partial-differential-equations-by-evans · Problem 12

Sobolev Spaces

Mathematics — Calculus · sobolev-spaces-not-lipschitz

Problem

Give an example of an open set

U \subset \mathbb{R}^n

and a function

u \in W^{1,\infty}(U)

, such that

u

is not Lipschitz continuous on

U

. (Hint: Take

U

to be the open unit disk in

\mathbb{R}^2

, with a slit removed.)