---
tag: 1GNKQ7NBAG
kind: concept
name: Lipschitz domain
msc: [31B25, 35J25, 26B35]
sources:
  - url: https://en.wikipedia.org/wiki/Lipschitz_domain
same_as:
  wikidata: Q1158876
  wikipedia: Lipschitz domain
---

## Summary

A Lipschitz domain is an open set whose boundary is locally the graph of a [[Lipschitz|0AW1E6YTJY]] function.
Near each boundary point the set lies on one side of the graph.
Such boundaries may have corners and edges, but no cusps or slits.
Lipschitz domains are the standard setting for boundary value problems with rough boundaries.
The divergence theorem, Sobolev traces and the theory of harmonic measure all work there.

## Explanations

### L1 human

Zoom in on any boundary point of the domain and turn your head suitably.
You see the domain lying above a graph whose slope is bounded.
A square is a Lipschitz domain: at a corner the boundary looks like the graph of $\abs{x}$ after a rotation.
A disc with a slit is not, because near the slit the domain lies on both sides of it.