${\displaystyle \mu }$ is a compactly supportedRadon measure. It satisfies the Poisson equation

$${\displaystyle \mathrm{\Delta }w=\mu }$$

in the sense ofdistributions. Moreover, when the measure ispositive, the Newtonian potential issubharmonic on${\displaystyle {\mathbb{R}}^d}$.

If${\displaystyle f}$ is a compactly supportedcontinuous function (or, more generally, a finite measure) that isrotationally invariant, then the convolution of${\displaystyle f}$ with${\displaystyle \mathrm{\Gamma }}$ satisfies for${\displaystyle x}$ outside the support of${\displaystyle f}$

$${\displaystyle f\ast \mathrm{\Gamma }(x)=\lambda \mathrm{\Gamma }(x),\lambda ={\int }_{{\mathbb{R}}^d}f(y)dy.}$$

In dimension${\displaystyle d=3}$, 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${\displaystyle \mu }$ is associated to a mass distribution on a sufficiently smooth hypersurface${\displaystyle S}$ (aLyapunov surface ofHölder class${\displaystyle C^{1,\alpha }}$) that divides${\displaystyle {\mathbb{R}}^d}$ into two regions${\displaystyle D_{+}}$ and${\displaystyle D_{-}}$, then the Newtonian potential of${\displaystyle \mu }$ is referred to as asimple layer potential. Simple layer potentials are continuous and solve theLaplace equation except on${\displaystyle S}$. They appear naturally in the study ofelectrostatics in the context of theelectrostatic potential associated to a charge distribution on a closed surface. If${\displaystyle \mathrm{d}\mu =f\mathrm{d}H}$ is the product of a continuous function on${\displaystyle S}$ with the${\displaystyle (d-1)}$-dimensionalHausdorff measure, then at a point${\displaystyle y}$ of${\displaystyle S}$, thenormal derivative undergoes a jump discontinuity${\displaystyle f(y)}$ when crossing the layer. Furthermore, the normal derivative of${\displaystyle w}$ is a well-defined continuous function on${\displaystyle S}$. This makes simple layers particularly suited to the study of theNeumann problem for the Laplace equation.

• Double layer potential
• Green's function
• Riesz potential
• Green's function for the three-variable Laplace equation

• Evans, L.C. (1998),Partial Differential Equations, Providence: American Mathematical Society,ISBN 0-8218-0772-2.
• Gilbarg, D.;Trudinger, Neil (1983),Elliptic Partial Differential Equations of Second Order, New York: Springer,ISBN 3-540-41160-7.
• Solomentsev, E.D. (2001) [1994],"Newton potential",Encyclopedia of Mathematics,EMS Press
• Solomentsev, E.D. (2001) [1994],"Simple-layer potential",Encyclopedia of Mathematics,EMS Press
• Solomentsev, E.D. (2001) [1994],"Surface potential",Encyclopedia of Mathematics,EMS Press

• Harmonic functions
• Isaac Newton
• Partial differential equations
• Potential theory
• Singular integrals

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