This article was downloaded by: [University of North Carolina]On: 05 October 2014, At: 15:02Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

Complex Variables, Theory and Application: AnInternational Journal: An International JournalPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/gcov19

A similarity principle for pascali systemsJ. L. Buchanan aa Department of Mathematics , U.S. Naval Academy , Annapolis, MD, 21402Published online: 29 May 2007.

To cite this article: J. L. Buchanan (1983) A similarity principle for pascali systems, Complex Variables, Theory andApplication: An International Journal: An International Journal, 1:2-3, 155-165

To link to this article: http://dx.doi.org/10.1080/17476938308814012

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”)contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensorsmake no representations or warranties whatsoever as to the accuracy, completeness, or suitabilityfor any purpose of the Content. Any opinions and views expressed in this publication are the opinionsand views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy ofthe Content should not be relied upon and should be independently verified with primary sources ofinformation. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands,costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly orindirectly in connection with, in relation to or arising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial orsystematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution inany form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Comp1e.x Variables, 1983, Vol. 1, pp. 155-165 0278-lO77/83/Ol02-Ol55 $18.50/0 0 Gordon and Breach, Science Publishers, Inc., 1983 Printed in United States of America

A Similarity Principle for Pascali Systems

J. L. BUCHANAN*

Department of Mathematics, U.S. Naval Academy, Annapolis, MD 21402

Communicated by Klaus Habetha

(Received July 4, 1982)

It is shown that when the matrices A and B have bounded support, any solution to the Pascali system a,w + A w + Bw = 0 of dimension n has the representation w = S+ where S is a nonsingular matrix continuous in the complex plane, and + is a vector analytic in the domain of w . Preliminary to this a generalization of Liouville's theorem is established.

A PascaIi system is an elliptic system in the plane which can be put into the normal form

for z E D, an open connected set. Here a, = (1/2)(a, + ia,,), A : = (aik);fk= B : = (bjk);fk= and w : = (w,),"= with all entries complex

valued. For the conditions under which the system (1) is the normal form for a 2n X 2n real elliptic system see Wendland [13]. The system (I) was first studied by Pascali [lo], [11] for arbitrary n. For n = 1, (1) is the Bers-Vekua equation (Bers [2], Vekua [12]). In this case (n = I), with suitable assumptions upon the coefficients A and B,

* The author gratefully acknowledges the support of the Naval Academy Research Council for this research.

155

Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Car

olin

a] a

t 15:

02 0

5 O

ctob

er 2

014

156 J. BUCHANAN

the similarity principle is valid: every continuous solution to (1) ( n = 1) has the representation

where 1C/ is an analytic function in D and w is continuous and bounded in C. In general w depends upon the solution w. When D = C the similarity principle implies Liouville's theorem: continuous solutions to (1) which vanish at infinity must vanish identically.

Pascali's papers [lo], [I I] contain attempts at proofs of the similar- ity principle for the case of general n, however they are incorrect. As pointed out by Habetha [8] only a local similarity is established. In [8] Habetha asserts that for the following Pascali system no similarity principle exists. Let D = C, := {z : IzI < 11, B - 0, and

However, Habetha's proof of this is also incorrect. What Habetha's example does establish is that Liouville's theorem, as stated above, is not valid for arbitrary n. Let A be extended to all of C by A = 0 in C - C,. Then (1) has the continuous solution

which vanishes at infinity. In this paper it will be shown that there is in fact a similarity

principle for Pascali systems, one which does not imply Liouville's theorem. For other instances in which similarity principles (local or global) obtain for various types of elliptic systems see Gilbert [4], Kiihn [9], and Begehr and Gilbert [I]. A Liouville theorem for a class of elliptic systems in the plane can be found in Gilbert and Hile [5]. All of these results assume that the matrices A and B have lower triangular form. Such a triangular form will not be assumed in this paper. For certain elliptic systems a function theory which is not predicated upon either the similarity principle or Liouville's theorem has been developed by Bojarski [3] and extended by Goldschmidt [6].

Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Car

olin

a] a

t 15:

02 0

5 O

ctob

er 2

014

PRINCIPLE FOR PASCAL1 SYSTEMS 157

1. THE LOCAL SIMILARITY PRINCIPLE

In this section we prove the local similarity principle of Pascali and Habetha. A consequence of this, the unique continuation property, is necessary for the global result. Preliminary to this we introduce the fundamental integral operator and its differentiability and embedding properties. The coefficients A and B are assumed to be in L P ( ~ ) , p > 2, where D is a bounded domain. Outside of D we set A, B - 0. Since the form of (1) is not affected by the transformation Z = kt, k a real number, we assume without loss of generality that A and B vanish outside of c o , the closed unit disk, Let w0 be a solution to (1) in the Sobolev sense (see Vekua [12]) which is continuous in C,. The equation ( I ) may be written

where

Thus it suffices to demonstrate the local similarity principle for

The Pompieu operator is

The domain G is assumed to be bounded.

