169
CHAPTER 3 GENERALIZED SOLUTIONS FOR LINEAR PARTIAL DIFFERENTIAL EQUATIONS H1.
PRELIMINARY REMARKS
As mentioned in Part 1. the early history of the linear theory of the Schwartz distributions had known quite a number of momentous events, both f o r the better and for the worse. One of the first major successes was the proof of the existence of an elementary solution for every linear constant coefficient partial differential equation, which was obtained in the early fifties by Ehrenpreis. and independently Malgrange. Soon after, in 1954. came the famous and improperly understood. so called impossibility result in Schwartz [2]. Another, rather anecdotic event is mentioned in Treves [4]. who in 1955 was given the thesis problem to prove that every m
linear partial differential equation with ‘4 -smooth coefficients not vanishing identically at a point, has a distribution solution in a neighbourhood of that point. The particularly instructive aspect involved is that the thesis director who suggested the above thesis problem was at the time, and for quite a while after, one of the leading analysts. That can only show the fact that around 1955. there was hardly any understanding of the problems involved in the local distributional solvability of linear partial differential equations with
m
‘4 -smooth coefficients, see Treves [4].
As mentioned in Part 1. Chapter 3. Section 1. a very simple and clear negatiue answer to the above thesis problem was soon given by Lewy in 1957. who showed that the following quite simple linear partial differential equation
cannot have distribution solutions in any neighbourhood of any
point class.
x € R3.
if
f
€
‘40(R3)
belongs to a rather
large
The solvability of linear partial differential equations with m
’4 -smooth coefficients failed to be achieved even when later. the Schwartz J ’ distributions were extended by other linear
spaces of generalized
functions, such as the hyperfunctions.
Sat0 et.al., Hormander [2]. stance in 1967, in Shapira.
That failure was proved for in-
E.E. Rosinger
170
A s mentioned in Part 1. Chapter 3. Section 1 , a sufficiently general characteriza~ion of solvability, and thus of unsolvability. for linear partial differential equations with m
'4 -smooth
coefficients has not yet been obtained within the framework of Schwartz's linear theory of distributions. And that inspite of several quite far reaching partial results which make use of rather hard tools from linear functional analysis as well as complex functions of several variables. In view of the above i t is the more remarkable that f o r the first time ever in the study of various generalized functions, Colombeau's nonlinear theory does yield local generalized solutions for practically arbitrary s y s t e m s of linear partial differential equations. Furthermore, under certain natural growth conditions on coefficients, one can also obtain g l o b a l generalized solutions for large classes of s y s t e m s of linear partial differential equations, systems which contain as parm
ticular cases most of the s o far unsolvable linear, '4 -smooth coefficient partial differential equations, see Colombeau C3.41. Without going into the full details - which can be found in Colombeau C3.41 - we shall present in Section 3 the main results and a few illustrations. The systems of linear partial differential equations whose generalized solutions will be obtained within Colombeau's nonlinear theory, contain as particular cases systems of the form (2.3.2)
DtUi(t.x) =
1
aijp(t.x)DPu X J (t.x) +
l
PEPij
+ bi(t.x).
1
<
i
<
t , (t.x) €Rn+l
1
<
i
<
t . x E Rn
with the initial value problem Ui(O.x) = ui(x).
(2.3.3)
C Nn are finite, while a ijp, bi and ui are i.i m '4 -smooth. It is well known, Treves [2], that under very general conditions, arbitrary systems of linear partial dif-
where
P
0
' -smooth coefficient and initial ferential equations with 4 value problems can be written in the equivalent form (2.3.2). (2.3.3).
Our aim is to find generalized functions (2.3.4)
U i , .. . .U E '4(Rn+l)
171
Solutions of Linear PDEs
which in a suitable sense a r e solutions of (2.3.2) and (2.3.3). or even more general systems, see (2.3.35).
A basic remark which conditions much of the way such generalized solutions are and can be found is the following.
