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Seria I : PRACE MATEMATYCZNE X (1966) ANNALES SOCIETATIS MATHEMATICAE POLONAE Serio I: COMMENTATIONES MATHEMATICAE X (1966)

usiałek

(Kraków)

Some properties o! zeros of solutions of second-order linear partial differential equations of elliptic type

П

i , k = l

n n

(2)

У Aik(X)«^(X) + ^ B t(X)n'l[ X ) + C { X M X ) =

i,k = 1 j = 1

П

(I)

У

(gik(X)u'4 )'H+f (X)u{X) = 0

i , k = 1

П

(II) У

(Ga (X) U'Xky4 + F ( X ) U(X) =

i,k = I

(2)

Definition 1.

b

Definition

Definition 3.

(u—1)-

Theorem 1.

(2)

(2)

\ X - P 0\ < r

Theorem

([2]).

Theorem 3.

(2)

<72

(2)

obtaining at an interior point of В its extremal value equal to zero; then u( X) = 0.

P) I.e. open, connected set.

(2) I.e. haying Lipschitzian derivatives of the first order.

(3)

(3)

Proposition 1 .

The following theorems from dimension theory will be needed:

Theorem 4 (Mazurkiewicz

dim A 2

Theorem

Th e o r e m 6.

Z

Z .

P t Z

r\ Z

r\ Z

V

r

Z )

Theorem 7.

(1)

for every X e D ,

(3) See [4], p. 343.

(4) Menger [7], p. 244.

(5) According to now common terminology a neighbourhood of P is a set V con taming P in its interior.

(C)

П n

a

i

(4)

hn kn-{-1

, — 7U < Xn < --- ---TU

Ex a m p l e

i,k = 1

i,fc= 1

(5)

(

)

=

n n

i,k= 1 j—1

n

7 = 1

.

Th e o r e m 8 .

(6)

г,/ = 1

i j - l

n n

1 = 1 ? = 1

n n

i = 1 i, 7 = 1

1ф1

n

7 = 1

1 = 1 l,7'= 1

гф1

n

7 = 1

(7)

П

* i= i

> / > 4 ^ »

П

i j — l

П

i= i

In the neighbourhood F 5 inequalities (13) are fulfilled and consequently C1( k , X ) < 0 in this neighbourhood. Now we choose a neighbourhood IF c F 5 such that o o sfcfy X ) in this neighbourhood. By Theorem 3 it follows that u( X) == 0 in the set D.

Prace Matematyczne X .l 3

(8)

Corollary 1.

7

8

Example

г=1

2

1/2

8 V/

H-

_L

n -

K m v m2,...,mn

(9)

Corollary 2.

(2),

Theorem

Proposition 2.

(2)

0,

Theorem 10.

(2)

0

= 0 .

position 2 we have и (X) == 0 in D, which contradicts the assumption.

(6) Tliis means that there does not exist a sequence of surfaces Г п, different from Г, contained in Z, and tending uniformly to Г.

(10)

under what regularity conditions imposed on coefficients of the equation (I) in the case n > 2, each solution of (I) vanishing in an open subset of the region D vanishes identically in D. In the paper [1] Aronszajn, Krzywicki and Szarski proved that it is the case when the coefficients g%k{X) are of class C0,1. A. Plis gave an example of an equation (I) with coefficients of lower class for which this is not longer the case.

References

[1] N. A r o n s z a j n , A . K r z y w ic k i and J. S z a r s k i, A unique continuation theorem for exterior differential forms on Biemannian manifolds, Arkiv for Matematik 4. 34 (1962), pp. 4 1 7-45 3.

[2] Ph. H a r t m a n and A . W i n t n e r , On a comparison theorem for self-adjoint partial differential equations of elliptic type, Proc. Amer. Math. Soc. 6 (1955), pp. 862-865.

[3] C. M ir a n d a , Equazioni alle derivate parziali di tipo ellittico, Berlin - Got­

tingen - Heidelberg 1955.

[4] K . K u r a t o w s k i , Topologie I I , Warszawa 1950.

[5] F. B a r a ń s k i, O własnościach oscylacyjnych i liniach węzłów rozwiązań pew­

nych równań różniczłcowych cząstkowych typu eliptycznego, Prace Mat. 7 (1962), pp. 71 -9 6 .

[6] E. K a m k e , Differentialgleichungen Lósungsmethoden und LSsungen, vol. I, Leipzig 1944.

[7] К . M en g e r , D i mensiontheorie, Leipzig -Berlin 1928.

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