Weakly harmonic Sobolev function
= Weakly harmonic Sobolev function
{title2=$\Delta h=0$}
An element $h\in H^1_{\mathrm{loc}}(U)$ is weakly harmonic on an <open set> $U$ when $\int_U\nabla h\cdot\nabla\phi=0$ for every <test function> $\phi\in C_c^\infty(U)$. This is the statement $\Delta h=0$ in <distributions>. It requires no regularity of the <domain boundary>. The <Weyl lemma>, a form of <elliptic regularity>, identifies such an element with a <smooth> <harmonic function>.