1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

3. HARMONIC MAPS NEAR THE IDENTITY OF MANIFOLDS WITH Re < O 269

That is, U so lves the equation

(34.43a)

(34.43b)

-6.U-Rc(U)=Q inM,

(V' NUh - II(U) = f on aM

in the weak sense, where N is the unit inward normal to aM.

We have the following reg ularity result for weak solutions of the elliptic bound-

ary value problem (34.43).

LEMMA 34.15. Under the hypothesis of Lemma 34.14, if in addition we have

Q E c= (TM) and f E c = (T(aM)), then the unique solution U to (34.43) is in

c =(TM).

The idea of the proof is as follows (for another proof, see Subsection 3.6 at

the end of this section). In a n eighborhood of a point on the boundary of M, let

(U, {xi}) be local coordinates satisfying

{xn = O} =UnaM c aU

and where N = a~n is the unit inward normal to aM. Write a vector field locally

n. a

as U = L.:;i=l U' axi. On aU we have

(V' N U)i = aUaxn i - L..__, ~ r i n1 .uj for i:::; n -^1

j:Sn-l

when un = 0, where the r~j denote the Christoffel symbols. Then the boundary

value problem (34.43), with U 1-= 0, lo cally takes the form

(34.44a)

L(U)i ~ L ake a~:~:e + L w ~~~ + L cjUj =<Pi in u for i ::::: n,

k,e::;n j,k::;n j'.Sn

(34.44b)

. aUi ~...
B(U)' ~ axn + L...., dj U^1 = 'lj;' on Un aM for i :::; n - 1,
j:Sn-l
(34.44c) B(U)n ~ un = 0 on Un aM,

where ake, bji, c;, <Pi for i , j, k , .e :::; n are smooth functions in U and dj, 'lj;i for

i, j :::; n - 1 are smooth functions on U n aM. Thus the boundary conditions are

of Neumann type for the components i :::; n - 1 and are of Dirichlet type for the
component i = n.
We now show, following Agmon, Douglis, and Nirenberg [4], that the regularity
theory of boundary value problems for elliptic systems applies to this situation.
The issue is to establish certain compatibility betwee n the elliptic system and the
boundary operator.
The principal symbol for the elliptic system in (34.44a) is

(34.45) uL (x ) (~)=A (0 idrxM : TxM--+ T x M, A(~)~ L ake (x)(k~e,

for ~ = L.:;i ~idxi E T; M. The adjoint of u L ( x) ( 0 is given by

adj (uL (x) (~))=A (~t-l idrxM·

k,e::;n