Newtonian potential

Inmathematics,theNewtonian potentialorNewton potentialis anoperatorinvector calculusthat 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 byconvolutionwith a function having amathematical singularityat the origin, the Newtonian kernel$\Gamma$which is thefundamental solutionof theLaplace equation.It is named forIsaac Newton,who first discovered it and proved that it was aharmonic functionin thespecial case of three variables,where it served as the fundamentalgravitational potentialinNewton'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$f$is defined as theconvolution $$u(x)=\Gamma *f(x)=\int _{\mathbb {R} ^{d}}\Gamma (x-y)f(y)\,dy$$ where the Newtonian kernel$\Gamma$in dimension$d$is defined by $$\Gamma (x)={\begin{cases}{\frac {1}{2\pi }}\log {|x|},&d=2,\\{\frac {1}{d(2-d)\omega _{d}}}|x|^{2-d},&d\neq 2.\end{cases}}$$

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

The Newtonian potentialwoffis a solution of thePoisson equation $$\Delta w=f,$$ which is to say that the operation of taking the Newtonian potential of a function is a partial inverse to the Laplace operator. Thenwwill be a classical solution, that is twice differentiable, iffis bounded and locallyHölder continuousas shown byOtto Hölder.It was an open question whether continuity alone is also sufficient. This was shown to be wrong byHenrik Petriniwho gave an example of a continuousffor whichwis not twice differentiable. The solution is not unique, since addition of any harmonic function towwill not affect the equation. This fact can be used to prove existence and uniqueness of solutions to theDirichlet problemfor the Poisson equation in suitably regular domains, and for suitably well-behaved functionsf: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 $$\Gamma *\mu (x)=\int _{\mathbb {R} ^{d}}\Gamma (x-y)\,d\mu (y)$$ whenμis a compactly supportedRadon measure.It satisfies the Poisson equation $$\Delta w=\mu$$ in the sense ofdistributions.Moreover, when the measure ispositive,the Newtonian potential issubharmoniconR^d.

Iffis a compactly supportedcontinuous function(or, more generally, a finite measure) that isrotationally invariant,then the convolution offwithΓsatisfies forxoutside the support off $$f*\Gamma (x)=\lambda \Gamma (x),\quad \lambda =\int _{\mathbb {R} ^{d}}f(y)\,dy.$$

In dimensiond= 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 measureμis associated to a mass distribution on a sufficiently smooth hypersurfaceS(aLyapunov surfaceofHölder classC^1,α) that dividesR^dinto two regionsD₊andD₋,then the Newtonian potential ofμis referred to as asimple layer potential.Simple layer potentials are continuous and solve theLaplace equationexcept onS.They appear naturally in the study ofelectrostaticsin the context of theelectrostatic potentialassociated to a charge distribution on a closed surface. Ifdμ=fdHis the product of a continuous function onSwith the (d− 1)-dimensionalHausdorff measure,then at a pointyofS,thenormal derivativeundergoes a jump discontinuityf(y) when crossing the layer. Furthermore, the normal derivative ofwis a well-defined continuous function onS.This makes simple layers particularly suited to the study of theNeumann problemfor the Laplace equation.

See also

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

References

• Evans, L.C.(1998),Partial Differential Equations,Providence: American Mathematical Society,ISBN0-8218-0772-2.
• Gilbarg, D.;Trudinger, Neil(1983),Elliptic Partial Differential Equations of Second Order,New York: Springer,ISBN3-540-41160-7.
• 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