5. Existence of Generalized Solutions

38 5.

J!XIS'I"CE OF GENERALIZED SOLUTIONS

First we start with two basic results on the existence and uniqueness of generalized solutions for nonlinear PDEs (2.1) on bounded domains n which are presented in Theorems 5.1 and 5.2 in this Section. They both hold f o r arbitrary continuous nonlinear partial differential operators (2.2). In the first result the right hand term f in (2.1) can be any continuous function on P . In the second result the right hand term f in (2.1) can be any discontinuous measurable function from the Dedekind order completion ?'(a) of the space Co(P) of continuous functions on P . Fortunately, the existence result in Theorem 5.1 extends in an easy way to unbounded domains fl as well, as seen in Corollary 5.1. The reason for that i s the fact that the spaces of generalized functions involved, namely

are flabby sheaves, see Remark 5.3 and Subsection 7.2. In this way, we are led to the powerful existence and uniqueness result in Theorem 5.3, a result which contains those in Theorems 5.1 and 5.2, as well as that in Corollary 5.1. This flabbiness of the respective sheaves involved is in fact one of the major advantages of the method of solving linear and nonlinear PDEs through order completion. Indeed, as is well known, Kaneko, neither the spaces C e ( f l ) , 1 E lN, of classical solutions, nor the space 'D'(n) of the Schwartz distributions enjoy the property of flabbiness, this being one of their major disadvantages.

It should be noted, however , that these existence and uniqueness results, as well as other similar ones, see for instance Theorem 8.1, are in part consequences of general existence results concerning the solution of abstract equations in order complete structures, presented in Theorems 9.1 and 9.2. Further comments in this regard are presented in Remarks 5.2 and 9.1.

39

Last but not least, it is important to note the following. The basic existence results in this Section, as well as those in Sections 8 - 11, are obtained through the use of the 'pull-back' type of partial orders defined on the domain of the respective nonlinear partial differential operators, see (4.7). This method, however, can be extended significantly by the use of more general, that is, non 'pull- back' partial orders defined on the domain of the nonlinear partial differential operators considered. A first presentation of such an extension is given in Section 13, where the corresponding basic existence result in Theorem 13.1 is an obvious extension of Theorem 5 . 1 . The first basic result on the existence and uniqueness of solutions of continuous nonlinear PDEs of type ( 2 . 1 ) , ( 2 . 2 ) in the order completion of spaces of smooth functions can be formulated as follows. Theorem 5.1

If n c IRn is a bounded open set, then

that is, for every

f

E C0(a),

there exists a unique

F

E

G(n),

such

that (5.2)

T(F)

=


which means that F is the unique generalized solution of the continuous nonlinear PDE in ( 2 . 1 ) , ( 2 . 2 ) (5.3)

T(x,D)U(x)

=

f(x),

x

E

n.

Proof

The uniqueness follows easily from the fact that the mapping T is injective. We take an arbitrary f

E

Co(fi).

in (4.16)

40

> 0.

By the property ( 2 . 2 2 ) , there exist a closed, nowhere dense such that l', c ll and U, E Cm(ll\l',),

Let

E

(5.4)

Let U,

<

f 5 T(x,D)u,

f

on ll\I',.

t E

be the equivalence class in $(ll)

generated by u,,

see (4.4).

Then, in view of (3.13), (3.5), (4.8) and (5.4), we have

Let us take now (5.6)

I

=

{UE

F

=

A e C $(ll).

E

c

> 0)

A

q(ll)

and (5.7)

Then, according to (A.16) and (A.12), it follows that (5.8)

F

E

q(ll).

Furthermore we have (5.9)

4 c F c #

#

q(ll).

Indeed, the first strict inclusion follows from (5.5) and Proposition 2.2, if we note that (2.15) gives V E C:d(ll) such that

T(V) 5 f in

P(n)

hence (5.5) results in p(ll) in the inequalities

T(V)

<

T (UJ,

f

>

0

41

thus V E F. The second strict inclusion follows from (5.7), (A.8) and the fact that in view of (5.6), we obviously have A # (. An important property is the following (5.10)

F

=

{U

E $(fl)

I

<

T(U)

f in p(fl)}.