The system (2.3.2). (2.3.3) cannot
(2.3.4) i f
D,, D E
in general have solutions
and the respective equality relation
=
are considered in 9(Rn+l) in the usual way defined in The argument for that is rather classical - see Chapter 1. for details Colombeau [3] - and i t is based on a contradiction between Holmgren type uniqueness and general nonuniqueness rem
sults in the C -smooth coefficient case, see also Treves [2] and Colombeau [4]. Furthermore, i f in view of Chapter 2. Section 1. we replace the equality relation = by the equivalence relation or in 9(Rn+l), the system (2.3.2). (2.3.3) will still fail to have solutions (2.3.4). see again Colombeau [3].
Z
A way out from this impasse is to replace the partial derivaby the m o r e smooth partial derivatives tives DE in 9(Rn+l)
Dp defined next in Section 2. The effect of such a replaceh x ment is very simple, yet crucial. Indeed, a s seen n Chapter 1 , the partial derivatives DE in '% coincide with the clasm
sical partial derivatives when restricted to '& -smooth functions. Therefore, they are n o t sufficiently smooth in the following
well
known sense
m
that a n Y -type bound m
on a
4'2
m
-
smooth function does not imply any Y -type bound o n its classical partial derivatives. Contrary to that situation, the more smooth partial derivatives the property (2.3.15). 82.
Dp
on
h x
9(Rn+l)
SMOOTH PARTIAL DERIVATIVES
Suppose given a function (2.3.5)
h:(O,m)
--f
(0.m)
with
lim h(e) = 0 B
lo
such a s for instance (2.3.6)
h(e) =
l/en(l/e)
if
B
E (0.1)
arbitrary otherwise
will have
E.E. Rosinger
172
Such a function rtuatton rate.
h
satisfying (2.3.5) will be called a de-
Before defining the smooth derivatives, we note the following property
9
V
@(Rn) : J, E @(Rn). E
3
(2.3.7)
€
*)
>
0 :
diam supp J, = 1
**)
# = JI,
and for given 9 . one obtains J, and B in a untque way. 9 In view of that, we shall in the sequel often exchange with J,,, according to condition **) in (2.3.7). Given
1
<
<
i
n.
we define the h-partial dertuattue
D
(2.3.8)
xi
:
'8(Rn)
4
Y(Rn)
as follows: i f
F = f + 9 E Y(R").
(2.3.9)
f E d
then
D
(2.3.10)
xi
F = g + 9 E Y(Rn), g E ~4
where
and * is the usual convolution of functions. I t is easy to see that the above definition is correct, since indeed g E d while D F does not depend on f E $ in (2.3.9). xi
Obviously (2.3.11) yields
1 h(~) 1
g(9.x) =
Dx
f(J,,.x-h(e)y)J,(y)dy
=
Rn
(2.3.12) =
1
f(+e.x-h(e)y)Dy
R"
i
J,(y)dy. WP(Rn).
xE R"
173
Solutions of Linear PDEs
Given now
p = (p l,...,pn) E N n ,
with
IpI
>
1,
we can de-
fine the h-partial derivative
Y(Rn)
hDp : Y(Rn)
(2.3.13) as the iteration (2.3.14)
hDp = (hDx1)".
..(
D
xn
)
Pn
of
the h-partial derivatives in (2.3.8). It is easy to see that the above definition is correct since the h-partial derivatives (2.3.8) are commutative. Obviously, the h-partial derivatives (2.3.8) and thus (2.3.13) are linear mappings. Remark 1 The essential smoothing property of the h-partial derivatives defined above is apparent in (2.3.12). Indeed, given a compact
where
K
C
Rn. we obviously have for
K' = K + h(E)supp
#
C
Rn
# E
@(Rn)
the estimate
is compact and
n
m
In this way, an 9 -type bound for a,
g(9.0)
tained in terms of an Y -type bound for ):,#(f
on
K
can be obon a bounded
neighbourhood of K. h-partial derivative
Therefore, a representative g of the D F is locally bounded by a represenxi tative f of F. It should be noted that in view of (2.3.7). we obviously have J I . c. F; and hence h(e) in (2.3.15) depen ding only on # and not on K. Finally, the justification of the definition (2.3.11) is in (2.3.12) which leads to (2.3.15). And all that is based on the classical property of convolution to commute with partial derivatives, see for instance (1.1.74).
