In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 at the origin, the Newtonian kernel
which is the
fundamental solution of the
Laplace equation. It is named for
Isaac Newton
Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
, 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
where the Newtonian kernel
in dimension
is defined by
Here ''ω''
''d'' is the volume of the unit
''d''-ball (sometimes sign conventions may vary; compare and ). For example, for
we have
The Newtonian potential ''w'' of ''f'' is a solution of the
Poisson equation
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
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 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
distributions. Moreover, when the measure is
positive, the Newtonian potential is
subharmonic 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 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
Lyapunov surface of
Hölder class ''C''
1,α) that divides R
''d'' into two regions ''D''
+ and ''D''
−, then the Newtonian potential of ''μ'' is referred to as a simple layer potential. Simple layer potentials are continuous and solve the
Laplace equation except on ''S''. They appear naturally in the study of
electrostatics
Electrostatics is a branch of physics that studies slow-moving or stationary electric charges.
Since classical antiquity, classical times, it has been known that some materials, such as amber, attract lightweight particles after triboelectric e ...
in the context of the
electrostatic potential associated to a charge distribution on a closed surface. If is the product of a continuous function on ''S'' with the (''d'' − 1)-dimensional
Hausdorff measure, then at a point ''y'' of ''S'', the
normal derivative undergoes a jump discontinuity ''f''(''y'') when crossing the layer. Furthermore, the normal derivative of ''w'' is a well-defined continuous function on ''S''. 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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{{Isaac Newton
Harmonic functions
Isaac Newton
Partial differential equations
Potential theory
Singular integrals