On eigenvalues and eigenfunctions of strongly elliptic systems of differential equations of second order

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

J a n B o c h e n e k (Kraków)

On eigenvalues and eigenfunctions of strongly elliptic systems of differential equations of second order

The purpose of the present paper is to determine eigenvalues and eigenfunctions for the so-called strongly elliptic systems of partial differ­

ential equations of second order and to generalize some theorems known for one equation to systems of equations. We shall nse the following notation: if В = {&i?}y=i is a matrix and V = ( % , . . . , un), V — (vt, . . . , vn) are vectors, then

B U — (&ii% + *• • + Ьщun, . .. , • • + bnnun) and UV = + unvn.

§ 1. Let G be a bounded Jordan-measurable domain in the space I f 1 of m variables X = (aq, . .. , xm). G may be approximated by an increasing sequence of domains Gn with regular boundaries (i.e., such that the bound­

ary F(G n) of Gn is a surface of class C£; for a definition of a surface of class Cl, see [3] p. 132). We do not require any regularity properties of the boundary of G. Domains with regular boundaries will shortly be called regular domains.

We shall consider a system of differential equations of the form ( 1 )

where

L(U) + f*P(X)U = 0,

- Q ( X ) U

is a self adjoint differential operator and [л is a real parameter. We make

the following assumptions: the coefficients of system (1) are symmetric

n x n matrices, P { X ) and Q(X) are positive definite and continuous in

G, Aij(X) — Aji(X) (i , j = 1, . . . , m) are of class C1 in G, and

172 J . B o c h e n e k

for all system of vectors £* = {hi, • • • , hn) (i = 1, m), a > 0 is a real constant, and TJ{X) = (иг{ Х ), un{X)). We shall also consider a gen­

eralized boundary condition (cf. [1]), which in the case where the bound­

ary F{G) is regular may be written in the form

(2) ^ Д - Я ( Х ) Е 7 = 0 on F ( G ) - r , £7 = 0 on Г , dv

where Г denotes an (m—1)-dimensional part of F{G) {Г being connected or not); in extreme cases Г may be the whole boundary of G, or the empty set. Here K ( X ) is a symmetric matrix which is continuous and positive definite in G, dU/dv is the transversal derivative of £7 with respect to system (1), i.e.,

(3) d v

dv —-cos(w , Xj),

OXi n being the internal normal to F(G).

We shall consider the eigenvalues and eigenfunctions corresponding to system (1) and condition (2); we shall shortly say: eigenvalues and eigenfunctions of problem (1) (2). The boundary condition (2) comprises as special cases all boundary conditions useful in applications.

Therefore we shall now explain the meaning of condition (2). Follow­

ing [1] we introduce some linear function spaces and some bilinear functionals on these spaces. Namely we put

(4) Я(Ф,Ф) = дгр

dXj + d>Q(X)'p\dX+ j 0K{X)WdS, -* F(G)-r

Н(Ф, W) = f ФР (Х)ШХ , о

where Ф = (<рг, . . . , <pn), W = (у)г, ipn). The functionals (4) and (5) are defined as follows. Given Ф and Ф, we consider an increasing sequence of regular domains Gn whose closures are contained in G and convergent to G. Suppose that expressions (4) and (5) are defined for Ф and W and every Gn, n = 1, 2, ... (In (4) the second integral is taken on F(G n) —Г п, where F n denotes a part of F(G n) such that Г п ~^ Г for n -> oo.) If these expressions have finite limits for every sequence {Gn}, these limits are the desired values of В and H , respectively. From (4), (5) and from the assumptions on the coefficients of system (1) it follows that the func­

tionals D and H are symmetric, i.e., В(Ф, W) — B(W, Ф) and Н(Ф, W)

= Н{Ф, Ф). Let В(Ф) = В { Ф , Ф) and Я(Ф) = Я(Ф, Ф). Observe that В(Ф) > 0 and Н(Ф) > 0 and

(6) Я(аФ + £Ф) = а2Я(Ф) + 2а£Я(Ф, Ф) + /52Я(Ф),

(7) Н ( а Ф Х ^ ) = а2-ЕГ(Ф) + 2 аД££(Ф, Ф) + £ 2Я(Ф).

The equality Н(Ф) = 0 may occur only if Ф = 0. (6) and (7) are valid for arbitrary real numbers a and (3 and for all functions Ф, W for which D and H are defined.