Now, we present several basic properties which show the COherence between the h-partial derivatives and the usual partial derivates. when both are applied to important classes of generalized functions in
Y?(Rn).
E.E. Rosinger
174
First, we establish a representation of type (2.3.12) for the arbitrary h-partial derivatives (2.3.13). In this respect, a n easy and direct computation yields the following. Suppose given
F = f + 9 E YI(Rn).
(2.3.16) and
n p = (p i,...,pn) E N ,
(2.3.17) where f o r
h
+
E
with
f E d
lpl
Dp = g + 9 E %(Rn).
@(R")
and
x
E
Rn
>
Then
1. g E
d
we have
with (2.3.19) Y = (Y
id
Il
. yij
E
Rn
and
(2.3.20)
a s well as (2.3.21)
=
1
yij
A first property easy to establish is that f o r any polynorntal P:Rn -B C and p E N". we have (2.3.22)
h
~ =P DPP~
in
YI(R")
However, the above identity between the h-partial derivatives
hDp
and the partial derivatives
Dp
cannot be extended even
to Ym-smooth functions, a s seen next in:
Solutions of Linear PDEs
175
Example 1 n = 1.
In the one dimensional case, when f E '&OD(R')
given by
(2.3.23)
Df - Df =
h
where (2.3.24) g ( 9 . x ) = ex
X
= e ,
f(x) g
+
1
(e-h(e)y
with
let us take
x E
9 E %(R'),
R'.
Then
g E
- 1)+(y)dy.
9 E @(R').
xER'
R'
m E N+ and + E @m(R1)\@m+l(R1), then applying the Taylor formula to the bracket under the integral in (2.3.24). we obtain. for 9 E @(R'). x E R' and suitable 0 < 8 < 1. the relation Assume given
But
8h(')Yym+1+(y)dy
+
R'
J
# 0 . when e +
ym+'+(y)dy
O
R'
and choosing h as for instance in (2.3.6). we have for any a > 0 the relation (2.3.27)
h(E)/ea
when
e +0
Thus (2.3.26). (2.3.27) yield (2.3.28)
g f 9
hence in view of (2.3.23) we obtain in general (2.3.29)
DeX # DeX h
in
%(R')
However, we have the following general coherence property.
E.E. Rosinger
176
Theorem 1
T E 59'(Rn)
If
and
( 2.3.30)
p E
Nn.
DPT
1
h
then
DPT
Y(R")
in
Proof
It follows from a direct application of (2.3.18). see details in Colombeau C3.41 0 I t should be noted that the above equivalence
ger hold f o r arbitrary
F E O(Rn).
Z
does no lon-
as seen next in:
Example 2 n = 1,
In the one dimenstional case, when (2.3.31)
h
D(b2)
# 26.D6
which follows easily f o r h tails in Colombeau C3.41.
in
we have in general
O(R')
in (2.3.6)
for instance, see de-
In connection with the linear systems of partial differential equations (2.3.2) the following coherence result is useful. Theorem 2 and T E b'(Rn) let us denote by x * T E Y?(Rn) the classical product in 9 ' , product in '8. Then .yT .y*T, see (2.1.178). Furthermore, for p E N" we have
For
.y E
Om(Rn)
-
.yT E J'(Rn)
and respectively the
(2.3.32) ,Dp(~T) 1 Proof.
,DP(x*T)
See Colombeau [3]
1
Dp(xT) z Dp(x*T) in Y(Rn) n
Corollarv 1
For
D
xi
,
K E Om(Rn) with 1
<
and T E J'(Rn). the h-part a1 derivat ve s i In. satisfy the following version of the
Leibnitz rule of product derivative
177
Solutions of Linear PDEs
(2.3.33)
D xi
(x*T) Z (Dx x0-T + x*(Dx T) in Y(Rn) i
i
Proof We have from (2.3.32) the relation
but
Dx
satisfies in i derivative, hence
Y(Rn)
Then (2.3.33) follows easily 53.
the Leibnitz rule of product
0
SOLUTIONS FOR DISTRIBUTIONALLY UNSOLVABLE EQUATIONS
The basic result which uses h-partial derivatives a n d leads to the existence of generalized solutions for systems containing those in (2.3.2). (2.3.3) is presented now. It suffices to formulate i t for one single linear partial differential operator of the type (2.3.34)
1 ap(x)DPU(x).
x €
Rn
PEP where
P C Nn
is finite, while
a
P
€
YZaD(Rn).
with
p E P.
