Weyl's lemma

Remarks

The hypothesis says that $\Delta u = 0$ in the sense of distributions, where $\Delta$ is the Laplacian. Proof idea: let $\eta_\varepsilon$ be a smooth radial mollifier supported in $B(0, \varepsilon)$. Let $u_\varepsilon = u * \eta_\varepsilon$, a convolution defined at the points at distance more than $\varepsilon$ from the complement of $U$. Then $\Delta u_\varepsilon = u * \Delta\eta_\varepsilon = 0$ by the hypothesis, so $u_\varepsilon$ is smooth and harmonic. By the mean value property, $u_\varepsilon * \eta_\delta = u_\varepsilon$, so $u_\varepsilon = u_\varepsilon * \eta_\delta = u_\delta * \eta_\varepsilon = u_\delta$ where both are defined. Thus the $u_\varepsilon$ fit together into one harmonic function $h$, and $u_\varepsilon \to u$ almost everywhere as $\varepsilon \to 0$ gives $u = h$ almost everywhere. The same holds for distributions: a distribution $T$ on $U$ with $\Delta T = 0$ is given by a harmonic function. Example: $u(x) = \abs{x}^{2-n}$ on $\R^n$, $n \ge 3$, is locally integrable and harmonic off $0$. But $\Delta u$ is a nonzero multiple of the Dirac delta, so the hypothesis fails on $U = \R^n$, and indeed $u$ has no harmonic version there.

Going deeper

For a student

The lemma is the starting point of elliptic regularity: weak solutions of elliptic equations are as smooth as the data allow. It is used, for instance, in Hilbert space methods for the Dirichlet problem, where Dirichlet's principle produces a weak solution, and Weyl's lemma upgrades it to a classical harmonic function. In complex analysis it shows that a distributional solution of the Cauchy–Riemann equations is holomorphic.