D efin it io n s . Ś8 denotes the space of all functions Ф of class <7°

in G such th a t Н(Ф) < oo (for a definition of a function of class Gn a (n ^ 0), see [1]). S? denotes the space of all functions Ф of class Cl in G such th at Н(Ф) < oo and В(Ф) < oo. S£x denotes the subspace of of functions Ф such th at Ф(Х) = ⁰ at all points of G whose distance from Г is less than or equall to some e > 0 . denotes the subspace of of functions Ф for which there exists a sequence ФРеЗ?х such th at Н(ФР— Ф) -> 0 and

Р(Ф„ — Ф ) 0 for v —> oo.

In the sequel the boundary condition U = 0 on Г will mean that U €&2. Let tF denote the subspace of ££ consisting of all functions Ф of class (72 in G such that Ъ{Ф)ев8.

We want to specify the meaning of the boundary condition dU/dr —

—K ( X ) U — 0 on F ( G ) ~ r . We consider a regular domain GB whose closure is contained in G, such that the distance between F(G e) and F(G) is less than e. In a way similar to that in [1] one can prove that

(8 ) r i dФ \

Я е(Ф, W) + H , { P - 1 L { 0 ) 9 m J v y — - К Ф ) ds = 0,

where Фе^-, We ЛР1, and Г е denotes the set of points of F{G e) whose distance from Г is less than or equal to e. Ds and H e are defined as func­

tionals D and H, respectively, by integration over the domain <re; F ~ l denotes the inverse matrix of P. Now let e —> 0 in (8). Since the limits

ЩФ, W) = ИтДДФ, V) and Л(Ф, W) = К т Я в(Ф, W),

e—>0 e—*-0

exist, there also exists the limit

J i dФ

dv № ) d S — lim . x ,

1 - W -r. ' * ^f w ( ^ - К Ф I dS.

F ( 0 ) - r

The involved limits are related by the formula

(9) Р(Ф, Ф )+ Я (Р -1Р(Ф) ЙФ

dv —КФ\ dS = 0.

One can prove that if (9) is valid for every ФеЗР and WeJ£x1 then it is

also valid for Ф е ^ and We S£2 (compare an analogous theorem for one

equation in [2]).

174 J . B o c h e n e k

The boundary condition d U f d v - K ( X ) TJ — 0 on F ( G ) ~ r for U is now defined by the requirement that the equality

( 10 )

I W

(°)-r

I dTJ

X&A -К (X) u

be valid for all The boundary condition (2) is defined by the requirements that TJ and (10) be valid for all We&2.

will denote the space of all functions Ф & satisfying (2) in the above sense.

Be ma r k 1. If the boundary F(G) of G is sufficiently regular (for instance, if F(G) is a surface of class Cl), if TJ is of class C1 in the clo­

sure G of G, and if U satisfies (2) in the ordinary sense, then TJ e^rK r (G).

Indeed, it is obvious that TJ satisfies (10) for all The proof of the fact that TJ еЛ?2 is analogous to the case of one equation (cf. [1]).

§ 2. Eigenvalues and eigenfunctions of problem (1) (2)

1. We define eigenvalues and eigenfunctions of problem (1) (2) in the following way (variationally): the first eigenvalue A of problem (1)(2) is

(11) ^. ЩФ)

mm--- Н(Ф)

and the first eigenfunction TJX is a function Ф at which the minimum (11) is attained. Having defined eigenvalues Xx, . . . , l n and corresponding eigenfunctions TJX, . . . , TJn we put

( ¹² ) _Ч 1+1 . ЩФ)

m]n — Фе*гп Н(Ф)

where X n is the subclass of ^j S?2 consisting of the functions Ф satisfying the orthogonality conditions Н(Ф, Ui) = 0 for i = l , . . . , w ; TJn+1 is a function ФеУГп at which minimum (12) is attained.

We shall need the following assumption:

H ypo th esis Z. Given (1) and (2), there exists a sequence of eigen­

values

(13) 0 < l x < A 2 < A 2 < . . . and a corresponding sequence of eigenfunctions (14) Ux(X), U2(X), Z7,( * ) , . . . which belong to SF.

We do not know whether the hypothesis Z is satisfied under our

assumptions on the coefficients of system (1).