Theorem 3
If U E B'(Rn) equivalent:
then
the
following
With the classical multiplication in (2.3.35)
1 apDPU = b
three
9J'(Rn)
conditions
are
we have
E J'(Rn)
PEP
For any family
(hplp
the multiplication in
€
P)
Y)(Rn)
o f derivation rates, we have with
E.E. Rosinger
178
)
(2.3.36)
DPU
a
f
PEP There exists a family
(hpl
B'(Rn)
€
P)
€
that with the multiplication in
DPU 1 aP hp PEP
(2.3.37)
b
hP of derivation rates, such
%(Rn)
f
b
f
DPU
we have
J'(Rn)
€
Proof (2.3.35) = > (2.3.36) In view of (2.3.30) we have DPU
hP and i t is easy to see then that in view of (2.1.158). we also have
D ~ zU a D ~ U hP P
a hence (2.3.38)
1 PEP
a
DPU hP
1 apDPU
f
in
%(Rn)
PEP
It follows that (2.3.36) holds for
b
€
J'(Rn)
from (2.3.35)
(2.3.36) = > (2.3.37) obviously. (2.3.37) = > (2.3.35) In view of (2.3.38). we obtain (2.3.39)
1 PEP
with
b
( 2.3.40)
a DPU
P
f
b
€
J'(R")
from (2.3.37). But obviously
1 apDPU E 91)'(Rn) PEP
and then the relations (2.3.39). (2.3.40) and (2.1.164) will 0 yield (2.3.35)
Solutions of Linear PDEs
In view of Theorem following:
3
179
above. we are naturally
led
to
the
Definition Suppose given the linear partial differential equation
1 aPDPU = b
(2.3.41)
PEP where
P
C
Nn
is fin te.
with
ap E qm(Rn).
p E
N.
and
b E J'(Rn). is called a C o l o m b e a u w e a k A generalized function U E YI(Rn) f and only if there exists a family solution of (2.3.41) (hplp E P) of derivation rates such that (2.3.42)
1a
PEP
DPU
Z
b
in
VI(Rn)
hP
Indeed with this definition we obtain: Corollarv 2
A distribution U E J'(Rn) is a Colombeau weak solution of (2.3.41) i f and only i f i t satisfies that equation in the sense of the classical operations in B'(Rn).
Proof
It follows directly froms Theorem 3
0
The above definition and corollary extend in an obvious way to linear systems such as (2.3.2). As is known for instance from the mentioned example of Lewy, linear partial differential equations (2.3.41) do not in general have distribution solutions.
However, as seen next, systems (2.3.2). (2.3.3) have Colombeau These weak solutions under rather general conditions. Rn+ 1 , that is in t and x. solutions are global on Theorem 4 and bi i .iP (2.3.2) satisfies the condition written generically f o r C
Suppose that each of
the coefficients
a
in
E.E. Rosinger
180
(2.3.43)
sup
<
ID:,,c(t.x)l
t a n
xER for every bounded
I C R'
and
q E Nn+l.
Further, let us suppose that each of the initial values
ui
in (2.3.3) satisfies the condition written generically for
u
p E N".
for every
Then there exists a Colombeau weak solution
U i , .. . .U, E '4(Rn+l)
(2.3.45)
for the system (2.3.2) which also satisfies the initial value conditions (2.3.3) in the sense defined in Chapter 2, Section 4.
Proof The proof is constructive and i t uses a n extension of a classical convergent iterative method which leads to a solution for differential equations in certain locally convex spaces. Indeed, one considers the initial value problem
X'(t) = F(t,X(t)). X(0) = xo where
T
>
t E
(-T.T)
0. X O € E, F:(-T.T)
x
V
-
E
is continuously dif-
is a suitable complet locally convex space ferentiable, E and V C E is open. A solution is constructed by the iterat ions
The choice of E and F obviously depend on the system (2.3.2). As expected, F will be linear in X. However, the above iterative method converges for a large clasas of nonlinear F a s well.
