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
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