THEOREM 1 (Vekua [12]) I f f E L P ( ~ ) , p > 2, then T$ is Holder continuous in C, analytic in C - c , and vanishes at infinity. Moreover aZTGf = f .

The L2-norm for vector functions will be denoted by

Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Car

olin

a] a

t 15:

02 0

5 O

ctob

er 2

014

158 J. BUCHANAN

Here "." means Euclidean inner product. For a matrix C = (c,,)~,=,

THEOREM 2 (Vekua [12]) For a matrix C E L P ( ~ ) , p > 2, the opera- tor

is a compact mapping of L ~ ( G ) into itself. Moreover

and the constant M ( G ) may be made as small as desired by shrinking the diameter of G. When f is continuous, PGf is Holder continuous in C, and when f E L ~ ( G ) there is an m for which P" is Holder continuous in C.

A solution to (3) which is continuous in a domain G must satisfy

where cp is analytic in G and continuous in G. Moreover, in view of Theorem 1 , (6) gives a continuous extension of w to any domain to which cp can be analytically continued. In particular if cp is the restriction to G of an entire function, then w is continuously extend- able to and is a solution to (1) with A , B - 0 outside of G. That any L'-solution to (6) is continuous in if cp is continuous in G follows from an iteration argument based upon the continuity results of Theorem 2. See Goldschmidt [7]. We now state and prove the local similarity principle.

THEOREM 3 (Pascali [lo], [ I I]; Habetha [8]) For C E Lp(Co), p > 2, and vanishing outside of Co let w be a continuous solution to (3). For a given zo E C, w has the representation

w = S$ (7)

Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Car

olin

a] a

t 15:

02 0

5 O

ctob

er 2

014

PRINCIPLE FOR PASCAL1 SYSTEMS 159

where S is a nonsingular matrix which is continuous in C, and + is analytic in a neighborhood of zo.

Proof Let zo be a given point in C and define D(zO,r) = {z : Iz - zOI < r). Let

For r sufficiently small the operator TD(zo,r)Cw is a contraction. Thus the equation .

where eJ = (0, . . . , 0,1,0, . . . , 0)' (t denotes transpose) has a unique solution d which is continuous in C. Let S be the n x n matrix (s', . . . , sn). It is easily verifiable that

in C and moreover that det S(w) = 1. Here "det" means determinant and "tr" the trace. From the similarity principle (2) for the Bers- Vekua equation it follows that det S # 0 in C. Since S is a solution to (3) in D(zo, r), it follows that a,(S -'w) - 0 in D(zo, r) which gives the local similarity principle. H

COROLLARY (the unique continuation property) If a solution w to (3) is continuous in a domain D and vanishes on an open subset of D, then w-Oin D.

Proof By continuity {z E D : w(z,F) = 0) is closed relative to D. By the local similarity principle (7) and the unique continuation property for analytic functions the same set is open. Hence it must be D. H

2. A GENERALIZED LlOUVlLLE THEOREM

As noted in the introduction a strict generalization of Liouville's theorem is not possible. In this section we give a version of Liouville's theorem which is valid for general Pascali systems and which implies the stricter result for n = 1.

Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Car

olin

a] a

t 15:

02 0

5 O

ctob

er 2

014

160 J. BUCHANAN

In general the equation

may have nontrivial solutions for $I -- 0. Habetha's example cited above illustrates this. Extended to C such solutions correspond to continuous solutions of (3) which vanish at infinity. Let A' denote the space of such functions. By Theorem 2 the Hilbert space Fred- holm theorem is applicable with the inner product

[ w ] := Zdo,.

The adjoint operator under this inner product is

where z E C, and t denotes matrix transpose. Following Bojarski [3] we establish a correspondence between the

null space of M* and solutions to the adjoint differential equation

which vanish at infinity by

Let M* denote the space of solutions to (I 1) which vanish at infinity. By the Fredholm theorem M and M* have the same dimension N over C.

Any element w of M is analytic in C - c, and hence has the expansion

where a > 1 and the vector a, # 0. The possibility of w - 0 in C - c, is precluded by the unique continuation property. The number a is no greater than N for if not { z % ) ~ , , would be an independent set of solutions in A' of dimension N + 1. We will refer to -a as the degree at infinity of w and to a, as the initial vector.

Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Car

olin

a] a

t 15:

02 0

5 O

ctob

er 2

014

1 PRINCIPLE FOR PASCAL1 SYSTEMS 161

LEMMA There are sets of function d : = {QJ)y: , c A" and d * : =

{ 8 * j ) y j 1 C M * , with NO + NO* < n, No, N,* < n, having degrees at infinity - 9 and - pi and initial vectors {a&)y: and { bi,)?! , respec- tively such that { Z ' % J ) ~ : ~ ~ ~ , and { z % * ~ ) r ~ , ~ ~ = ~ , are bases for Jt/ and M * over C. Thus 2:: , a, = Xyi j3, = N. The vectors { a:, . . . , a?, b:, . . . , b,No*) form a linearly independent subset of Cn with a$ . b,k = O , j = 1 , . . . , N o ; k = I, . . .,NO*.

Proof Let - y > - N be the minimal degree at infinity possessed by any element of A". The set of all elements having degree - y form a subspace M - , of M . Let MI_, be a complementary subspace of M - , . The elements of MI_, possess minimal degree - y + 1 and the elements of degree - y + 1 form a subspace M - ,+ of ML, which then must have a complement ML , + , in M L , . Proceeding in this manner we obtain the decomposition M = M - , d3 . d3 M - , . If W E N is of degree - a , then W E N - , $ . . . @ N - , . Let 9 := { w - ~ ~ ~ ) J ' k / , , ~ = , be a basis over C for M where { w - J , ~ ) $ = , is a basis for M P j . Proceeding in the order (- y, l), . . . , (- y, k,), ( - y + 1, l), . . . , ( - 1, k , ) we eliminate from 9 all the elements - l,m which can be written

where pj,, is a polynomial of degree j - 1. We denote the set which remains by & : = { $J )y' .

Consider the set {a&~,~ )$ , , , , , of initial vectors of elements in 9. Suppose that = 0 for some set of complex con- stants {c j k ) with ch # 0. Then

is in M - , G3 . . . d3 M - ,. Consequently w has a representa- tion of the form (14) and is thus not in &; hence the initial vectors {a&}y:, of the elements of & form a linearly independent set.

We can carry out a corresponding construction for M * . We will denote the set obtained by &* := { $ * J ) y j , and the set of initial vectors by {bi,);!, . Let { - aj)y:, and { - j3,)yj I designate the de- grees at infinity of the elements of & and &* respectively. It is easily

Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Car

olin

a] a

t 15:

02 0

5 O

ctob

er 2

014

162 J. BUCHANAN

seen from (3) and (1 1) that for w E M , w* E N*, a,(w w*) - 0 in C. Since both vectors vanish at infinity, Liouville's theorem implies w . w* - 0 in C. In C - c, both w and w* have expansions of the form (13) and thus a i . bA=0 for j = 1 , . . . , No; I = 1, . . . , No*. Hence No + N,* < n. Since neither No nor N,* is zero unless both are

N o by the Fredholm theorem, No, No* < n. Moreover {a:, . . . , a, , b i , . . . , b,N"*) must be a linearly independent set.

Finally let us show that PP := {z%J)Y: , is a basis for M . Every k w E JY has the representation w = CJ, ,Ci= ,c,,w-J-~ for some c,,

E C. By virtue of (14) we can write w = CY2,p,wJ for some set of polynomials { p,);: I with the degree of p, less than 9 Upon rear- rangement we conclude that w = C ~ : , C ~ : ~ ~ ~ Z % J for some set of complex numbers (4,). Thus 2 spans A". To establish linear independence suppose that C?: ,C%&Z%J - 0 in C for some set (4,). In C - c, each 8J has an expansion of form (13). Thus in C - C,

From the linear independence of the set {a{);:, we conclude that for j = 1, . . . , No, 4 , q - k = 0 for k = 0,1, . . . , ( ~ j - 1 successively.

From the Lemma the generalized Liouville theorem follows at once. This result is similar to those of Vinogradov [13], [14], for the dimension of the space of solutions of polynomial growth for certain elliptic systems.

THEOREM 4 (generalized Liouville theorem) Every w E ~ f " has the representation

where $J E &, p, is a pOlynomial of degree less than 9 and no < n.

Note that No < n implies that No = 0 if n = 1. Hence w - 0 which is Liouville's theorem in the Bers-Vekua case.

Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Car

olin

a] a

t 15:

02 0

5 O

ctob

er 2

014

PRINCIPLE FOR PASCAL1 SYSTEMS 163

3. THE SIMILARITY PRINCIPLE

THEOREM 5 Let w be a continuous solution to (3) in some domain D with C = 0 outside of C,. There is an n X n matrix S which is continuous and nonsingular in C and a vector + which is analytic in D such that w has the representation

Proof Let s J = G J E ~ f o r j = 1 , . . . , N o . Let {ad}::,, {b6}y i1 , 9, and p, be as in the Lemma. A set of vectors { P ~ } J ~ ~ may be chosen to satisfy the following conditions:

N 1) { p & } ~ ; ~ { + , forms a basis for {a:, . . . , a, O , b:, . . . , b p } ' 2) for j = 1, . . . , N , * , p& E { a ; , . . . , a f o , b ; , . . . , bd-I ,

bg+ ', . . . , b p , p P + ' , . . . , P [ - ~ O } ~ .

