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Matrix criterions for the extension ofsolutions of a general linear system ofpartial differential equationsLe Hung Son aa Department of Mathematics, D. H. B. K. Institute of Technology,Hanoi, VietnamVersion of record first published: 29 May 2007.

To cite this article: Le Hung Son (1992): Matrix criterions for the extension of solutions of a generallinear system of partial differential equations, Complex Variables, Theory and Application: AnInternational Journal: An International Journal, 20:1-4, 265-276

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Matrix Criterions for the Extension of Solutions of a General Linear System of Partial Differential Equations

LE HUNG SON

Department of Mathematics, D. H. B. K. Institute of Technology, Hanoi, Vietnam

AMS No. 47A&, 47A62,35A(?i Communicated: K. Habetha (Recehzd 4pril 6. 1988)

1. INTRODUCTION

In this paper we shall consider the following linear system of partial differential equations.

m m e ) dui L ( ~ ) ( u ) : = C ~ A ~ ~ ~ = O , e = i ,..., L,

i=1 j=1 xi (1.1)

( e ) where A;; = const. for all i = I,. .., m, j = 1, .. ., n, C = 1, .. ., L. ui = ui(xl,. .., x n ) are (real) analytic on xl,. . . , x , and u = (ul ,..., urn) is the unknown function. For the class of solutions of such a system the following Uniqueness Theorem holds.

UNIQUENESS THEOREM Let u = (u l (x) , . . . , urn(x)) be a (real analytic) solution of (1.1) in a domain G c Wn, furtiter let a be a nonempty open subset of G. If

u = O for x E a then

u=O for all x E G.

Vtize system is elliptic, then every c3-solution of (1.1) is (real) analytic on XI,. . ., x,,. This means tizar the Uniqueness Theorem holds for sufficientIy smooth solutions of

(1.1)

DEF~NITION 1 Let u = {ul (x) , . . . , urn(x)) be a solution of (1.1) in domain G and ii = {ii,(x), . . . , iirn(x)) he a solution of the same syslem in another domain G with G 5 G 2 W n . Then ii is called a continuous extension of u if ii = u in G.

From the Uniqueness Theorem it follows that the extension ii of u is uniquely determined. In this paper we shall consider the following problem:

EXTENSICIN PROBLEM A Let G be a domain in Rn, C be an open neighbourhood of dG. When can every solution u oJ (1 .1) in C be extended continuous& to a solution of the same system in the whole of G?

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The solvability of problem A depends on the coeficients A!:) and on n, m, L.

We shall give criterions for the solvability of Problem A.

2. TWO MATRIX CRlTERlONS FOR THE SOLVABILITY OF PROBLEM A

At first we denote

A([) = (A,, (e) ),,,, e = l , . . . , ~ (2.1)

with

A(e' is a matrix of type m x n and Xi is a L-dimensional vector. If the vector Ai is given then we define the following matrix IUI' type rn x n:

Further we define the matrix

and the matrix

c = of type m2 x n.

THEOREM 2.1 Suppose that m 5 n and there exist m vectors X I , . . . , A, such that the following conditions hold:

(i) rankVi = 1 foralli= 1, ..., m (ii) rank23 = m

(iii) rankC = m

Then every (real) analytic solution u of (1.1) in C can be extended continuously to a solution of the same system in the whole of G .

Proof At first assume that there exist the injective linear mappings

and u' = A1(u),

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MATRIX CRITERIONS

where

If we denote

then we can see easily that G' is a domain in Rt'(J1, . . .,El,) and C' is an open neigh- bourhood of aG'. Since the functions ul(x), . . . ,um(xj are given in C and (real) an- alytic on XI;. . . , x, it follows that u;(<), . . . , LL;(~) are given in C' and (real) analytic on < l , . . . , L

From

it follows that u: depend on t i . Since uf(J) is given in a neighbourhood C' of G', one can extend this function to a real analytic function in the whole of G'. We denote these extensions of u:({) by

4 (0 , . - . , n k ( 0 (2.9)

and consider ic = h,'(D1),

where D = (61, .. .,fi;), ic = (iil,. . .,fim). It is easy to show that the functions tci are defined in the whole of G and real analytic on XI,. .., x,. On the other hand it is

Since u = (ul, .. . ,urn) is a solution of (1.1) in C, it follows

Further it is obvious that L(~)(D) is real analytic in XI,. . . , x,. Because of the Unique- ness Theorem and of (2.12) we get

~ ( ~ ) ( i i ) = 0 in the whole of G for C = 1,. . . , L. (2.13)

The condition (2.13) means that u = (iil,. ..,urn) is a solution of (1.1) in the whole of G. Hence, D is the extension of u to the whole of G.