Solutions of Linear PDEs
181
In this way the proof has also implications for the numerical approximation of the solution (2.3.45). Details which are quite lengthy can be found in Colombeau [3] and involve results from Colombeau [ 5 ] . 0 As a n easy consequence we obtain the following local existence
result which does not require any boundedness conditions on co efficients or initial values.
Corollarv 3 In every strip
A = R'
(2.3.46)
with
>
L
x {x E
RnI 1x1 < L} C Rn+l
there exists a Colombeau weak solution
0.
U,,....Ue
(2.3.47)
A
which satisfies (2.3.2) in value conditions(2.3.3)
for
%(A)
€
and also satisfies the initial
x E: Rn.
1x1
< L.
Proof Given
L
>
0.
assume
r
E B(Rn)
~ ( x )= 1
such that
for
x E Rn. 1x1 < L. We can apply now Theorem 4 to the system obtained from (2.3.2). (2.3.3) where all the coefficients a and b i a s well a s the initial values u have been i i JP multiplied by y 0 Remark 2 The power of the local existence result in Corollary 3 above can easily be seen a s i t yields Colombeau weak solutions in e u e r y s t r i p of type (2.3.46) for various distributionally un00
solvable linear partial differential equations with Y: -smooth coefficients. For instance, Lewy's equivalent form
equation
(2.3.1) c a n be written in the
(2.3.48) D t U = -iDx U + 2i(t+ix,)Dx 1
which
will
A = R'
x {X E R211xl
Similarly. written a s
have
Colombeau
Grushin's
<
weak
L) C R3,
equation
2
U + f. tER'.
solutions
with
L
>
x=(x,.x2)ER2
in
every
strip
be
equivalently
0.
(1.3.10) c a n
E.E. Rosinger
182
t
hence i t will
A = R'
x
+ f.
D U = -itD U
(2.3.49)
[-L,L]
X
have Colombeau weak C
R2,
L
with
The same applies to the Cauchy-Riemann equation
(2.3.51)
U(0.x)
>
R'.
x E R'
solutions
in every
t E R'.
= UO(X).
value
problem
even locally i f
u o E '&OD(R')
solutions
of
for
the
x E R' x €
R'
which, a s is well known, cannot have distribution distribution z = t + ix.
strip
0.
initial
D t = -iDxU.
(2.3.50)
t E
solutions
is not analytic, since the only
(2.3.50)
are
analytic
in
I t should be noted that Lewy's equation (2.3.48) does not satisfy the boundedness conditions (2.3.43) owing to the coefficient 2i(t+ixl). Hence, with the methods in this Section we cannot obtain for i t global Colombeau weak solutions. On the other hand, Grushin's equation (2.3.49) satisfies (2.3.43) i f and only i f f satisfies that condition, in which case we have global Colombeau weak solutions for i t . The Cauchy-Riemann equation (2.3.50) obviously satisfies (2.3.43). T h u s , i f uo satisfies (2.3.44). then (2.3.50).
(2.3.51) will have global Colombeau weak solutions. Remark 3
1)
Concerning the coherence between the Colombeau weak solutions obtained by the method in the proof of Theorem 4 and known classical or distributional solutions, a series of examples of familiar linear partial differential equations are studied in Colombeau C3.43. Here we mention the following coherence results. 1.1
I f (2.3.2) is constant coefficient hyperbolic, the Colombeau weak solutions coincide with the classical ones.
1.2) I f (2.3.2). (2.3.3) is analytic, the Colombeau weak solutions coincide with the classical analytic ones. 1.3) Similar results hold for classes of parabolic o r elliptic equations.
Solutions of Linear PDEs
183
As mentioned, see also Treves [2]. one cannot expect uniqueness results in Theorem 4 or Corollary 3. since sysm
tems (2.3.2). (2.3.3) can even have nonunique % -smooth solutions. The method of proof for Theorem 4 can be extended to systems (2.3.2). (2.3.3) with more general coefficients and m
initial values, which are n o longer '8 -smooth but c a n be distributisons or even generalized functions. In fact, the method of proof for Theorem 4 is nonlinear, thus i t can yield Colombeau weak solutions for nonltnear systems of partial differential equations, see Colombeau C3.61.
