Harmonic Extensions By

Marston Morse* Institute for Advanced Study, Princeton, N. J., USA Dedicated to Professor Emeritus P a u l F u n k

(Received February 25, 1963) w 1. Introduction. In the last three years Schoenflies extension problems have been defined and solved under varied conditions. There has resulted a theory which ranges from the topological through the differentiable to the analytic ease. In this paper we are concerned with the analytic case, or more percisely with a classical theorem on harmonic extension which is essential in the analytic case and in particular in a proof of Theorem 1.1. Theorem 1.2 is a natural and powerful complement of Theorem 1.1, established, however, by different methods. Theorem 1.3 is proved here while Theorems 1.1 and 1.2 are presented as background without proof.

Notation. Let (x 1, . . . , xn) be rectangular coordinates of a point x in a euclidean n-space E, n > 1. One may regard x as a vector with components (xl, . . . , x~) and length ][ x H. Let S be the (n -- 1)-sphere on which II x II---- 1. Set B =

Ill x rl > 1}.

(1.0)

Let R be the axis of reals. Theorem 1.1 is proved in [3]. Theorem 1.1. A real analytic diHeomorphism [ o / S into E which can be extended by a C2-diffeomorphism F into E o] an open neighborhood N o] B can also be extended by an analytic diffeornorphism q5 o[ some open neighborhood N' c N o] B. For a proof of Theorem 1.2 see [1]. * This research was supported by the Air Force Office of Scientific Research.

318

M. Morse

Theorem 1.2. Let P be an arbitrary point interior to S or on S. A real analytic di]/eomorphism o] S into E admits an extension over E which is a homeomorphism of E onto E and which is a real analytic dif/eomorphism of E - - P onto E - - / ( P ) . See Theorem 1.3 of [3]. The extension F in Theorem 1.2 may fail to be differentiable at P. There are examples when n > 6 where F must fail to be differentiable at P. Hypotheses of Theorem 1.3. Let B 0 be the ball B with the origin to deleted. Let GObe a non-empty open subset of S. Let G be an open connected subset of B o such that Go o G is open relative to B. Let T be the inversion of E -- ~o in S. Then the set G u Go u T ( G ) = G

(1.1)

is open relative to E and connected. Theorem 1.3. Let a mapping x -+ U(x): G -+ R

(1.2)

be given such that U is harmonic on G and admits an extension G u Go which is continuous on G o Go and vanishes on Go. Then U admits an extension U ~ which is defined and harmonic on f, and such that

= -

IIx ]r -n U(r(z))

r(a)).

(1.3)

Theorem 1.3 is used in [2], p. 173. (In line 30, loc. tit., n should be replaced by n + 1). Corollary 5.1 of Theorem 1.3 is used in [3]. The following is a sketch of a possible proof of Theorem 1.3. Set I] x I1 = r. One can show that the first partial derivatives as to r of V and of the right member of (1.3) take on the same continuous boundary values on Go. Moreover the right member of (1.3) is harmonic on T(G). Finally all of the first partial derivatives of U and of the right member of (1.3) take on continuous boundary values on Go. The n-dimensional generalization of Theorem VI, p. 261 of [5], will then suffice to establish Theorem 1.3. We propose to establish Theorem 1.3 by a new a n d simpler method t h a t does not need to evaluate the boundary values of the above partial derivatives. This new method is more formal but also more revealing. We are concerned with the case n > 2. A special case. We begin by verifying Theorem 1.3 for the special case in which Go ---- S, G = B 0 and U is replaced by the mapping x --~ V(x) = 1] x [I2-'~ -- 1

(x e Bo)

(1.4)

tIarmonic Extensions

319

That V is harmonic is readily verified. For a simple proof see [4], p. 711. This proof shows that the mapping

Ilxll

-

-i

n>2)

is harmonic over all of E - - co. Thus V admits an extension Ve defined b y the right member of (1.4), harmonic over all of E - - co, and vanishing on S.

