A R K I V F O R M A T E M A T I K B a n d 5 n r 18
C o m m u n i c a t e d 13 N o v e m b e r 1963 b y AKE PLEIJEL a n d L ~ s G)~RDING
A t h e o r e m o f the P h r a g m 6 n - L i n d e l / i f t y p e f/~r s e c o n d - o r d e r elliptic o p e r a t o r s
By LARs LITHNER
]. Introduction and notations
Let R ~ be the real n-dimensional Euclidean space with coordinates x = (x 1, x~ . . . Xn), Ix [ = Vx~ .4- x~ .4-... -4- x~. C denotes the set of all complex-valued infi- nitely differentiable functions on R n with compact supports and L2(~) is the Hilbert space of all complex-valued square integrable functions on the set ~ .
I n R 1 let D be the domain {x I x >~a} where a is a r b i t r a r y and let L be the differential operator
- ~ +L ~>0-
The solutions of L u = 0 arc
u(x) = Cl e ~ + C2e- Y ~x,
where C 1 and C 2 are arbitrary constants. F r o m this we conclude t h a t if a so- lution is bounded in the domain D or if it belongs to L2(D) then it decreases like e - r ~ when x tends to infinity and the same holds for its derivative. I n particular, u d '~: and ( d u / d x ) d 'z belong to L~(D) if ~u< l/~.
I n this paper we shall extend this result to second-order elliptic differential operators in R n,
L = - ~ a,k(x)D,~ .4. ~ bk(x)Dk §
t, k = l k = l
where D~k=~2/~x~xk, Dk=O/axk and aik(x)=ak~(x) (for simplicity we confine our- selves to the real domain).
Giving the result of the general case at the end of the paper we start with the operator L = - A + a(x), where A is the Laplace operator in R n and where a is positive and continuous or, more generally, locally bounded and Borel measurable. Then we can prove t h a t if u is a solution of L u = 0 outside some compact set K and if u belongs to L2(R n - K ) t h e n , in the same sense as above, u and its first derivatives decrease exponentially like e -v(x) when Ix[ tends to infinity. ~(x) is the geodetic distance from the origin to the point x in the metric ds 2 = a(x) (dx~ -I- dx~ § dx~n).
L. LITHNER, A theorem of the Phragmdn-Lindel6f type 2. The special case
L e t D be the domain {x I1 x I~> R}, where R is a positive number and let B be the boundary of D. L is the operator - A + a where the function a is strictly positive in R n. L e t ~(x) be the geodetic distance from the origin to the point x in the metric ds 2 = a(x) (dx~ + dx~ + ... + dx2n) t h a t is, ~(x) is the greatest lower bound of
] frVa~)Vdy~ +dy~ +...
+dy~ , Y=(Yt, Y, . . . y,),where F is a piecewise continuously differentiable curve starting at the origin and ending at x. Putting further conditions on a ~ will be continuously differ- entiable.
Lemma 1. ]grad ~(x) I,.< ] / a ~ .
Proo[. I t is evident from the definition of ~ t h a t
where F 0 is the straight line segment joining x and x + Ax. This gives the in- equality.
Lemma 2. (Carleman [1].) I[ u belongs to L2(D) and is a solution o / L u = 0 then ]/au and ]grad u I belong to L2(D).
Proo[. L e t ~ be a positive function in C. Then we have
o=fou(x) (x)Lu(x)dx=fa(x)w(x)u2(x)dx-fo lU, (x)u(x) (x)dx,
O2u
where u~k = Ox~Oxk"
B y partial integration we get
L f"
o= M(u)~ + ~ u~ (xho(x)dx +
J D 1
fo
+ ui(x)y~i(x)u(x)dx+ a(x)y~(x)u2(x)dx, (1)
where M(u) contains u and first derivatives of u and where d8 denotes the surface element. In the third integral we can integrate b y p a r t once more and get
~u~(x)y~t(x)u(x)dx= M'(u)ds - 89 vAi(x)u2(x)dx, (2) where M ' is an analogue of M.
ARKIV F 6 R MATEMATIK. B d 5 n r 1 8
F r o m (1) a n d (2) we get the following estimates
f y'(x)grad~u(x)dx+fa(x)y,(x)u~(x)dx<~
<~ I ~ (M(u) + M'(u))d8 + 89 ~ ~'y~,,(x)'u2(x)dx.