The problem is now reduced to proving the existence of the injective linear map- pings A and Al.

We denote Q = [D!,!] l=l....,m . (2.14)

J =I, ..., n

From the supposition (i) of this theorem we have

rank231 = 1. (2.15)

Hence, it follows that there exists at least an element of Dl, which doesn't vanish.

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268 L. H. SON

Without loss of generality we may assume that

This means

where all # 0, all f 0.

Further we denote

From (2.15) it follows that there exist constants ?;!') such that

Here [D:), . . . , D ~ I is the kth row of the matrix Dl. From (2.17), (2.19) and (2.20) it follows

= ('1 = a , for k = I , . . . , rn, J'kj Yk Dlj (2.21)

j = 1, ..., 12.

Denoting (li 7k all = alk7 k = 1,. . ., m

we get

D:) = alkajl, k = 1, ..., m, j = I , . . . ,n . (2.23)

We have thus proved the existence of two m-tuples

and (a l l , a12,. . . t Ql l rn ) ,

which satisfy the conditions (2.18) and (2.19). By an argument analogous to the previous one, we can prove the existence of 2(m - 1) tuples of the following form

which satisfy the conditions

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From (2.25), (2.23) and the assumption (ii) of Theorem 2.1 it follows

where 1 < j e < m , e = 1 ,..., L. Because of (2.26) we get

det

From (2.231, (2.25) and the assumption jiiij it bl:ows

alklQll alkla21 . . '

'lk2@12 ' lk2a22 . ' ' rank .

where 1 5 kt < m, e = 1,. . ., m. By denoting

Q1 = (~11 , a21 ,...,@,I)

Q2 = (~12 , a 2 2 1 - . - 1 an2)

we get then a system of m vectors with n components

The condition (2.28) means that the vector systems {crl,cr2, .. . , a m ) is linear in- dependent. Because of m 5 n we may add to this system (n - m) vectors (with n-components)

0rn+1,.-.,0n

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such that the resulting system

{al,--.,arn.arn+~,...,an}

is lincar indepeiideni. Hence, i t foiiows that

det

where

We have thus proved the existence of n2 numbers di;, i, j = 1, ..., n, which satisfy the conditions (2.25) and (2.31).

Now we shall define the following coordinate transformations

Because of (2.27) and (2.31) the transformations (2.33) and (2.34) are injective lin- ear transformations.

From (2.33) and (2.34) it follows

Particularly

holds.

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MATRIX CRITERIONS 271

We now apply (2.35) with k playing the role of the role of j and j playing the role of e and get

Comparing (2.35") with (2.25) and (2.23) we get

The coefficients of the right hand side in (2.36) are the elements of the matrix Vi. Because of (2.3) the right hand side of (2.36) is a linear combination of L(')(u), . . . , L(',) j ill. Hence

We have thus got the injective mappings cf the type (2.4) and (2.5) which satisfy the condif on (2.8).

Thus the proof of Theorem 2.1 is complete.

If m, n are arbitrary we have the following

THEOREM 2.2 Assume rhal there &st m 1,ecroi-s XI , . . . , A, such that the subordinate muli-ik ;it, 3 and C mti.fy the following suppositioitr:

(i) rankVi = 1 (ii) rank B = m

(iii) rankC = 1 for i = 1,. ..,m.

Then every (real analytic) solution of (1.1) u = (ul ( x ) , . . . , um(x)) in C can be ex- tended conti~zuousij to a solution of the same system in the whole of G.