Does (1.3) hold when U ---- V and U e = V~? In vectorial notation

-

It

II r(x)II = I1 9 II

#

(1.5)

( x e T(Uo) )

(1.6)

B y virtue of (1.4) and (1.5)

V(T(x)) = ]l x [Ia-: -- 1

The right member of (1.3), with U replaced by V, takes the form -- [[ x I[=-" [[t x []"-~ -- 1] ---- l[ x [[~-"-- 1

(xs T(Bo))

(1.7)

We draw the following conclusion. (h) The right member o/ (1.3), with U replaced by V, evaluates the

harmonic V e o / V correctly on T(Bo). This fact will be useful in w 4 in proving Theorem 1.3. A working hypothesis. Without loss of generality we can suppose that GO is a proper subset of S. If Go were in fact identical with S, the truth of Theorem 1.3 in the case in which Go is an arbitrary proper open subset of S would easily imply the truth of Theorem 1.3 in the case in which Go = S. w 2. Inversions. If v is an inversion in an ( n - 1)-sphere 27, the center of 2: will be called the center of ~. If x is not the center of v, v(x) is well-defined. B y an inversion in an (n - - 1)-plane ~ is meant a reflection in ~. If v is an inversion in an (n - - 1)-sphere 27, or (n - - 1)-plane 7, and ff ~(x) is well-defined, then x and v(x) are termed inverse relative to 27 or ~] respectively. Let 2:1 be an (n - - 1)-sphere meeting p. An inversion ~ with center p will map 271 - - P onto an (n - - 1)-plane ~, and conversely map V onto 271-- P. By abuse of language we say then that ~ carries 271 into ~ and into 271. With this understood we state a known lemma. Lemma 2.1. I] an inversion 7: carries an (n -- 1)-sphere 271 (or (n -- 1)plane ~1) into an (n--1)-sphere 272 (or (n -- 1)-plane ~2) then ~ carries

320

I~L Morse

points inverse relative to Z 1 (or rtl), neither o] which is a center o~ v, into points inverse relative to Z 2 (or ~2). The inversion T. Let a = ( a l , . . . , an) be an arbitrary point in E, and ~ an arbitrary positive number. Let

z = {~iI II 9 - a II = d .

(2.1)

The inversion in Z is a mapping T of E - - a onto E - - a in which e2(x -

T(x) - - a - -

a)

II x - a I[2

(11 x - - a II 4= 0).

(2.2)

The inversions T, T'. Consistent with the notation of w 1 let u---= (-- 1, 0, . . . , 0) and set s = {~ 111 9 tl -

S' = { 9 Ill ~ -- u II = 2}.

1}

(2.~)

The (n - - 1)-spheres S and S' intersect in the point -- u = (1, 0, . . . , 0). Let T and T' be respectively inversions in S and S'. Then T' carries S into the (n -- 1)-plane ~ on which x 1 = 1. Let R be the reflection of E in ~. We shall verify the following. Lemma

2.2. T(x) = T'(R(T'(x)))

(x 4= o9, x 4= u).

Proo/o/Lemma 2.2. Let y be any point of E such that neither y nor R(y) is the center u of T'. Recall t h a t T' carries S into ~. I t follows from Lemma 2.1 t h a t the points y and R(y), inverse relative to g, are carried by T' into points T'(y) and T'(R(y)), inverse relative to S. Thus T(T'(y)) = T'(R(y)).

(2.4)

Set i~ =-- x. Then x 4= ~o, since T(x) equals the finite left member of (2.4). Nor is x = u, since u is the center of T'. If T'(y) = x, then y =- T'(x), and (2.4) takes the form given in the lemma. Apart from the conditions that x =4=r and x ~= u, x is arbitrary in E. Thus Lemma 2.2 holds as stated. w3. The Kelvin transtorm. According to Lord Kelvin, if U is harmonic on an open subset D of E which does not contain the origin, then the function V with values

Xl

Xn )

v(~)=ll~ll=-~u i1~11~, , i l 7 ] i ~

(xer(D))

(3.1)

is harmonic in T(D). A proof of this statement in case n = 3 is given in [5] on p. 232. The proof for a general n > 2 is similar. Relation (3.1) m a y be written in the form

Harmonic Extensions V(x) = li x IL~-',

321

U(T(x))

(xe T(D)).

(3.2)

We shall define a mapping denoted by T . U. Set ( T . U) (x) ---- II x [12-~ U(T(x))

(x e T(D))

(3.3)

and call T . U the T-trans/orm of U. DeIinition. More generally let T be the inversion with center a given in (2.2). Suppose that U is defined and harmonic on an open subset D o / E which does not contain a. We set

~ Ir-"U(t(x))

(T.U)(x)=II

(xeT(D))

(3.4)

and term T * U the T-trans/orm o / U . We shall verify the following.

Lemma 3.1. I/ U is de/ined and harmonic on an open subset D o / E whivh does not contain the center a o/an inversion T, then T * U is harmonic on T(D). Let H be the translation x -> x + a and K the mapping e~(x - a) [] x -- ai[ ~

x --> K(x) --

(~ ~= a)

(3.5)

Then T(x) = H(K(x)) for x 4= a, so that by definition of a T-transform x--a

-~

( T . U)(x) --II

]1~-~ u ( g ( g ( x ) ) )

[xsT(D))].

