Newtonian potential

In

.

The Newtonian potential of a

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

where the Newtonian kernel Γ in dimension d is defined by

$${\displaystyle \Gamma (x)={\begin{cases}{\frac {1}{2\pi }}\log {|x|}&d=2\\{\frac {1}{d(2-d)\omega _{d}}}|x|^{2-d}&d\neq 2.\end{cases}}}$$

Here ω_d is the volume of the unit

)). For example, for ${\displaystyle d=3}$ we have ${\displaystyle \Gamma (x)=-1/(4\pi |x|).}$

The Newtonian potential w of f is a solution of the

Poisson equation

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

which is to say that the operation of taking the Newtonian potential of a function is a partial inverse to the Laplace operator. Then w will be a classical solution, that is twice differentiable, if f is bounded and locally who gave an example of a continuous f for which w is not twice differentiable. The solution is not unique, since addition of any harmonic function to w will not affect the equation. This fact can be used to prove existence and uniqueness of solutions to the Dirichlet problem for the Poisson equation in suitably regular domains, and for suitably well-behaved functions 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

when μ is a compactly supported Radon measure. It satisfies the Poisson equation in the sense of on R^d.

If f is a compactly supported continuous function (or, more generally, a finite measure) that is rotationally invariant, then the convolution of f with Γ satisfies for x outside the support of f

In dimension 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 μ is associated to a mass distribution on a sufficiently smooth hypersurface S (a

for the Laplace equation.

See also

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

References

• ISBN 0-8218-0772-2
  .
• Gilbarg, D.;
  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