1238 CHAPTER 35. WEAK DERIVATIVES

Lemma 35.3.6 Let U be an open set, ψ ∈ C∞ (U) and suppose u,u,i ∈ Lploc (U). Then

(uψ),i and uψ are in Lploc (U) and

(uψ),i = u,iψ +uψ ,i.

Proof: Let φ ∈C∞c (U) then

(uψ),i (φ) ≡ −∫

Uuψφ ,idx

= −∫

Uu[(ψφ),i−φψ ,i]dx

=∫

U

(u,iψφ +uψ ,iφ

)dx

=∫

U

(u,iψ +uψ ,i

)φdx

This proves the lemma.Recall the notation for the gradient of a function.

∇u(x)≡ (u,1 (x) · · · u,n (x))T

thusDu(x)v =∇u(x) ·v.

35.4 Morrey’s InequalityThe following inequality will be called Morrey’s inequality. It relates an expression whichis given pointwise to an integral of the pth power of the derivative.

Lemma 35.4.1 Let u∈C1 (Rn) and p > n. Then there exists a constant, C, depending onlyon n such that for any x, y ∈ Rn,

|u(x)−u(y)|

≤C(∫

B(x,2|x−y|)|∇u(z) |pdz

)1/p(| x− y|(1−n/p)

). (35.4.1)

Proof: In the argument C will be a generic constant which depends on n. Consider thefollowing picture.

xU W Vy

This is a picture of two balls of radius r in Rn, U and V having centers at x and yrespectively, which intersect in the set, W. The center of U is on the boundary of V and the