partial-differential-equations-by-evans · Problem 16

Sobolev Spaces

Mathematics — Calculus · sobolev-spaces-composition-with-c1

Problem

Assume

F: \mathbb{R} \to \mathbb{R}

C^1

, with

F'

bounded. Suppose

U

is bounded and

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

for some

1 < p < \infty

. Show

v := F(u) \in W^{1,p}(U) \quad \text{and} \quad v_{x_i} = F'(u) u_{x_i} \quad (i=1,\ldots,n).