T H E O R E M 3.1. A necessary and sufficient condition /or the boundary conditions (0.3) to be hypoelliptic with respect to the hypoelliptic operator P (D) o/ determined type
#, ~ and co, is that
Im ~-->oo i/ ~-->~ in ~t satisfying C(~)=O. (3.1) The analogy between this condition and the condition (0.4) for the hypoellipticity of an operator is obvious. Note t h a t the only geometric property of ~ and ~o which is involved in (3.1) is the direction of the interior normal of ~ on co.
T H E O R E M 3.2. A necessary and sufficient condition for the boundary conditions (0.3) to be elliptic with respect to the elliptic operator P (D) of determined type /~, and co, is that with some constant M
C(r when IRe r r (3.2)
M may be assumed so large that all ~ satisfying the latter condition are in A.
This is clearly analogous to the characterization of elliptic operators given by Lemma 2.1. The following theorem connects it with the condition (0.5) (cf. also the results of Petrowsky [13] concerning elliptic systems).
T HV, OREM 3.3. Let QO and po be the principal parts of Q, and P, which we assume elliptic: Let C o be the characteristic function o/ the boundary problem defined by po and QO. Then
(a) I f C~ for real ~:vO, the boundary conditions Q, are elliptic with re- spect to P.
(b) I / the boundary conditions Q, are elliptic with respect to P and n > 2, we have either C o = 0 identically or C ~ (~):~0 for real ~ :~ O.
242 LARS HORMANDER
Proof.
(a) Denote the degree of Q, b y m,. Then, as is immediately verified, CO (~) is a homogeneous function of ~ of orderM = m z + - - . + m ~ - 1 - 2 . . . (/~- 1).
Clearly it will be bounded from below in a complex neighbourhood of the real unit sphere. From this the result follows easily if we argue in the same way as in the proof of Lemma 2.1.
(b) Let $:~0 be real and such t h a t C ~ We have to prove t h a t C O must then vanish identically. Let // be a real vector and note t h a t
s-MC(s(~+wil))-->CO(~+wil)
when s-->c~ and is real, (3.3) provided t h a t J wJ is sufficiently small (cf. (a) above). Now b y assumption the func- tion on the left hand side of (3.3) is analytic in w and is =~0 if s is real and positive ands ( J ~ J - J / l J J R e w D > ~ M ( l + s J / / J J I m wJ), hence if JwJ ~< (J ~ J - M s - ' ) / ( 1 +M) J/1J.
I t t h u s follows t h a t the limit
C~
when s-->~ is either identically zero or never zero when J w J< l$ J/(1 + M)]/1J. B u t b y assumption this function vanishes whenw=O,
so t h a t it must vanish in the whole circle. If J/l] < J~J/(1 + M), the circle containsw=
1 and we obtain C O ($ +/1) = 0 . B u t since C O is analytic and vanishes in a neigh- bourhood of $i it must vanish identically, which completes the proof.The proof of Theorems 3.1 and 3.2 will be given in sections 4-6. In this sec- tion we shall only illustrate the results with a few examples.
Example
1. Let the boundary conditions be the Diriehlet conditions~Vu/ST~=O
in o~, ~---0, 1 . . . / ~ - 1 , (3.4) where T is a direction transversal to (o, i.e. ( T , N ) * 0. A simple computation shows t h a t C(r is a constant =~0. Thus the Dirichlet boundary conditions are (hypo-) elliptic ifP(D)
is (hypo-)elliptie. Note t h a t it m a y occur t h a t # = 0 so t h a t no boundary conditions are present.A remarkable feature of this example is t h a t the Diriehlet boundary conditions are {hypo-)elliptie with respect to all (hypo-)elliptie operators of type ju, in spite of the fact t h a t conditions (3.1) and (3.2) involve P also. We are going to s t u d y the boundary conditions which have this property.
O N T H E R E G U L A R I T Y O F T H E S O L U T I O N S O F B O U N D A R Y P R O B L E M S 243 D E F I N I T I O N 3.1. The # boundary conditions (0.3) are called completely elliptic i/
they are elliptic with respect to all elliptic operators P (D) o/ determined type ~.
