- WEAK SOLUTION FORMULATION 365
LEMMA 7.112 (Hausdorff dimension of a Lipschitz graph). Let (.Mr, § 1 )
and (.M~i, §2) be two Riemannian manifolds. If f : M1 --+ M2 is locally
Lipschitz in the sense that for any p E M1, there is an open neighborhood
Up of p and a constant Gp such that dg 2 (f (q1), f (q2)) :S Cpdg 1 (q1, q2) for
any qi, q2 E u;, thenHdim { (x, f (x)) : x E M1} = n,
where Hdim denotes the Hausdorff dimension. In particular, the (n + m)-
dimensional Riemannian measure vanishes:measMixM 2 { (x, f (x)): x E M1} = 0.
Later we shall recall some more basic results, especially about convex
functions, as we need them.Now we give a proof that integration by parts holds for locally Lipschitz
functions. We say a vector field v on M is locally Lipschitz if for any
p EM and local coordinates {xi} in a neighborhood of p, we have for each
i that the function vi (x) is locally Lipschitz, where v (x) ~vi (x) 8 ~i. It is
well known that integration by parts holds for Lipschitz functions.
LEMMA 7.113 (Integration by parts for Lipschitz functions). Let f be a
locally Lipschitz function on M and let v be a locally Lipschitz vector fieldon M. Suppose that at least one off and v has compact support. Then
JM f div~dμ?J =~JM v · V'f dμfJ.Here div v and v · V' f are defined with respect to g.PROOF. We prove the lemma in the case where v has compaet sup-
port; the other case can be proved. similarly.^15 .. Rademacher's Theorem
says that both derivatives div v and \7 f exist almost everywhere. Since
v has support in some compact set IC, we have lvl, ldivvl E L= (JC), and
f, IV' f I E L~c (M) , so the integrals in the lemma make sense. We can
choose a smooth partition of unity { cPa} ~=l where the support of each cPa is
contained in a local coordinate chart. Clearly ¢av is locally Lipschitz. To
prove the lemma, it suffices to show
r, fdiv(cf)av)dμh=- r, cPa.V·\i'fdμh.
fM ' fM '
Hence without loss of generality we may assume the support of v is contained
in some local coordinate chart (U, {xi}).
15see the exercise below.