35.5. RADEMACHER’S THEOREM 1243

and at every Lebesgue point, x of ∇u

limy→x

(1

m(B(x,2 |x−y|))

∫B(x,2|x−y|)

|∇u(z)−∇u(x) |pdz)1/p

= 0

and so at each of these points,

limy→x

|u(y)−u(x)−∇u(x) · (y−x)||x−y|

= 0

which says that u is differentiable at x and Du(x)(v) = ∇u(x) · (v) . See Page 117. Thisproves the corollary.

Definition 35.5.3 Now suppose u is Lipschitz on Rn,

|u(x)−u(y)| ≤ K |x−y|

for some constant K. Define Lip(u) as the smallest value of K that works in this inequality.

The following corollary is known as Rademacher’s theorem. It states that every Lips-chitz function is differentiable a.e.

Corollary 35.5.4 If u is Lipschitz continuous then u is differentiable a.e. and ||u,i||∞ ≤Lip(u).

Proof: This is done by showing that Lipschitz continuous functions have weak deriva-tives in L∞ (Rn) and then using the previous results. Let

Dhei

u(x)≡ h−1 [u(x+hei)−u(x)].

Then Dhei

u is bounded in L∞ (Rn) and

||Dhei

u||∞ ≤ Lip(u).

It follows that Dhei

u is contained in a ball in L∞ (Rn), the dual space of L1 (Rn). By Theorem35.1.3 on Page 1232, there is a subsequence h→ 0 such that

Dhei

u ⇀ w, ||w||∞ ≤ Lip(u)

where the convergence takes place in the weak ∗ topology of L∞(Rn). Let φ ∈ C∞c (Rn).

Then ∫wφdx = lim

h→0

∫Dh

eiuφdx

= limh→0

∫u(x)

(φ (x−hei)−φ (x))h

dx

=−∫

u(x)φ ,i (x)dx.

Thus w = u,i and u,i ∈ L∞ (Rn) for each i. Hence u,u,i ∈ Lploc (R

n) for all p > n and so u isdifferentiable a.e. by Corollary 35.5.2. This proves the corollary.