Properties of nodal lines of eigenfunctions of a certain systemof partial differential equations of second order

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I : PRACE MATEMATYCZNE X I I (1968)

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I : COMMENTATIONES MATHEMATICAE X I I (1968)

Jan B ochenek (Kraków)

Properties of nodal lines of eigenfunctions of a certain system

of partial differential equations of second order

Let G be a bounded domain in the м -dimensional Euclidean space F m of real-valued X = {xx, . . . , x m), measurable in the sense of Jordan.

We assume that the domain G can be approximated from inside by an increasing sequence of domains Gn with regular boundaries (G = £ G n);

П it means that the boundaries are surfaces of the class Cl, where the defi­

nition of such surfaces can be found in [4], p. 132. We make no assump­

tion converning the boundary F(G) of the domain G.

We shall consider the system of differential equations of the form

(1)

with the

( 2 )

(P = 1 , -

m

da^f aii^ ^ 1 + IftQpk(X) — qp]c(X)]uk — 0 i,j

° 1

1 -*

boundary condition of the form

du n

— 2- = V Тсрг{Х)щ = 0 on F ( G ) ~ r , up = 0 on Г

dv Ы

., n) where

(3) dup

dv

VI и Up

2 j aV (X ) cos (n 1 >

i,j=

1

n denotes here the interior normal to the boundary F(G) of G. We make the following assumptions concerning the coefficients appearing in the system ( 1 ) and boundary condition ( 2 ) (which we shall for simplicity call problem ( 1 )( 2 )): ац{Х) — ац{Х) (i , j = belong to the class C1 in G, qpi ( X ) , qPi (X ), kpi(X) (p, l = 1 , ..., n) are defined and continuous in the domain G. Besides, we assume that the matrices {qPi(X)}

and {kPi(X)} are positive-definite in G, and the matrices {а#(Х)} and

32 J . B o c h e n e k

{ qvi {X)} are uniformly positive-definite in О. In the boundary condition ( 2 ), Г denotes a certain (m —1)-dimensional part of the boundary F(G), not necessarily connected. In particular, Г may consist of the whole boundary F(G) or it may be empty. On the other hand, the boundary condition should be understood in the generalized sense, as presented in papers [1] and [3]. The problem (1)(2) is thus a particular case of the problem in paper [3] for more general system of differential equations.

The object of this paper is to study some properties of nodal lines of eigenfunctions connected with system ( 1 ) and with boundary condi­

tion (2). The definition of eigenvalues and eigenfunctions of problem (1) (2) is given in paper [3].

Let U (X) — (ux{X ) , . . . , un{X)) be an eigenfunction of problem (1) ( 2 ).

Using the results proved in [2] we shall prove the following

L emma 1 . I f 1 ° U (X0) = 0 for X qz G) 2 ° there exists a ball К c: G such that X 0e F ( K ); 3° U(X) belongs to the class C ¹ in the closure К of the

П

ball К ; 4° TJ{X) Ф 0 for X e K ) then J^grad2% > 0 at the point X 0.

i=

Proof. Denote B {X ) = [u\{X)-\-...-\-u2n{X)'f,21 and denote by e(X) the unit vector parallel to, and of the same direction as, the vector U (X) for U(X) Ф 0. For X e K , by assumption 4° of the lemma, we have B (X )

= U (X )e*(X ), where e*(X) is the transpose to vector e{X). Assume П

that ^ grad2щ = 0 at the point X 0. Since U(X) belongs to the class G1 i= 1

at X 0, we have dUjdl = 0 for every direction l starting from X 0 and entering the ball K , or

( 4 ) lim

A_>X 0

U ( X ) - U ( X o)

= o>

where X e l . On the other hand, since B ( X 0) = ⁰ and U (X0) ^{= 0} we have [ B ( X ) - B ( X 0)]/X X Q = [ U ( X ) - U ( X 0)]e*(X )IX X 0

which implies, together with (4) that

(5) lim B ( X ) - B ( X 0)

x x ^{0 = 0 (X el).}

Equality (5) contradicts the assertion of Theorem 2 from paper [2].

Let us denote bj Z the set of zeros of U(X) in the domain G (nodal lines of function U(X)), and by Zp the set of zeros of the function

П uv(X) (p = l , . . . , n ) , Z = f ) Z „ .

P=1

Т ^беоеем 1. I f X 0 is an arbitrary point of the set^Z, then in every

neighbourhood of the point X 0 there are points from at least one of the sets Zv.

Properties of nodal lines 33 Proof. Suppose that there exists a neighbourhood O{X0) of X 0 such that for every p — 1, .. ., n, we have up(X) Ф 0 for all XeO (Xf) and X ф X 0. It follows that every function iip(X) assumes its local extremum

П

at the point X 0, hence ^ grad2% = 0 at X 0. However, this equality

г=1

contradicts the fact that U (X) Ф 0 for Х е О (Х 0) and X Ф X 0.

