Potential theory

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• ConceptPotential theory

  Potential theory is the study of harmonic functions, subharmonic functions and the potentials generated by distributions of mass or charge. It began with the gravitational and electrostatic potentials of physics, which satisfy the Laplace equation away from the masses. Its main topics include the Dirichlet problem, Green's functions, capacity, harmonic measure and the boundary behaviour of harmonic functions. In the plane it is tied closely to complex analysis, since the real part of a holomorphic function is harmonic and $\log\abs{f}$ is subharmonic.

• ConceptLaplacian

  The Laplacian of a twice differentiable function on $\mathbb{R}^n$ is the divergence of its gradient. It measures how the value of the function at a point differs from its average value on small spheres around the point.

• ConceptHarmonic function

  A harmonic function is a twice continuously differentiable function on an open subset of $\mathbb{R}^n$ whose Laplacian is zero at each point. Harmonic functions describe steady states, such as equilibrium temperatures and electrostatic potentials in empty space.

• ConceptHarmonic at a point

  A function on an open subset of $\R^n$ is harmonic at a point when it is harmonic on some neighbourhood of that point. This local notion lets one say where a function is harmonic without fixing a domain in advance. A function is a harmonic function on an open set exactly when it is harmonic at each point of the set.

• ConceptHarmonic polynomial

  A harmonic polynomial is a polynomial in $n$ real variables whose Laplacian is the zero polynomial. Harmonic polynomials are the simplest harmonic functions on all of $\R^n$. The homogeneous ones of degree $k$ form a finite-dimensional space, and their restrictions to the unit sphere are the spherical harmonics. Every harmonic function on a ball is a locally uniform limit of harmonic polynomials, so they play the role that monomials play for power series.

• ConceptFundamental solution of the Laplacian

  The fundamental solution of the Laplacian is a radial function $\Phi$ on $\R^n \setminus \set{0}$ with $-\Delta \Phi = \delta$, the Dirac delta at the origin. It equals $-\frac{1}{2\pi} \log \abs{x}$ in the plane and $\frac{1}{n(n-2)\omega_n} \abs{x}^{2-n}$ in dimension $n \ge 3$, where $\omega_n$ is the volume of the unit ball. It is the fundamental solution of the Laplace operator that is used throughout potential theory, up to sign. Convolving it with a source $f$ solves Poisson's equation, and it is the building block of Green's functions and of the Newtonian and logarithmic potentials.