2. ESTIMATES ON THE p-LAPLACIAN 15

for each v e C°l(B(0,R)) with v = 0 on dB(0,R). According to Lemma 2.2 the vector

fields {APfc}^1 are bounded in L°°; and so, passing if necessary to a further subsequence,

we have

APk -^ A weakly * in L°°.

Thus

(2.37)

(2.38)

2. In addiltion

compute

JB(0,R)

|A|

I

A

SB(O,R)\DUP^

dz liming•^oo

Dv dz = vf dz.

JB(0,R)

Pkdz = JB(0,R)UPkf dz

-* A«ug

uf dz

= IB(O,R)

A

•

Du dz

JB{0,R) l^-Pfcl ^

lim^oc (/B{ofl) \Duptf" dz)

~Pk

\B(0,R)\\

= SB(0,R) A • D u dz by (2-38)-

Since \Du\ 1 a.e., it follows that

| A| = A • Du a.e. (2.39)

Thus for a.e. z we can write

A(z) = a{z)Du{z), (2.40)

where a € L°°, a 0. For a.e. point z such that Du(z) exists and |£^(z)| 1. (2.39) implies

a(z) = 0. Finally (2.40) and (2.37) show —div(a Du) = f in the weak sense. Assertion (i)

is proved.

3. To verify assertion (ii) we first take any w € C°^(B(0, R)) with w = 0 on dB(Q, R),

\Dw\ 1 a.e. In light of (2.6)

- / fupdz

-\Dup\p

- fup dz I

-\Dw\p

- fw dz.

JB{Q,R) JB(0,R) P JB{Q,R) P

Let p = Pk —• oo:

/ fudz fw dz.

JB(0,R) JB(0,R)

If \Dw\ 1 a.e. but we do not have w = 0 on dB(0, R), we can modify w on 5(0, R)—B(0, S)

(i.e., outside the support of / ) to reduce to the previous case. To do so, we note first that