П

T

2. I f X 0eZ and ^ g r a d ф 0 at the point X 0, then: 1°

i= 1

at least one of the functions up (X) changes its sign in every neighbourhood of X 0; 2 ° at least one component of the set Zp which cuts the set G passes through X 0.

Proof. 1° Suppose that there exists a neighbourhood O(X0) of the point X 0 such that for every p = 1 , .. ., n the function up(X) preserves its sign for X e O (X 0). Since up(X0) = ⁰ for every p — ¹ , . . . , n every function up(X) assumes its local extremum at X 0, which contradicts the assumptions of theorem 2 .

2° The second assertion of Theorem 2 follows directly from the first part and from the continuity of function up(X) in G.

П

T heorem 3. I f X QeZ and ^ g ra d 2Wi = 0 at X 0, then the point X 0

г

is a ubranching” point of the set Z in the sense that every ball contained in G, whose surface contains X 0, contains points of the set Z.

Proof of Theorem 3 follows directly from Lemma 1.

To study further properties of nodal lines of eigenfunctions of prob­

lem ( 1 ) ( 2 ) we make further assumptions concerning the coefficients.

We shall assume that qPi(X) = бргд {Х ), hpi(X) = дргТс(Х) (p , l = 1 , . . . , n), and the elements of the matrix { qpi (X)} are constants.

Since the matrix P — { ^qpi } is, by assumption, symmetric and positive- definite, there exists an orthogonal transformation (see [5])

( 6 ) W(X) = V U (X )

such that the matrix VPV~1 — P = {6 piq p}, where qp > 0 (p = 1, .. ., n).

We see easily that the transformation ( 6 ) reduces the system (1) to a system of n independent equations, and the boundary conditions ( 2 ) to n independent conditions; it can be written in the form

v i д Г m dw pT

(7) ^^ ~Q% I ^ ~~д%~ I ^ ^ p P fiQpwp — 0,

i,j

( 8 ) —h{X)wp = 0 on F { G ) - r , wp = 0 on Г {p = 1, ..., n ) . dv

Roczniki PTM — P ra ce M atem atyczne X II 3

34 J . B o c h e n e k

Problem (7) ( 8 ) can be treated as n independent problems separately for every p = 1 , . . . , n. Each of these problems coincides with the problem treated in paper [1]. Using the theorems of this paper we give some properties of eigenfunctions of problem (7) ( 8 ). Let ns denote by

(9) A lp, Xlp, Xip , ... {p = 1 , •.. , n)

the sequence of eigenvalues of problem (7) ( 8 ), and the corresponding sequence of eigenfunctions by

(10) wlp( X ) , w2p( X ) , w3p( X ) , ... ( p = l , . . . , n ) .

The scalar functions of sequence (10) are eigenfunctions for one equation considered in paper [ 1 ], corresponding to eigenvalues of the sequence (9). On the other hand, to the eigenvalue Xkp there corresponds a vector eigenfunction Wkp{X) = (0, wkp(X), . . . , 0), with all compo­

nents except the ^-th equal to zero. Under transformation ( 6 ), to the function Wkp{X) there corresponds the function Ukp(X) = V ^ W ^ X ) . We see easily that the function Ukp(X) may be written in the form Ukp(X)

= Apwkp{X), where A p denotes the vector, whose components are ele­

ments of the p -th row of matrix V~1.

Using the mentioned theorems of paper [ 1 ] we obtain the following properties of the sequence of functions Ukp(X):

1° Every function Ulp(X) (p = l , . . . , n ) does not vanish in the domain G;

2° The function Ukp(X) (p = 1, .. ., n) vanishes along the nodal lines which cut the domain G into at most к nodal domains;

3° In every nodal domain of the eigenfunction Ukp(X) there are nodal lines of all eigenfunctions UiP{X) for l >

4° If to the same eigenvalue Xkp there correspond two eigenfunctions V\4(X) and f7g(X) which are linearly independent, then the zeros of these functions cross each other.

R eferen ces

[1] J . B o c h e n e k , On some problems in the theory of eigenvalues and eigenfunc­

tions associated with linear elliptie partial differential equations of the second order>

Ann. Polon. Math. 16 (1965), pp. 153-167.

[2] — On the Dirieldet's problem for a class of the elliptic systems of differential equations of the second order, Zeszyty Naukowe U J, Prace Matematyczne 11 (1966), pp. 21 -2 6 .

[3] — On eigenvalues and eigenfunctions of strongly elliptic systems of differential equations of second order, this fasc., pp. 171-182.

[4] M. K rz y ż a ń s k i, JRównania różniczkowe cząstkowe rzę,du drugiego I , W ar­

szawa 1957.

[5] A. M o sto w sk i i M. S ta r k , Algebra wyższa, Warszawa 1953.