LINEAR SCATTERING We present one more linear application of Colombeau's nonlinear theory this time dealing with the so called scattertng operator in quantum field theory. One of the mathematical interests for us is in the fact that a sufficiently general treatment of the scattering operator does involve a n t n f t n t t e dtnenstonal extenston of Colombeau' nonlinear theory, extension in which the generalized functions still defined on Euclidean spaces will take values in arbitrary Banach algebras. Particular cases of these Banach algebras, such a s given by the bounded operators on a Hilbert space, will suffice for the study of scattering operators. However, the study of Banach algebra valued generalized functions has its own interest among others in the resulting noncommutattutty of multiplication of the respective generalized functions. I t should be mentioned that such a n extension of Colombeau's nonlinear theory is rather immediate.
I t should also be mentioned that a wider range application of Colombeau's nonlinear method to quantum field theory leads to generalized functions which have values a s u n b o u n d e d linear operators on Hilbert spaces. Hence they c a n go beyond Banach algebra valued generalized functions. However, they c a n be dealt with within the framework of generalized functions with values in bornologtcal algebras and the extension to that situation of the theory in Chapter 1 is again rather straightforward. Details on the above can be found in Colombeau [l]. As is known, there also exists a n interest in another kind of infinite dimensional extension of a theory of generalized func tions, extension in which the Euclidean spaces on which the generalized functions are defined are replaced by infinite dimensional vector spaces with or without topologies. As seen in Colombeau C2.43. such t n f t n t t e dtnenstonal extensions o f
184
E.E. Rosinger
Colombeau's nonlinear theory can also be made in a rather easy way, unlike the usual similar extensions of Schwartz's linear theory of distributions, see Kuo and the literature cited there. Let u s now recall some of the relevant facts concerning linear scattering. T h e so called scattering operator is obtained from the following type of heuristic computation, rather familiar in quantum field theory. Suppose given the initial value problem in the Fock space 8 . see Colombeau C2.43 (2.3.52)
S'(t) = -igH(t)S(t).
(2.3.53)
S(t0) = I
t E R1
where g E R'. H:R1 + Y ( 8 ) is the interaction Hamiltonian and I E Y(0) is the identity operator, while Y ( 0 ) denotes the space of operators on the Fock space F which is a Hilbert space. We are looking for a solution
S t o:R*
Y(0)
of (2.3.52).
(2.3.53). in which case the scattering operator will be defined by
S(g) E Y(8)
(2.3.54)
whenever such a l i m i t exists. The basic difficulty comes from the fact that
H
in (2.3.52)
cannot in fact be assimilated with a mapping H:R' + Y(8) since i t has a more involved structure, such a s for instance a n operator valued distribution. The usual heuristic approach is to assume some asymptotic expansion at g = 0 (2.3.55)
S(g) = 1 +
)
gmsm.
sm
E
re(%)
m>l
motivated by the fact that obviously
(2.3.56)
S(0) = I
However, when replacing (2.3.55) in (2.3.52) and (2.3.53) in Sm, one encounters divergent integrals order to obtain all whose possible regularizations a r e the object of quantum renormalization theory.
185
Solutions of Linear PDEs
For clarity, let us consider the following scalar version of (2.3.52). (2.3.53)
(2.3.57)
X'(t) = ig6'(t)X(t).
(2.3.58)
X(t) = 1.
t
<
t E
R'
0
with 6 E 9'(R') being the Dirac distributions and X being the unknown function, distribution or generalized function. A heuristic formal solution of (2.3.57). (2.3.58) is given by (2.3.59) which has the formal series expansion
The difficulty with (2.3.57). (2.3.58) is now apparent, owing to the powers 6m of the Dirac distribution which occur in (2.3.60). powers which cannot be dealt with in a satisfactory manner within the Schwartz linear theory of distributions. The aim of this Section is to show the way the above heuristic and formal computations can be given a meaning within Colombeau's nonlinear theory of generalized functions. I t will be convenient to consider several versions of problem
(2.3.52)-(2.3.54) and do so in a n increasing order of gene-
ral i ty.
