954 CHAPTER 26. INTEGRALS AND DERIVATIVES

Now |∇un| = |∇u∗φ n| by Lemma 26.5.2 and this last is bounded. Also, by this lemma,∇un (z)→ ∇u(z) a.e. and un (x)→ u(x) for all x. Therefore, we can pass to a limit in theabove and obtain 26.5.13.

Note you can write 26.5.13 in the form

|u(x)−u(y)| ≤ C(

1|x−y|p

∫B(x,2|x−y|)

|∇u(z) |qdz)1/q

|x−y|

= Ĉ(

1mp (B(x,2 |x−y|))

∫B(x,2|x−y|)

|∇u(z) |qdz)1/q

|x−y|

Before leaving this remarkable formula, note that if you are in any situation where theabove formula holds and ∇u exists in some sense and is in Lq,q > p, then u would need tobe continuous. This is the basis for the Sobolev embedding theorem.

Here is Rademacher’s theorem.

Theorem 26.5.5 Suppose u is Lipschitz with constant K then if x is a point where ∇u(x)exists,

|u(y)−u(x)−∇u(x) · (y−x)|

≤C(

1m(B(x,2 |x−y|))

∫B(x,2|x−y|)

|∇u(z)−∇u(x) |qdz)1/q

| x− y|. (26.5.14)

Also u is differentiable at a.e. x and also

u(x+tv)−u(x) =∫ t

0Dvu(x+ sv)ds (26.5.15)

Proof: This follows easily from letting g(y) ≡ u(y)− u(x)−∇u(x) ·(y−x) . As ex-plained above, |∇u(x)| ≤√pK at every point where ∇u exists, the exceptional points beingin a set of measure zero. Then g(x) = 0, and ∇g(y) =∇u(y)−∇u(x) at the points y wherethe gradient of g exists. From Corollary 26.5.4,

|u(y)−u(x)−∇u(x) · (y−x)|= |g(y)|= |g(y)−g(x)|

≤ C(∫

B(x,2|x−y|)|∇u(z)−∇u(x) |qdz

)1/q

|x−y|1−pq

= C(∫

B(x,2|x−y|)|∇u(z)−∇u(x) |qdz

)1/q 1|x−y|p

1q|x−y|

= C(

1m(B(x,2 |x−y|))

∫B(x,2|x−y|)

|∇u(z)−∇u(x) |qdz)1/q

|x− y|.

Now this is no larger than

≤C(

1m(B(x,2 |x−y|))

∫B(x,2|x−y|)

|∇u(z)−∇u(x)|(2√pK)q−1 dz)1/q

|x− y|