L e t D (vl . . . 3,, 8) be the principal p a r t of the polynomial
QI( +3N) . . . ,
where 31 . . . 3p and 8 are considered as independent variables. We shall call D the characteristic /unction o/ the boundary conditions (0.3).
T H E OR E M 3.4. A su/]icient and, i / n > 2, also necessary condition/or the boundary condition (0.3) to be completely elliptic is that
D (V 1 . . . T t z , ~) * 0 i/ I m 3j > O, j = "1 .. . . . /z, and 8 is real, ~ =v O. (3.5) A n equivalent condition is that the polynomial in
n (31 + ~ 3 ~ .. . . . 3, + 2 3~, 8) (3.6) has only real zeros i/ 3 ~ . . . 3 ~ and 31 . . . Tz, ~ are real, ~ * 0 . (When ~ = 0 the polynomial either vanishes identically or else it has only real zeros.)
Note t h a t if D (v I . . . 3,, 0) does not vanish identically, this means precisely t h a t D is hyperbolic with respect to all vectors (31 . . . 3~, 0) with all 33>0. (For the definition of hyperbolic polynomials cf. Gs [2].)
Proo/ o/ Theorem 3.4. The sufficiency of (3.5) is quite obvious. Indeed, if P (D) is an elliptic operator of determined t y p e ju and po its principal part, we have
o = D (31 . . . 8) + 0 (J
where M is the degree of D and 31 . . . 3, are the zeros of the polynomial po (8 + 3 N) with positive imaginary part. As in Theorem 3.3 (a) this implies t h a t (3.2) is fulfilled.
N e x t assume t h a t n > 2 and t h a t the b o u n d a r y conditions (0.3) are completely elliptic. Take 8' real, $' * 0 , and 3~ . . . 3, with positive imaginary parts. We have t
to prove t h a t
D ( ' T 1 , . . . , T ~ , ' 8') : ~ 0 .
t t t t
Since D does not vanish identically and n > 2, we can find 31 . . . 3~ with positive imaginary parts and a real ~" such t h a t $' and $" are linearly independent a n d
i !
D (3I(, . . . . 3~, 8") * O. L e t P (8) be a homogeneous positive definite polynomial of degree 2/~ such t h a t P (~' + 3 N) is divisible b y 1-I (3 - 3~) and P (8" + 3 N) is divisible
1
2 4 4 LARS HORMANDER
It
b y l~(27-27~'). The existence of such a polynomial will be p r o v e d when /~= 1 in
1
L e m m a 3.1; in the general case one only has to m u l t i p l y /~ such factors. N o w we obviously h a v e
lim s -M C (s ~) = D (271 . . . Vl, , ~ ) , s-~r
if 1" 1 . . . Tt~ n o w d e n o t e t h e zeros of P (~+27 N) with positive i m a g i n a r y parts, a n d arguing as in t h e proof of T h e o r e m 3.3 (b), we can t h u s conclude t h a t t h e right h a n d side either vanishes identically or is ~:0 for all real ~ with ~:~0. N o w b y a s s u m p t i o n it does n o t vanish when ~ = ~", a n d hence n o t when ~ - ~' either. This proves (3.5). Since t h e equivalence between t h e two conditions in t h e t h e o r e m is obvious, it o n l y remains to prove the following lemma.
L E M M A 3.1. Let ~' and ~" be real, ~' and ~" linearly independent, and let 2, and 2" be two non real numbers. Then there is a positive definite quadratic /orm S (~) such that S (~' + 2' N ) = S (~" + 2" N ) =O.
Proo/. Since ~' + 2' N, ~" + )," N a n d N are linearly independent, we can find a complex v e c t o r y E G + i G so t h a t
(y, ~ ' + ) , ' N } = ( y , ~ " + 2 " N } = 0 , (y, N } = l .
Write S (~) = ((y, ~} + (9, ~})2 + s($);
S (~) will h a v e t h e desired properties if s (~) is positive definite in ~ a n d s (~') = - ((y, ~' + 2' N } + (y, ~' + ~ N})* = - ((9, (2' - 2 ~) N } ) 2 = 4 ( I m 2') *, s (~") = 4 ( I m 2") 2.