2 . It is clear that the eigenvalues form an increasing sequence of non-negative numbers. In order to investigate further properties of eigen­

values, we shall now give another definition of eigenvalues of problem (1) (2). "

Let тГп denote a set of n functions V x(X), . .. , Vn(X) belonging to the space Ś8 and let

d [ r n-] . D(U) mm--- и * я н Ю

where J t n is the subclass of consisting of functions U satisfying the orthogonality conditions E ( U , Vi) — 0 for i — 1, n.

T h e o r e m 1. I f hypothesis Z and the above assumptions are satisfied, then

Л&+1 — sup {d \ fn\:

The proof of this theorem is quite similar to the proof of an analogous theorem in the case of one equation (cf. [2], p. 405 or [4], p. 289). The proofs of the following theorems are also similar to those for one equation and are omitted.

T h e o r e m 2. I f {pn}, {Яп} and {vn} are the sequences of eigenvalues of the system (1) with boundary condition U = 0 on F(G), with boundary condition (2), and with boundary condition dTJjdv = 0 on F(G ) (in the generalized sense), respectively, then

Vn ^ 4г ^=5 Pn -- 1 ) 2 , ... ) .

T h e o r e m 3 . Let {4^} be a sequence of eigenvalues of the system (1 )

with boundary condition TJ = 0 on F( G X), where Gx c G, then

where {4t} is the sequence of eigenvalues of problem (1) (2).

T h e o r e m 4 . I f the matrix P 2(X )—P 1(X) is positive definite in G and if { ? $ } and {4^} are the sequences of eigenvalues of problem (1) (2) where P ( X ) = P 2(X) and P ( X ) — P x(X), respectively, then 4^ < 4P (n = 1 , 2 , . . . ) .

T h e o r e m 5 . I f the matrix Q2(X) — Q1(X) is positive definite in G and if {4^} ап<% {4^} are the sequences of eigenvalues of problem (1)> (2), where Q ( X ) = Q 2(X) and Q ( X ) — Q1(X), respectively, then 4^ ^ 4»*

(n = 1 , 2 , . . . ) .

§ 3 . Lagrange’s Lenuna. I f the function F ( X ) — (fx(X), . . . , / » (X)) belongs to the space Ś8 and if

(1 5 ) j F ( X ) 0 ( X ) d X = ^{0 ,}

a

1 7 6 J . B o c h e n e k

where Ф(Х) — {срх{Х), . .. , cpn{Xj) is an arbitrary function of =W2, then F ( X ) = 0 in domain G.

Proof. Let Ф(Х) — (0, . . . , 0, (fi(X), 0, . .., 0) belong to Then (15) may be written in the form

(16) / Л ( х м х ) < г х = о.

a

It is known ([4], p. 247) that (16) implies fi{X) = 0 in G.

T h e o r e m 6 . I f hypothesis Z is satisfied, then each function Un(X) of sequence (14) satisfies equation (1) with p = Xn (n = 1 , 2 , ... ). More­

over, Un( X ) e ^ K>r(G).

Proof. To begin with, observe that if Ф{Х)е&гКгГ{&) satisfies (1) for p = t, then by (9)

(17) П(Ф, Ф ) - Ш { Ф , W) = 0,

for every W e 3 ? 2. We shall now show that if (17) holds for a fixed Ф { Х ) е & г

and for every WeSP2, then Ф { Х ) satisfies (1) for p — t and Ф { Х ) satis­

fies (10) for every We£t?2. Indeed, by (9) and (17) we have

Я (<Ф +Р-1(Х)Х(Ф) dФ

dv -К {Х )Ф \ ds = o.

Since W in is arbitrary, Lagrange’s Lemma yields L{Ф)-\-tP{X)Ф = 0 and F ( G ) - r / ^W

1йФ

\ I h К(Х)Ф) dS = 0.

Therefore all that remains to be shown is that

(18) D (U n, W ) - l nH ( U n, W) = 0, n = 1 , 2 , . . . ,

for any W e We shall prove it by induction. If n — 1, let W be an arbitrary function of S£2, and let r be an arbitrary real number. Put ф = TJ-^Ą-rW. Then by (11) В{Ф) > ХхН{Ф). Therefore by (6), (7) and by the definition of TJ1 we get

2 r [ D ( U 1, W ) - X xH { U 1, W ) ] + r 2[ B ( W ) - ^ H ( W ) ] > 0 . Since r is arbitrary, this is possible only if

B ( U 1, W ) - ? l 1H ( U 1,W) = 0.