Proof By an argument analogous to that used for the proof of Theorem 2.1 we shall find the injective linear mappings of the form (2.4) and (2.5) such that these mappings satisfy the condition

du' A=(, ,..., % L o . Rl as1

Let ul, . . . , u, be the given real analytic functions in C, then the functions u;, . . . , uh are defined in C' and real analytic on (1,. . .,&. From (2.37) it follows that u;, . . ., uk do not depend on tl. Hence, ui, . . . , uk can be extended to real analytic functions in the whole of G'. We denote the extensions of u;, . . . , uk by a;, . . ., iih.

In the same way as in the proof of Theorem 2.1 we may show that ii = ~ , ' ( k ' ) is the extension of u in the whole of G.

From the assumption (i) (of Theorem 2.2) we have the existence of (all,alt,. . . , al,) and (all ,a21,. . . , ctnl), which satisfy

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and

a11 # 0, all # 0. (2.18')

Analogously because of assumption (i) we get 2(m - 1) tuples of the form

which satisfy the condition

The assumption (iii) means that

rank

From (2.38) it follows

where i = 2,. . ., m; e = 1,2,. . ., rz, 7( i3e) are constants. Because of (2.39) we obtain

f o r i = 2 ,..., m ; C = l , ..., m a n d k = l , ..., 12. Noting

Then, because of (2.40), we get

for i = 2,. . . ,mi 4 = I , . . ., m; k = 1 , . . . , i t . Because of the assumptions (i) and (iii) we thus obtain the numbers

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From (2.23') and (2.25') it follows that these numbers satisfy the condition

a i la j l = D!) for i, k = 1,. . . , m ; j = 1, .. ., n.

From the assumption (ii) we get

We may add to the number found above (n" n) numbers a j k , j = 2, ..., i z ; k = I , . . . , n such that

det # 0. (2.45)

Lanl (inn J

Further we define the coordinate transformations analogous to (2.33) and (2.34). Because of (2.44) and (2.45) these transformations are linear injective transforma- tions.

Because of these coordinate transformations and of (2.43) we obtain

for i = 1,. . . , m. (2.46)

The right hand side of (2.46) is a linear combination of c(')(u), . . ., L(=)(u). Hence, we get (2.37). The theorem is thus proved.

Remark 1 Theorems 2.1 and 2.2 remain valid for solutions of the system

where f ( e ) are given r'uncfons in G and real analytic on x l , . . . , x n . The proof is analogous to proof of Theorems 2.1 and 2.2.

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3. APPLICATION

In order to show an example where Theorem 2.1 can be applied we consider the system

I dul au2 -+ -=O 8x1 8x2

dul du2 --.--=o dx2 8x1

au, t3u2 - + - = o 8x2 8x4

dul du2 - + - = o \ ax4 ax3

where u; = 2,(xir xi, x3,x4) , t = : ,2 ,3 ,4 . . . . The system (3.1) has the form of (1.1) with rn = 2, ,r = 4, L = 4. From definition

(2.1) it follows

Because of m < n we may apply Theorem 2.1. From (2.3) it follows here

We can denote Dl by D and D2 by D' with

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From the condition (i) of Theorem 2.1 it follows rank23 = 1, this means

Hence, we choose A = [O, 0,1,11,

A' = [O,O,l,-I].

Then there is

It is rank23 = rankD1 = 1 and the condition (i) of theorem 2.1 is valid. Since

it follows

rankB = rank

rankC = rank

The conditions (3.6) and (3.7) mean that the assumptions (ii) and (iii) of Theorem 2.1 are valid. Thus we may apply Theorem 2.1. Hence it follows that every given real analytic solution u = (ul, u2, ug, uq) ~f (3.2) in C can be extended continuously

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References

[I] L. Hormander, An Introduction to Complex Analysis in Several Variables, Princeton, 1967. [2] Le Hung Son, Ein Fortsetzungssatz fiir raumliche holomorphe Funktionen, Math. Nachr, 1982. [3] Le Hung Son, Fortsetzungsdtze fiir holomorphe Vektorfunktionen und verallgemeinerte holomorphe

Vektorfunktionen, Tagungsbericht: Komplexe Analysis und ihre Anwendung auf partielle Differential- gleichungen, Halle, 1980.

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