Editing Harmonic function (section)

===Weakly harmonic function===
A function (or, more generally, a [[distribution (mathematics)|distribution]]) is [[weakly harmonic]] if it satisfies Laplace's equation
<math display="block">\Delta f = 0\,</math>
in a [[weak derivative|weak]] sense (or, equivalently, in the sense of distributions).  A weakly harmonic function coincides almost everywhere with a strongly harmonic function, and is in particular smooth.  A weakly harmonic distribution is precisely the distribution associated to a strongly harmonic function, and so also is smooth.  This is [[Weyl's lemma (Laplace equation)|Weyl's lemma]].

There are other [[weak formulation]]s of Laplace's equation that are often useful.  One of which is [[Dirichlet's principle]], representing harmonic functions in the [[Sobolev space]] {{math|''H''<sup>1</sup>(Ω)}} as the minimizers of the [[Dirichlet energy]] integral
<math display="block">J(u):=\int_\Omega |\nabla u|^2\, dx</math>
with respect to local variations, that is, all functions <math>u\in H^1(\Omega)</math> such that <math>J(u) \leq J(u+v)</math> holds for all <math>v\in C^\infty_c(\Omega),</math> or equivalently, for all <math>v\in H^1_0(\Omega).</math>