Let us now assume that (18) holds for n = l , . . . , s and for any

W e £ ? 2. Because of (12) we have В ( Ф ) > Х3+1Н ( Ф ) , Ф = На+1 + тЖ, г

being an arbitrary real number and W an arbitrary function in . Hence as in 1°,

B ( U 8+ 1, W ) A,8+1H ( U S+ 1, TP) — 0.

We have to show that this equality holds also for every Indeed, given let aly ..., as be real numbers such that W = WĄ- аг TJx-\-...

...-\-as Us belongs to X s. This is always possible if we denote щ = — H(W, Ui)IH(Ui), i = 1, For such W, by the induction assumption and the symmetry of T> and H, we get

D {U S+x, W)-Xs+lH { U s+l, V) = 0, Q.E.D.

§ 4. Completness of eigenfunctions of problem (1) (2)

Th e o r e m

7. I f hypothesis Z is satisfied, the sequence (13) of eigen­

values of problem (1) (2) tends to infinity for n -> oo .

Proof. By Theorem 2 it suffices to prove that the sequence {vn}

of eigenvalues of Neumann’s problem (i.e., with boundary condition dU/dv = 0 on F ( G )) tends to + oo. By Theorems 4 and 5, and by the assumption on quadratic form of system (1) we get В(Ф)1Н(Ф)

>X>0(Ф)1Н0(Ф), where

r vn дФ дФ

г

Д,(ф) = а > — — dX, Н 0(ф) = Р ФФdX,

% ) C/OCd (Jt/Uf

О ^{г = 1 г} 1 G

a is the positive constant appearing in the assumption on quadratic form of system (1); P is a positive constant satisfying the inequality ФР{Х)Ф sśC ФРФ, and Ф is arbitrary function in IP. Let us observe that the functionals D0 and H 0 correspond to the following system of equations (19) AfpiX p — (pi = P 0, i = l , . . . , n

a

with the boundary condition dqpijdn = 0 on F(G), i — 1 n. Therefore the eigenvalues of Neumann’s problem for system (1) are not less than the corresponding eigenvalues of Neumann’s problem for the system (19).

Consequently every eigenvalue of system (19) is an eigenvalue of a single equation of the form

(20) AuĄ-p — и — 0 P

a

with boundary condition du/dn = 0 on F(G). We see that the sequence of eigenvalues of the system (19) may be received from the sequence of eigenvalues of equation (20) by repeting each eigenvalue of equation (20) a finite number of times, respectively.

It is known ([2], p. 424) that the sequence of eigenvalues of equation (20) tends to + ^{o o} for n -> oo, therefore the sequence of eigenvalues of system (19) also tends to infinity. Denoting by {vn} the increasing sequence of eigenvalues of system (19) with boundary condition of Neu­

mann’s type on F(G), we infer that vn < Xn, n = 1 , 2 , ... This yields Theorem 7.

Roczniki PTM — P race M atem atyczne X II ¹²

178 J . B o c h e n e k

Now we shall use Theorem 7 to prove the following theorem:

T heorem 8 . The set of eigenfunctions of problem (1) (2) form a complete orthogonal system in the domain G.

Proof. Let {TJn{X)} be the sequence of eigenfunctions of problem (1) (2) normalized so that H(U n) = 1 , n = 1 , 2 , . . . and let {Xn} be the corresponding increasing sequence of eigenvalues. Let F ( X ) be any con­

tinuous function in G. Let cn {n = 1 , 2 , ...) denote the Fourier coeffi­

cients of F { X ) with respect to {Un(X)} with weight P(X), i.e., let c„ = j F ( X ) P ( X ) V n(X) dX, « = 1 , 2 , . . .

We shall prove Parseval’s identity:

00

/ F ( X ) P ( X ) F ( X ) d X = ^ e l ,

n = 1

or, in another form, H ( F ) = ]?с2п. Let F ( X ) be a function of SFX. Let

П

Wn = F ( X ) — CiUi(X). We shall prove that limffC^) = 0. If к < n we have %=i

П

B(Wn, Vk) = H { F , Vk) + 2 « i B { V t, V*) = e » - « i = 0.

i—l

Hence H(Wn, JJk) = 0 for к < n. In virtue of (17), for any function Ф(Х) ejSfj we have В{Ф, U к) — ХкН(Ф, TJk) = 0, к = 1 , 2 , ... Consequently D{Wn, TJk) = 0 for к < n. By definition of eigenfunction Uk(X) and by (12) we get

(21) f l ( PnK l / V i D W .