First we prove the inclusion '2'.

Let U

in p(fl). Then there exists u with T(x,D)u

<

E

T(x,D)u 5 T(x,D)uE which means that for

E

A1

=

$(n)

U and I' C fl

be such that T(U) 5 f closed, nowhere dense,

f on fl\r

But then (5.4) gives for all

therefore U

E

E

E

> 0 on

n\(r

u

r,)

> 0, we have

F.

For the converse inclusion 'g' in (5.10), let U E F and u E U. In view of (5.7), for every E > 0 there exists a closed, nowhere dense P; 5 fl

such that (5.11)

T(x,D)u 5 T(x,D)uE

<

f+

E

on fl\I''E

4

where the second inequality follows from (5.4). Assume that T(U) f in p(fl). Then, for every closed, nowhere dense r C fl, there exists x0 E fl\r, such that

42

Now, due t o t h e continuity properties of T(x,D), x0 E A E n\r, A open and 6 > 0 , such t h a t T(x,D)u 2 f

t

6 on

which contradicts ( 5 . 1 1 ) ,

and

f,

we can f i n d

A

and completes the proof of ( 5 . 1 0 ) .

Finally, we show t h a t , with indeed (5.12)

u

P

constructed i n ( 5 . 7 ) - ( 5 . 1 0 ) , we have

T ( F ) =
We note t h a t (5.10) yields

thus, i n view of ( 4 . 1 4 ) , (A.14) and (A.18), we obtain

+(F) = ( T ( F ) ) E~ ~< f l u e = < f ] . Therefore, i n order t o prove ( 5 . 1 2 ) , it only remains t o show t h a t t h e inclusion holds (5.13)


E

T(F).

Let us take g E Co(fi), (5.14)

g(x) < f ( x ) ,

such that x E fi.

It follows t h a t f o r s u i t a b l e

E

> 0 , we obtain

Then, according t o Proposition 2 . 2 , we can f i n d dense and v E C m ( n \ r ) , such t h a t (5.15)

g 5 T(x,D)v 5 f on

n\r.

l' 5

n

closed, nowhere

43

Let V be the equivalence class of v in J(!(fl),

see ( 4 . 4 ) . Then ( 5 15)

and ( 4 . 8 ) yield (5.16)

g

<

T(V)

<

p(fl).

f in

In particular, owing to ( 5 . 1 0 ) , we obtain

V

E

F.

Now, according to Lemma 5 . 1 below, we have in p(fl) (5.17)

f

=

sup {g

But the embedding p(fl)3 h

in (5.14)}.

-


E

p(n)

preserves infima and suprema, see Corollary 3 . 1 . ( 5 . 1 6 ) , we have in p ( n ) the relations

Therefore, in view of


I

the last inequality resulting from ( 5 . 1 0 ) . and obtain
I

U E F}

=

We apply now (A.37) in p(fl)

T(F)

which ends the proof of ( 5 . 1 3 ) , and also of ( 5 . 1 2 ) . In the proof of Theorem 5 . 1 above, we used: Leua 5.1

For every f

E

Co(n),

we have in

<

p(n) the relation

44

(5.18)

f =

SUP

[

1

g

*) g E co(n) **) vx E n: g(x) < f ( x )

I

Proof Obviously, we have t h e inequality '2'. W e prove now t h e converse inequality '<'. For t h a t , assume it i s f a l s e . Then t h e r e e x i s t s H E P(n) such t h a t we have vg E

co(n) :

while at t h e same time (5.20)

f

4 H.

Take h E H and l' c n closed, nowhere dense, such t h a t Then i n view of ( 5 . 2 0 ) , it follows t h a t

Hence t h e r e e x i s t s xo E

E n\r,

A

open and

A

6 > 0,

h

E

CO(n\I').

such t h a t

h < f - 6 on A. Let us now take

6

g = f - 2,

then

g

satisfies

*)

and

**)

in (5.18),

while on t h e other hand (5.21)

h(x) < g ( x ) , x

E A.