The simplest case, which in view of (2.1.67) will nevertheless contain the problem in (2.3.57). (2.3.58). is the following. Suppose given (2.3.61)
A E
Y(R')
real valued and with compact support
which means the existence of a representation (2.3.62)
A = f
+ 9
E Y(Rn)
where for a certain finite interval [a.b] C R'.
w e have
E.E. Rosinger
186
SUPP f ( 4 . O )
(2.3.63)
+
and of every real valued
[a.b].
C
E @,
E (0
we have + R'
f(+.*) : R'
(2.3.64)
+
We consider the problem
X' = ig A X
(2.3.65) ( 2.3.66)
%(R')
in
= 1
in
%((--,to))
to)
with given
g.to E
R'.
<
to
a.
Let us define ( 2.3.67)
h:@xR'
4
R'
by
to
then i t is easy to see that (2.3.69)
h € d
and
(2.3.70)
B = h + 9
does not depend on have
f
%(R')
in (2.3.62)-(2.3.64).
Moreover, we
B' = A
(2.3.71) (2.3.72)
E
Bl ( --.
= 0
in
%((--,to))
to)
In view of (2.3.68). B is also a real valued generalized function. Therefore. according to Section 8 . Chapter 1 , we can define (2.3.73)
X = eigB
E
%(R')
187
Solutions of Linear PDEs
(2.3.73) gives a solution for can state the following first
and i t is easy to see that (2.3.65). (2.3.66). Now we result. Theorem 5
For
every
real
function
A
X E '%(R')
for
E
valued
and
%(R').
compactly
there
exists
supported a
generalized
unique
solution
X' = ig AX
X = 1
at the left of
supp
A
supp
A
and this solution is given by X = e ig B where
B
Y(R')
E
and
B' = A B = 0
at the left of
Proof The
existence
follows
(2.3.67)-(2.3.73).
from
For
the
uniquenesas. let us assume Y E %(R') satisfying (2.3.65). (2.3.66). Similar to (2.3.73) we can also define
X
*
E '8(R1)
= e- i g
Let us take
Z = X*(Y-X)
E
%(R')
Then obviously 2'
= -ig AX*(Y-X)
+ X*(ig A(Y-X))
while = O
in
%((--,to))
I t follows that
Z' = 0 Z
in
%(R') = 0
in
%((--.to))
= 0
E.E. Rosinger
188
and
thus an easy
'B(R').
computation will
yield
2 = 0
that
in
0
In view of (2.3.54). the following consequences of Theorem 5 are of interest. Corollarv 4 If
A = 6.
then x(c.) = eig E
(2.3.74)
If A = T E J'(R') distribution, then
c
is a real valued, compactly
X(a) = e iJ3 T(1) E
(2.3.75)
supported
c
Proof In view of (2.1.67) we obtain 6 = f + 9 E V?(R')
where
= a(x)$(-x).
f($.x) with given
a E
J(R'),
Then, for suitable
=
x E R'
such that
on a neighbourhood o f
a = 1
h(#,.t)
$ 6 @,
to
<
m,
0 E R'
(2.3.68) yields
a(es)$(-s)ds.
$ E @,
t
>
0, B
>
0
to/€
hence, f o r each
$ E @,
(2.3.76)
we have
h($€.t)
= 1
if t > 0 is sufficiently large and B > 0 is sufficiently small. And obviously, (2-.3.76). (2.3.70) and (2.3.73) Yield (2.3.74).
If
A = T E B'(R')
is a
real valued, compactly supported
distribution, then (2.3.71). (2.3.72) imply that is also a real valued distribution and
B E J'(R')
Solutions of Linear P D E s
B
(2.3.77)
= T(l) (ti
for t, > 0 (2.3.75). We
present
in
189
%'((ti."))
.")