On the orher hand П D {F ) = D[4>n+ 2 °V i)

г-1

n n n

D(Wn) + 2 2 t CiD('Fn, Vi) + 2 c*D ^lTt'> = B C F n) + 2 c*Xi-

г—1 г=1 г=1

n

0 ^ D { W n) = B ( F ) - 2 » U i < I > ( F ) , i—l

for Xi > 0, i = 1 , 2 , ... From (22) it follows that the sequence {D(Wn)}

is bounded. By Theorem 7, lim/lw+1 = + oo, hence by (21), limH(Wn) = 0.

This equation is equivalent with Parseval’s identity for {Un(X)}.

Therefore

(22)

Let now F ( X ) be any continuous function in G. We can approxi­

mated it by a function Ф{Х)е£Р1 in such a way that (23) H ( F — Ф) < —, e > 0.

4

For this purpose it is sufficient to use the construction which is given in [4], p. 305, for each component of Ф{Х). Let yn be the Fourier coeffi­

cient of Ф{Х) to respect {Un(X)} and let П

c , = ] f y k Vk{X).

k = 1

In virtue of the first part of the proof there exists a number N such that Н (Ф —оп) < e/4 for n > N. Thus, by (23) it follows

H ( F — Sn) X H { F — on) < e for n > N , П

where Sn = ^C iU i(X ). Hence limH ( F —Sn) = 0. This implies Parseval’s г ^{= 1}

identity for the function F ( X ) with respect to the sequence of eigenfunc­

tions of (1) (2).

The following statements are simple consequences of Theorem 8.

C orollary 1. The sequence of eigenvalues of problem (1) (2) contains all eigenvalues of this problem.

C orollary 2. Every eigenvalue of (1) (2) has finite multiplicity.

C orollary 3. Every function F ( X ) continuous in G can be expanded in a series of eigenfunctions {Un{X)} of problem (1) (2), which converges in the mean with weight P (X ), i.e.,

(24)

П

lim h ( F ( X ) - У с к и к(Х)\ = 0.

Еещ агк 2. The above expansion (24) may seen to depend on the matrix P (X ). Yet, since P (X ) is continuous and uniformly positive definite in G, the equality

n 1 n

lim f I f ( X ) ~ У ck Uk(X)] P (X ) { F i x ) - У ck Uk(X)] dX = 0,

n ^ c o G * = 1 jfcTl

is possible only if

П П

lim f p ( X ) - Y c k Uk(X)\ [ .F ( X ) - Y c k Vk(X )]d X = 0.

tZi tZi J

Now, let <Pk{X) = a1?41)(X) + . . . -f anU^iX), where Uk{X ) =

— ( r ^ ( Y ) , . .. , (Y )), Tc — 1 , 2 , . . . , are the eigenfunctions of (1) (2),

and oq, ..., an are real numbers fulfilling the condition a\X- • - X al, > 0.

180 J . B o c h e n e k

Let {ipn{X)} denote a sequence obtained from the {(pn{X)} by removing functions which are linearly dependent on remaining functions and or- thogonalization.

T h e o r e m 9 . The sequence {грп(Х)} is a complete system in the class of continuous functions in domain G.

Proof. Let f (X ) be any function continuous in G, and orthogonal to each function y>n(X). Then the function/(X) is also orthogonal to each function <pn(X). The function F ( X ) — (cq/(X),..., anf(X )) is obviously continuous in G and

J F ( X ) U n(X )d X = 0, n = 1 , 2 , ...

о

This means that F ( X ) is orthogonal to, all eigenfunctions of (1) (2).

Since {U n{X)} is a complete system in the class of continuous functions, F (X ) = 0 in G. Thus /(X ) = 0 in G. This concludes the proof (cf. [6], p. 164).

