35.4. MORREY’S INEQUALITY 1239

center of V is on the boundary of U as shown in the picture. There exists a constant, C,independent of r depending only on n such that

m(W )

m(U)=

m(W )

m(V )=C.

You could compute this constant if you desired but it is not important here.

Define the average of a function over a set, E ⊆ Rn as follows.

∫−

Ef dx≡ 1

m(E)

∫E

f dx.

Then

|u(x)−u(y)| =∫−

W|u(x)−u(y)|dz

≤∫−

W|u(x)−u(z)|dz+

∫−

W|u(z)−u(y)|dz

=C

m(U)

[∫W|u(x)−u(z)|dz+

∫W|u(z)−u(y)|dz

]≤ C

[∫−

U|u(x)−u(z)|dz+

∫−

V|u(y)−u(z)|dz

]

Now consider these two terms. Using spherical coordinates and letting U0 denote the ballof the same radius as U but with center at 0,

∫−

U|u(x)−u(z)|dz

=1

m(U0)

∫U0

|u(x)−u(z+x)|dz

=1

m(U0)

∫ r

0ρ

n−1∫

Sn−1|u(x)−u(ρw+x)|dσ (w)dρ

≤ 1m(U0)

∫ r

0ρ

n−1∫

Sn−1

∫ρ

0|∇u(x+ tw) ·w|dtdσdρ

≤ 1m(U0)

∫ r

0ρ

n−1∫

Sn−1

∫ρ

0|∇u(x+ tw)|dtdσdρ

≤ C1r

∫ r

0

∫Sn−1

∫ r

0|∇u(x+ tw)|dtdσdρ

= C1r

∫ r

0

∫Sn−1

∫ r

0

|∇u(x+ tw)|tn−1 tn−1dtdσdρ