---
tag: 9CWM3F4M04
kind: statement
role: lemma
name: Weyl's lemma
msc: [31B05, 35B65, 35J05]
uses: [0DEB0JMN0W, HS65DW0HXH, 0E0B6RA8K0, 0HYPR64JM2, 2Z29JN3VES]
about: [0DEB0JMN0W]
sources:
  - url: https://en.wikipedia.org/wiki/Weyl%27s_lemma_(Laplace_equation)
same_as:
  wikidata: Q7990327
  wikipedia: Weyl's lemma (Laplace equation)
---

## Statement

Let $U \subseteq \R^n$ be an open set and let $u$ be a locally integrable function on $U$.
Assume that
$$\int_U u(x) \, \Delta\varphi(x) \, dx = 0$$
for every infinitely differentiable function $\varphi$ with compact support in $U$ (a [[test function|HS65DW0HXH]]).
Then there is a [[harmonic function|0DEB0JMN0W]] $h$ on $U$ with $u = h$ almost everywhere.

## Remarks

The hypothesis says that $\Delta u = 0$ in the sense of distributions, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]].
Proof idea: let $\eta_\varepsilon$ be a smooth radial mollifier supported in $B(0, \varepsilon)$.
Let $u_\varepsilon = u * \eta_\varepsilon$, a [[convolution|2Z29JN3VES]] 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|0HYPR64JM2]], $u_\varepsilon * \eta_\delta = u_\varepsilon$, 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.

## Explanations

### L1 human

Weyl's lemma says that it does not matter how weakly one solves Laplace's equation.
Suppose a function is only known to satisfy it "on average", tested against smooth bumps.
Then it is a genuine smooth harmonic function after a change on a set of measure zero.

### L2 human

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.