§ 5. Multiplicity of eigenvalues of problem (1) (2) and a theorem of Sturm’s type. In the case where the system (1) reduces to a single equa­

tion, the first eigenvalue of the problem is simple (see [1]). This means that all the eigenfunctions corresponding to first eigenvalue are linearly dependent. Moreover, there are examples (see [2]) of eigenfunctions problems in which all eigenvalues are simple. It is easy to give an example of a problem of the form (1) (2) for which all eigenvalues are R-fold at least. To this purpose let us denote X i?(X) — а^(Х)Е (i , j = 1, ...,m), Q(X) = q { X ) E , P (X ) = q { X ) E , K { X ) = k { X ) E , where a*,(X) (i , j

= 1, . . . , m) , q{X ), q (X), k(X) are the scalar functions sufficiently regular in domain G, and E is the unit matrix of the rank n. Then (1) can be written in the form

( 2 5 )

i,j = 1

[ dus

aij{X)~dXj q{X)us + juQ(X)us = 0 with the boundary condition

du*

dv k (X )u s = 0 on F (G )—r , us = 0 on Г , where s = 1, ... , n. Let Я be an eigenvalue of the equation

vw d f m d u ~I

(

2 6

2 j \ - q ( X ) u + y g { X ) u =

г,7 = ^{1 1} L ¹ -I with the boundary condition

(27) - U- — k (X )u = 0 on F (G )—r , и — 0 on Г,

dv

and let u(X) e the corresponding eigenfunction. Then A is also an eigen­

value of (25) corresponding to the eigenfunctions

Ui = (u, 0, 0), U2 = (0, u, 0, . .. , 0), TJn = (0, ..., 0, u) which are linearly independent.

In [1] there are some theorems of Sturm type for the problem (26) (27) which correspond to the following theorem:

T h e o r e m 10. I f 1° Як, Хг are eigenvalues of problem (1 ) ( 2 ) and Kk < A*, 2° to the eigenvalue Хг there correspond n linearly independent functions Uifi{X) (i = 1, . .. , n), 3° the matrix V composed of the functions и ^ (Х )

( ^{Ibb ^ i}

У i y - — У""1 (i = l , . . . , n ) j=i ufyI

are symmetric, then in every nodal domain of eigenfunction Uk there exists a zero-point of det V.

Proof. Since each function Uiti(X) (i = 1, . .., n) satisfies (1) and (2) for p — h we get

(28) n 9 Г dV l

i,i= l

r IdV \ „

and J ¥/{—---K V \dS = 0 for any function 4* and V . Similary

F(G)~ Г ^' f

(29)

n а г

duk~\

г,j = l 1

Let Gk denote the nodal domain of the function Uk and let det V Ф 0 in the closnre Gk of Gk. Then the assumption 3° of Theorem 10 is satisfied in domain Gk. Multiplying both sides of the equality (28) by V~l TJk from the right and by Uk from the left, multiplying the equality (29) by Uk from the left, and combining the resulting the equations we obtain

+ ( h - h ) U kPUk = 0.

By assumption 3° of Theorem 10, we get -Ay--- V~ dV

1 dxi ■"*)]

dV

dxi v -luk

+ (Afc —

— 0.

182 J. Bochenek

Integrating both sides of this equation over Gk and applying the boundary conditions for the function Uk and for the matrix V we obtain the equality

dV , \

— V ' l Uk\dX +

dxj /

+ ( 4 - Ai ) J ^{VkP V ki X = 0.}

Gk

The left-hand side of this equation being negative, we get contradiction.

Therefore there exist a zero-point of the d etF in Gk.

Е е mark 3. In the proof of Theorem 10 we applied a method of L. M. Kuks [5], who used it in a proof of a similar theorem for a more special strongly elliptic system with boundary condition of Diriehlet’s type.

R eferen ces

[1] J . B o c h e n e k , On some problems in the theory of eigenvalues and eigen- junctions associated with linear elliptic partial differetial equations of the second order- Ann. Polon. Math. 16 (1965), pp. 153-167.

[2 ] R. C o u ra n t und D. H ilb e rt, Methoden der mathematischen Physilc I I , Berlin 1937.

[3] M. K r z y ż a ń s k i, Równania różniczkowe cząstkowe rzędu drugiego I, Warszawa 1957.

[4] — Równania różniczkowe cząstkowe rzędu drugiego I I , Warszawa 1962.

[5] L. M. K u k s, Теоремы качественной теории сильно эллиптических систем второго порядка, Uspehi Mat. Nauk 17, 3 (1962), pp. 181-184.

[6] S. S a k s, Zarys teorii całki, Warszawa 1930.