Let

where the vectors p/, are as just indicated and m, = pj for j =

1, . . . , N,* ; m, = 0 for j = N,* + 1, . . . , n - No. The remaining coef- ficients of thepJ will be chosen so that Mw = pJ has solutions. By the Fredholm theorem the necessary and sufficient condition for this is [p j ,u] = 0 for each solution to M*v = 0. By Green's theorem, the Lemma, and (12) this is equivalent to

For (15) to hold we must have

Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Car

olin

a] a

t 15:

02 0

5 O

ctob

er 2

014

164 J. BUCHANAN

whenever k - /3, + mj - s = -1. Since mj = 0 for j = N,* + 1, . . . , n - No, this is possible only if k = /3, - 1 , s = 0. The require-

ment (16) is then p$ . 6,' = 0 which is satisfied by the choice 1) of p/, made above. For j = 1, . . . , N,*, m, = /3,. When s = 0 and I # j , (16) becomes p/, . b,' = 0 which is in keeping with 2) of the criteria for choosing Finally when s = 0, j = Ik - /3, + mj - s = k 0 and hence (16) is met vacuously. Since the b,' are linearly independent we can now solve

successively for pi , . . . , ph . Thus (16) can be satisfied. Let sJ+ NQe a solution to M w = pj in C,. Since pJ is entire, each

sJ+ has a continuous extension to all of C. Moreover sJ+ N o = pJ + O ( t - ' ) in C - co. Let S := ( s ' , . . . , s n ) . Then S is a matrix solution to (3). Moreover in C - co

and thus, by the choice of pJo, j = 1, . . . , n - No, detS(co) is a nonzero constant. By the same argument as in Theorem 3, det S # 0 in C. Thus if w is a solution to (3) in D, then a,(S -'w) = 0 in D.

References [I] H. Begehr and R. P. Gilbert, Boundary value problems associated with first order

elliptic systems in the plane, Contemporary Math., 11 (1982), 13-48. [2] L. Bers, Theory of Pseudoanalytic Functions, Courant Institute, New York, 1953. [3] B. B. Bojarski, Theory of generalized analytic vectors, Ann. Pol. Math. 17 (1966),

28 1-320, (Russian). [4] R. P. Gilbert, Pseudohyperanalytic function theory, Gesellschaft fur Mathematik

und Datenverarbeit, No. 77, Bonn (1973), 53-63. [5] R. P. Gilbert and G. Hile, Generalized hypercomplex function theory, Trans.

Amer. Math. Soc., 195 (1974), 1-29. [6] B. Goldschmidt, Funktiontheretische Eigenschaften verallgemeinter analytischer

Vectoren, Math. Nachr. 90 (1979), 57-90. [71 -- , Regularity properties of generalized analytic vectors in Rn, Preprint der

sektion Mathematik, No. 41. Martin-Luther-Universitat Halle-Wittenberg (1980). [8] K. Habetha, On zeros of elliptic systems of the first order in the plane, Function

Theoretic Methods in Partial Differential Equations, R. P. Gilbert and R. J. Weinacht, eds., Pitman, London, 1976, pp. 45-62.

Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Car

olin

a] a

t 15:

02 0

5 O

ctob

er 2

014

PRINCIPLE FOR PASCAL1 SYSTEMS 165

[9] E. Kiihn, ~ b e r die Funktionentheorie und das ~hnlichkeits~rinzip einer Klasse elliptischer Differentialgleichungssysteme in der Ebene, Dissertation, Universitat Dortmund (1974).

[lo] D. Pascali, Vecteurs analytiques generalises, Rev. Roumaine de Math. Pures et Appliquees, X (6) (1965), 779-808.

1111 - , Sur la representation de premiere espkce des vecteurs analytiques gene- ralises, Rev. Roumaine de Math. Pure et Appliqubes, XI1 (5) (1967), 685-689.

[12] I. N. Vekua, Generalized Analytic Functions, Pergamon Press, Oxford, 1962. 1131 V. S. Vinogradov, Power growth solutions of systems of elliptic type (Russian),

Complex Analysis and its Applications (Russian), Nauka, Moscow, 1978, pp. 120-125.

[I41 - , Uber Liouvillesche Satze fur Systeme von Gleichungen verallgemeinerter holmorpher Vektoren (Russian), Komplexe Analysis und ihre Anwendungen auf partielle Differentialgleichungen, Halle, 1980, 2, pp. 167-168.

1151 W. L. Wendland, Elliptic Systems in the Plane, Pitman, London, 1979.

Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Car

olin

a] a

t 15:

02 0

5 O

ctob

er 2

014