Since 4 ( I m 2') 2 > 0 , 4 (Ira 2") 2 > 0 a n d $' a n d ~" are linearly independent, one can find a form s (~e) with t h e desired properties.
E x a m p l e 2. L e t P ( D ) be the Laplace operator, t h a t is, if we i n t r o d u c e co- ordinates so t h a t N = (0 . . . 0, 1),
P (~) = (~', ~') + ~ ,
where ~' = ( ~ 1 . . . ~ n - - 1 ) a n d (~', ~') = ~12 + --. + ~n-1. We have tt = 1 and, changing if 2
necessary t h e b o u n d a r y condition with a multiple of P (D), we m a y assume t h a t Q (D) = qo (D') u - i ql (D') ~ u / a x n, (3.7)
ON THE REGULARITY OF THE SOLUTIONS OF BOUNDARY PROBLEMS 2 4 5 where q0 and ql are polynomials in ~'. If ~=(~', ~ . ) w e can obviously identify with ~' and then have
C (r = qo ($') + i ($', $')t ql (~'), (3.8) where (~', ~')89 denotes the square root with positive real part. The condition (3.2) /or elliptieity is equivalent to the ellipticity o/ the polynomial
F (~') = % (~,)2 + (~,, ~,) ql (~,)z. (3.9) Indeed, if this polynomial is elliptic, (3.2) follows from Lemma 2.1 because C is a factor of F. On the other hand, assume t h a t (3.2) is fulfilled. If n = 2 it follows if we apply (3.2) to positive and negative ~1 respectively, using (3.8), t h a t
q0 (~1) ~--- i ~1 ql (~1) ~ 0,
and hence the product _~ of these two polynomials does not vanish identically which proves the assertion since all polynomials ~ 0 in one variable are elliptic. If n > 2 we decompose the principal part Q0 in a form similar to (3.7) and obtain
CO (6) = qO (~,) + i (~', ~')89 q0 (~,).
This cannot vanish identically since (~', $')~ is not a rational function when n > 2 . Hence, according to Theorem 3.3, the boundary condition being elliptic, we have C O ( 6 ) * 0 for real 6 * 0 . Multiplying C~ and CO( - 6) together, noting t h a t qo and ql are homogeneous, qo of one degree higher than ql, and t h a t ( - ~ ' , - ~ ' ) } = (~', ~')89 we find t h a t
q0 0 (~,)~ + (~,, ~,) q0 (~,)~. 0 for real ~ 0.
But this means precisely t h a t the principal part of F (~') satisfies the definition (0.5) of an elliptic polynomial, so t h a t the assertion is proved.
In particular, the result shows t h a t the conditioa /or ellipticity with respect to the Laplace equation does not depend on whether ~ is situated in the hal/ space x ~ > 0 or xn<O.
The latter conclusion does not hold in the hypoelliptic case. Indeed, let n = 3 and Q (~) = i $~ + $a.
Then C ($') = i ~ + i ~/~+ ~
if ~ is situated in the half space x" > 0 and
(3.1o) (3.11)
(3.12)
2 4 6 LARSHORMANDER
if ~ is situated in the half space x " < 0. (Note t h a t the square root is defined so that it has a positive real part.) Now (3.12) vanishes if $1 a n d ~2 are real and satisfy the equation ~ = ~ + ~ . This curve has a real infinite branch, so t h a t the b o u n d a r y condition (3.10) is not hypoelliptic with respect to the Laplacean if ~ is situated in the lower half space. On the other hand, if (3.11)vanishes, we m u s t h a v e Re ~12~0 in view of the definition of the square root. Hence
If a bound for IImr is prescribed, this gives an estimate of ]Re ~{ when C ( ~ ' ) = 0 , thus according to (3.11) we get an estimate of [ ~ [. Thus (3.10) defines a hypoeIliptic b o u n d a r y condition with respect to the Laplacean if ~ is situated in the upper half space.