Obviously, (2.3.77) implies
sufficiently large. 0
a
first
extension
of
the
previous
results
in
Theorem 5. This time A can be a generalized function on R' which takes values in a given Banach algebra U ( 8 ) of conFor tinuous linear operators on a certain Hilbert space 8 . the time being however, we shall again assume that A is compactly supported and has as values self adjoint operators, the latter condition being the appropriate extension of the earlier requirement according to which A was real valued. Before we proceed in detail, the following remark is important. Given any Banach algebra ( X . I I 1 1 ) . a trivial extension of the construct ons in Chapter 1. will lead us t o noncoaautattue dif f eren t ial algebras of X valued generalized functions
(2.3.78)
Y(R,x), R c R". R
open
Naturally, i f the multiplication in X is commutative, then the same will hold for the multiplication in %(R,X). as seen in Chapter 1. which can be considered to correspond to the particular case when
(X.II
II) = C ' .
Coming back to our problem we can now reformulate i t more precisely, as an extension of (2.3.61)-(2.3.64). Indeed, we assume given
(2.3.79) A E %(R',U(%))
self adjoint and with compact support
that is, the following representation holds
(2.3.80)
A = f + 9 E Y(R'.!t(%))
where for some finite interval
(2.3.81)
supp
f(o.0)
and for every real valued
(2.3.82)
f(9.t) E
9
[a.b]
C
[a.b].
E
@J
U(8)
and
C
R'
9 E
CJ
t E
R'.
self adjoint
we have
E.E. Rosinger
190
Theorem 6 The problem (2.3.83)
X ' = ig
(2.3.84)
X =
where g E R'. is a unitary (2.3.73).
I
AX supp A
at the left of
has a unique solution X E 'S(R'.Y(f)) which operator valued generalized function, see
Proof
It is a direct extension of the proof for Theorem 5. let us define
Indeed,
k:bxR' + Y ( f )
(2.3.85) by ( 2.3.86)
(9.t) =
1
t
ingn]
nEN
to
. . . f(9.sn)ds where
H
...]
t
H(s,-s,)
. . .H(s
.).
n-l-sn)f(O.
to
9 E @,t€ R'
'...ds,.
denotes the usual Heaviside function and
to
<
a.
In view o f the extended version of (2.1.26) which corresponds to the case of the algebras in (2.3.78). we obviously have
3
+ € a : c > o
P
tER'
tr
llf(#.t)ll
<
c
hence (2.3.81) implies that the series in (2.3.86) is convergent in
U(f).
since the coefficient of
ingn
has the norm
bounded by Kn/n!. where K > 0 only depends on on n. In this way (2.3.85) holds.
It is easy to see from (2.3.86) that for have
9 E b
9
and not
fixed, we
191
Solutions of Linear P D E s
and that f E d = > k € d Therefore, we can define (2.3.87)
X = k + 9 E %(R'.%(at))
which will not depend on
f
in (2.3.80).
+
Further, a direct computation shows that, for
E @,
we have
In this way, we obtain (2.3.83) and (2.3.84). Details concerning Colombeau [2].
the rest of
the proof
can be
found
0
in
Concerning the existence of the scattering operator (2.3.54). we have the following result. Corollarv 5 Under the conditions in Theorem 6. if sidered as a function of ( 2.3.88)
Proof.
X(@)
g
€
R'.
X(m)
E %(at)
is con-
then
E YI(R',%(%)).
See Colombeau [2].
0
Remark 4 1)
Obviously (2.3.74) and (2.3.75) are particular cases of (2.3.88).
2)
If A has no compact support, the above situation becomes more involved. Indeed, let us consider A = 1 E YI(R')
in which case (2.3.65) and (2.3.66) are replaced with
X' = ig X X(t0)
for a certain is
in
P(R')
= 1
to € R'.
The unique, classical solution
E.E. Rosinger
192
hence lim
Xto(t)
does not exist
t-m tot-
A
Finally, we consider now the case of port. More precisely, suppose given (2.3.89)
A E %(R'.%'(%))
without compact sup-
self adjoint
which means the existence of a representation (2.3.90)
A = f
+ 9 E %(R',%'(%))
such that for every real valued (2.3.91)
9 E
Q,
and
t E
R',
we have
f(+,t) E 9(%) selfadjoint
Theorem 7 The problem (2.3.92)
X ' = ig AX
(2.3.93)
X(t0)
where
g.to E R'
= I
a r e given, has a unique solution
X E %(R'.Y(f)). Proof The solution is constructed by the same method a s in (2.3.85). (2.3.86). see for details Colombeau [2].