L e t t i n g yJ t e n d to 1 in such a w a y t h a t ~0ii is bounded we conclude t h a t the l e m m a is true.
We are now going to p r o v e T h e o r e m 1 below. As before, let
uEL~(D)
be asolution of L u = 0 and let v be a function with locally s q u a r e integrable first derivatives such t h a t ~aav a n d ]grad v[ belong to
L~(D).
An integration b y p a r t s givesf +
where
M(u, v)
containsu,
grad u a n d v. I n fact, this identity is true if we re- place v b y yrv, where yJs C. Letting y; t e n d to 1 in such a w a y t h a t {grad ~0{tends to zero, we get the general case.
Now let 0 < e < l a n d p u t
9
= m i n(e 2(i-e)~(x),
-~),where N is a positive n u m b e r a n d where ~ is the function in L e m m a 1. I t follows f r o m L e m m a 1 a n d L e m m a 2 t h a t
~aa/Nu
a n d {grad (/~u)] belong to L2(D). Thus we can substitute /Nu for v in the formula (3) a n d geto= fBM(u, /~u)ds+ f /N(x)grad2u(x)dx+
+
fDU(x)gradu(x).grad/N(x)dx
+fva(X)/N(x)u2(x)dx.
(4)F r o m L e m m a 1 it follows
I grad/~(x){ <~ 2(a
- e) ]gradr l/~(x ) <~
2(1 -- e) ] / a ~ / N ( x ) . F r o m (4) a n d (5) we getf /N(x) grad2 u(x)dx + f Da(x) /~(x)u2(x)dx <<-
(5)
L. LITHNER, A theorem of the Phragmdn-Lindel6f type Usiiig S c h w a r z ' s i n e q u a l i t y we g e t
f M(u,/ u)ds
F o r large N t h e r i g h t m e m b e r is a c o n s t a n t a n d thus, letting N t e n d to in- finity, we h a v e p r o v e d t h e following
Theorem 1. I / u belongs to L2(D) and is a solution o/ L u = 0, then/or every posi- tive number e
~aa(x)u(x)e (1-e)~(z) and [grad u(x) [ e (1-~)~(z) belong to L2(D).
Consider t h e o p e r a t o r
3. T h e general case
n
L = - ~ aik(x)D,~ + ~bk(x)Dk + a(x),
t , k f f i l 1
aik(x) = ak~(x). W e suppose t h a t a, besides t h e conditions in T h e o r e m 1, satisfies a(x) ~ d > 0 outside some c o m p a c t set.
T h e o p e r a t o r L is supposed t o be u n i f o r m l y elliptic in D, t h a t is
n
a i k ( x ) ~ k >1 0 ~ for all x in D,
L k = l 1
where ~ is a positive n u m b e r . F u r t h e r we suppose t h a t ~a~k(x)/Ox~ a n d bk(x), i, /c= 1, 2 . . . . , n, t e n d to zero w h e n Ix] tends t o infinity.
W i t h t h e same t e c h n i q u e as in t h e proof of T h e o r e m 1 we c a n p r o v e
Theorem 2. I / u belongs to L2(D) and is a solution o / L u = O, -then/or every positive e ue 0-~)~ and I g r a d u l e O-')~ belong to L2(D),
where qD(x) is the geodetic distance/rom the origin to x in the metric
ds2=a(x) ~ a[k(x)dx~dx~
i, k = l
(d[k(x)) is the matrix which is inverse to (aik(x)).
To show t h a t our t h e o r e m s in a certain sense are t h e best possible we give t h e following
Example. L e t b a n d c be t w o positive numbers. T h e f u n c t i o n u(x, y) = e-t(v~ ~'+~'~u ')
ARKIV FOR MATEMATIK. B d 5 n r 18
satisfies an equation
-- A u + a(x, y) u = O,where
a(x, y)/bx e + cy2--> lwhen ] / x ~ y2 tends to infinity. If ~o(x, y) denotes the geodetic distance from (0, 0) to (x, y) in the metric
ds 2= ( b x 2 + c y 2 ) ( d x ~ § 2)it is easy to see that ~(x, y ) = 89
+ V~y2).
R E F E R E N C E
][. CARLEMAN, T. Sur la th~orie m a t h ~ m a t i q u e de l ' ~ q u a t i o n de Schroeding(r. A r k i v m a t . i astr.
och fysik 24, (1934).
Tryckt den 1O april 1964
Uppsala 1964. Almqvist & ~Viksells Boktryckeri AB
