a subharmonic function

Harmonic functions sit exactly at their local average. Subharmonic functions are the ones that sit at or below their average — they lean upward, bowing beneath the harmonic ideal the way a convex curve lies below the chord, except here the comparison is against the averaging over spheres. They are the half-relaxed cousins of harmonic functions, and they are the key technical device for building solutions on domains too irregular for any formula.

Precisely, a function u is subharmonic on a region if it is at most its average over every small sphere centred in the region: u at the centre is less than or equal to the spherical average around it. Equivalently, where u is twice differentiable, Laplacian u is greater than or equal to 0 — the borderline Laplacian u = 0 is harmonic, and a strictly positive Laplacian bends u up. Convex functions of one variable are the one-dimensional model. The crucial property is a one-sided maximum principle: a subharmonic function attains its maximum only on the boundary (it can dip below a harmonic comparison but never rise above it), though, unlike harmonic functions, it need not obey any minimum principle. Superharmonic functions are the mirror image, with Laplacian u less than or equal to 0.

Subharmonic functions are the engine of the Perron method, which solves the Dirichlet problem on essentially arbitrary domains. The idea is to take the family of all subharmonic functions that stay below the boundary data, and define the solution as their upper envelope (their pointwise supremum); under mild conditions this envelope is exactly the harmonic solution. Because subharmonicity is preserved under taking maxima and is defined by an inequality rather than an exact equation, the family is flexible enough to build a solution where no explicit formula exists. They also pervade complex analysis (log|f| is subharmonic for analytic f) and the modern theory of viscosity solutions.