(3.6)

The mapping x - + U(H(x)) is clearly harmonic on H-I(D). Set U(H(x)) = F(x) and H - t ( D ) = X. The mapping x--a

x -- a

~

lr-=~(K(z))

x ~

l1

[

(x 1 -- al)

[ 1 ~ _ ~ [ i ~,

(3.7)

(x~ -- an)

',

lix_.[i

~

is harmonic on K-a(X), as one infers from the harmonicity of V in (3.1). But the right member of (3.7) equals the right member of (3.4), thereby showing that T . U is harmonic on K-~(X) = K-1(It-I(D)) = T-I(D) =

= T(D). This completes the proof of Lemma 3.1. We continue with the following. Lemma 3.2. Under the hypotheses o] Lemma 3.1 Monatshefte ftir Ma~hema~ik. Bd. 67/4.

21

322

M. Morse

(3.9)

T.(T.U)----U.

To establish (3.9) one first notes that T(x)

--

a

(x -

a)

--

Q IIx--all

T(x)

II

--

a

Q

II = fix Q

If.

(3.10)

By definition of a T-transform, one has for x e T(T(D)) ~- D x--a

(T. (T.

U)(x)

=

II--ii

T(x) - II2-n II

x--a

-Since (3.10) holds and

T

(T. U)(T(x)) a

II2-n U(T(T(x)).

(3.11)

is an involution, (3.9) follows.

w 4. Proof of Theorem 1.3. Theorem 1.3 has a simple analogue in the case in which S is replaced by an (n - - 1)-plane. We shall consider the case of the (n -- 1)-plane ~ on which Xl ---- 1. Let E' and E " be, respectively, the open subsets of E on which x 1 < 1 and xl > 1. As previously, let R be the reflection of E in ~. Let X be an open connected subset of E " , and X 0 a non-empty open subset o f ~ such that X u X 0 is open relative to Cl E". The set

X

u

Xo

u

R(X) = X

(4.1)

is then open relative to E, and connected. The following lemma is known. I t is easily established with the aid of the Poisson integral. Lemma 4.1. Let a continuous mapping x -+ W(x); X o u X -> E

(4.2)

be given such that W is harmonic on X and vanishes on X o. Then W admits an extension W e which is defined and harmonic on the set X and such that We(x) -~ -- W(R(x)) (x e R(X)). (4.3) Completion o/proo/o] Theorem 1.3. Let U be given as in Theorem 1.3. The mappings T and T', defined as in w 2, are inversions in S and S' respectively, and R is the reflection in ~. Recall that U is given as continuous on G u Go and vanishing on Go. Without loss of generality in proving Theorem 1.3 we can assume that GOis a proper subset of S, and that rectangular coordinates with origin o) have been chosen so that u is not contained in Go. Under this assumption

Harmonic Extensions

323

it follows that T' is defined and continuous at each point of G u G0. We then set X--U

= ( T ' . U)(x) = II

(z i'(a)).

It

(4.4)

Noting that the right member is defined and vanishes for x e T'(Go), we set Z(x) = 0 for x s T'(Go). So defined Z is harmonic on Z'(G) by Lemma 3.1, continuous on T'(G) o T'(Go) and vanishes on T'(Go). Since T'(Go) c ~ we can apply Lemma 4.1, identifying T'(G) with X and T'(Go) with Xo. According to Lemma 4.1 Z admits an extension Z e, defined and harmonic on the open set T'(G) u T'(a0) u R(T'(G)) = / 2

(4.5)

and such that Ze(x) = - - Z(R(x))

(x e R(T'(G)))

(4.6)

The relation (4.6) may be written equivalently in the form ge(T'(x)) = - - Z ( R ( T ' ( x ) ) )

(x e 2(G))

(4.7)

since the condition T ' ( x ) e R(T'(G)) is equivalently x e T ' ( R ( T ' ( G ) ) ) and, by Lemma 2.2, reduces to the condition x e T(G). We need the following fact. (i) The set ~ does not contain the center u o/ T'. The point u is the image under T' of no point of E. Hence neither T'(G) nor T'(Go) contains u. Nor does R(T'(G)). For if there were a point x in G such t h a t R ( T ' ( x ) ) = u, then x = T'(R(u)). But the latter point is the origin, and so not in G by hypothesis. This establishes (i). I t follows from (i) that T'(Q) is well-defined. Using Lemma 2.2 we find t h a t T ' ( 9 ) = G u Go o T(G) = G (4.8) Statement (ii) will now be proved. (ii) The T'-trans/orm T' ~ Z e is de/ined on the set T ' ( ~ ) and is a ]~armonic extension o[ U. Since Z e is defined on 9 , and 9 does not contain the center u of T', T' ~ Z ~ is defined on T'(9). Since Z ~ is a harmonic extension of Z, T' ~ Z ~ is a harmonic extension of T' ~ Z. Cf. Lemma 3.1. But T" ~ Z = T" ~ ( T ' ,

as one sees on using (4.4) and Lemma 3.2.

