Weyl's lemma

Weyl's lemma states that a distribution with zero Laplacian on an open set is represented there by a smooth harmonic function.

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Distribution: A continuous linear functional on a space of smooth test functions, generalizing locally integrable functions. Weyl's lemma begins with harmonicity expressed for this generalized notion of function.
Laplacian: The differential operator that sums a function's second partial derivatives in Euclidean coordinates. Weyl's lemma assumes this operator annihilates the distribution.
Hermann Weyl: A German mathematician and physicist whose work ranged across analysis, geometry, and mathematical physics. The lemma bears Weyl's name, though its result belongs to a wider history of harmonic analysis.
Laplace equation: The partial differential equation stating that the Laplacian of an unknown function is zero or equals prescribed data. Weyl's lemma shows distributional solutions of its homogeneous form are smooth.
Distributional derivative: A derivative defined by transferring differentiation from a distribution to smooth test functions. The distributional Laplacian is assembled from these generalized derivatives.
Harmonic function: A twice-differentiable function whose Laplacian is zero. The lemma concludes that the distribution is represented by a function of this kind.
Schwarz's theorem on harmonic functions: A nineteenth-century result establishing regularity properties of functions satisfying the Laplace equation. Earlier classical results on harmonic functions form the setting for Weyl's distributional extension.
Maximum principle: A principle constraining interior extrema of solutions to certain elliptic differential equations. Once smoothness is established, classical harmonic-function principles apply to the representative.
Mollifier: A smooth, compactly supported function used to approximate other functions by convolution. Convolving with mollifiers produces smooth approximations that retain harmonicity locally.
Open set: A subset of a topological space in which every point has a neighborhood contained in the subset. The regularity conclusion holds locally throughout the domain where the Laplacian vanishes.