U) = U

(4.9)

324

M. Morse

Statement (ii) follows. The proof of Theorem 1.3 will be completed by Lemma 4.2. Lemma 4.2. With Z defined as above and extended on ~ by Z e as in (4.7) (T' ,Ze)(x) -~ -- [[ x [[~-" U(T(x))

( x s T(G))

(4.10)

Since Z * is defined on Q and harmonic, it follows from the definition of a T'-transform that

(r'

X--ll

= Ir-y-

(4.11)

Now T(G) c T'(~2) by (4.8), so that it follows from (4.11) and (4.7) that $--11

(T' .Z~)(x) = -- I ] - - ~

[xe T(G)]

t]~-'Z(R(T'(x)))

(4.12)

In accord with (4.4) and Lemma 2.2 --.

Z(R(T'(x))) = II

2

11~-" U(T(x))

(4.13)

provided R(T'(x)) e T'(G), that is provided (4.14)

x s T'(R(T'(G))) = T(G)

From (4.12) and (4.13), with x e T(G) therein, it follows that (T' * g~)(x) = M(x) U(T(x))

(x e T(G))

(4.15)

where x ~ M(x) is an analytic mapping of T(G) into the axis of reals, independent of the choice of the harmonic mapping U of G into E. Evaluation o / M ( x ) . To complete the proof of (4.10) it suffices to show that M(x) = -- I1 x II~-~ [xs f ( a ) ] (4.16) For that purpose we turn to the harmonic mapping V defined in (1.4), and let / be the restriction of V to G . / t follows from (A) of w 1 that / has a harmonic extension f over G such that f(x) = H ~ I]~-~/(T(x))

[x~ T(a)]

(4.17)

Since the harmonic extension of ] over an open connected set is unique it follows from (4.15) that f ( x ) ---- M(x) /( T(x)) (x s T(G)) (4.18) Now /(T(x)) never vanishes for x e T(G) so that (4.16) follows from (4.17) and (4.18), thereby establishing Lemma 4.2.

Harmonic Extensions

325

B y virtue of (ii), T . Z e is defined on the set T(G) and is a harmonic extension U e of U. Relation (1.3) now follows from (4.10). w 5. A corollary of Theorem 1.3. In Theorem 1.3 U takes on zero boundary values on Go. In the following extension of Theorem 1.3, U takes on analytic boundary values on Go. Corollary 5.1. Let a mapping U : G--~ R be given such that U is harmonic on G and admits an extension U* on G u Go which is continuous and such that U* [Go is analytic. Then U admits an extension which is harmonic on a neighborhood o~ G u Go which is open relative to E. Set ] = U* I Go. By hypotheses / is analytic on Go. B y virtue of the Cauchy-Kowalewski theorem, [6], p. 656, there exists a harmonic extension F os / defined on a neighborhood X of Go, open relative to E. The mapping x ~ U ( x ) - F(x) (x ~ X n G) is harmonic on X n G, takes on zero boundary values on Go, and so, b y Theorem 1.3, admits a harmonic extension V on a neighborhood Y of GO open relative to E. The mapping x -~ V(x) + F(x)

(x ~ x n y )

is harmonic and reduces to U on (X n Y) n G. The corollary follows. References

[1] Huebsch, W., and Marston Morse. Schoenflies extensions without interior differential singularity, Annals of Mathematics, 76 (1962), pp. 18--54. [2] Royden, H. L. The analytic approximation of differentiable mappings, Math. Annalen, 139 (1960), pp. 171--179. [3] Huebsch, W., and Marston Morse. The Sehoenflies extension in the analytic ease, Annali di Matematica, 54 (1961), 359--378. [4] Carathdodory, C. On Dirichlet's problem, Amer. Jour. of Math., 59 (1937), pp. 709--731. [5] Kellogg, O. D. Foundations of Potential Theory. Berlin, Julius Springer, 1929. [6] Goursat, E. Cours d'analyse math6matique, Tome II, Paris, GauthierVillars, 1929.