But i n view of ( 5 . 1 9 ) , we obtain (5.22)

g 5 h

on

n\r'

f o r a c e r t a i n closed, nowhere dense ( 5 . 2 2 ) c o n t r a d i c t each o t h e r .

l"

c n.

Now obviously ( 5 . 2 1 ) and

45

Remark 5.1

1) As seen in the proof of Theorem 5 . 1 , the essential property of the unique generalized solution F in ( 5 . 2 ) ) is that it satisfies the relation ( 5 . 1 0 ) , in other words (5.23)

F

=

{U E $(n)

I T(U)

5 f in

p(n)}.

This obviously means that F is the totality of classes of subsolutions of the continuous nonlinear PDE in ( 2 . 1 ) or (5.3). The way this situation is related to the nonuniqueness of solutions, or to the usual initial and/or boundary value problems associated with the PDEs in ( 2 . 1 ) is discussed in Section 6 next, as well as in Part I1 of this work. As seen easily, the essential ingredient involving the nonlinear PDE in (2.1), and which is needed in the existence result in Theorem 5 . 1 , is the abundance of local, classical subsolutions granted in Lemma 2 . 2 . The rest is but a quite simple and natural construction related to partial orders. In this way, the solution of rather large classes of continuous nonlinear PDEs is reduced to tuo separate steps, both of them quite straightforward: first , the construction of polynomials which provide sufficiently many local, classical subsolutions, followed by an order completion of spaces whose elements are functions patched up from locally smooth functions. This method in which the global solution of continuous nonlinear PDEs is reduced to the provision of many enough local, classical subsolutions, is in essence an extension to continuous nonlinear PDEs of the best possible general result for analytic nonlinear PDEs, which is also only local, as granted for instance by the Cauchy-Kovalevskaia theorem. In view of that, it may be more easy to understand now, why such an 'order first' approach is simpler, and at the same time more powerful, than the customary functional analytic methods, which from the very beginning, try to find nothing less than a global generalized solution, and do so as if forgetting about what in fact can be expected at best, even in such nice cases as those of analytic nonlinear PDEs. 2)

46

As mentioned in (1.9) - (1.13), the basic problem facing a theory of generalized solutions for linear or nonlinear PDEs is to reverse the strict inclusion (1.11) into an inclusion (1.12) or even (1.13), by a suitable choice of the space X of generalized functions. As seen in (5.1) in Theorem 5.1 above, one can indeed obtain an inclusion of the type (1.12) for the rather large class of continuous nonlinear PDEs in (2.1) and (2.2), by choosing

In Theorem 5.2 next, we show that with the same choice of X as given in (5.24), the result in (5.1) can be further strengthened in a significant manner by replacing Co(fi) with the much larger space Co(fi). The extent to which Co(fi) is indeed larger than Co(fl) can be seen in Theorem 7.2 in the sequel. In particular, in view of (7.71), to(fi)contains a large class of discontinuous measurable functions on fl. In this way one obtains an inclusion of the type (1.13) with an T which is significantly larger than Co(fi). Theorem 5.2

If n c lRn (5.25)

is a bounded open set, then

T(q(fl)) 1 ?'(a)

which means that for every A E CO(fi),

there exists a unique H E

q(n),

such that (5.26)

T(H)

that is, H

=

A in ?(n)

is the unique generalized solution of the nonlinear PDE in

(2*1), (2.2)

where the right hand term A E Co(fi)

can be discontinuous and measurable

47

Proof

In view of (3.13) we have the order isomorphical embedding

which according to Proposition A.l in the Appendix, can be extended to the order isomorphical embedding

(5.28)

EO(ii) A t

LP(n) 'l aw

where in view of (A.12) and (A.37), we have

A

=

Auc C Co(n)

as well as (5.29)

i(A)

{e(f) If

=

E

A}"

sup j&v

=

{<(p(f)]

If

E

A} .

Therefore, given f E A, from (5.1) we obtain Ff

T(Ff)

=

<(o(f)]

in

P(0).

Then (5.29) gives

i(A)

= SUP {T(Ff)If

PP) thus

E

A}

E

$(n),

such that

48

where the second inequality follows from the fact that T is increasing, see ( 4 . 1 6 ) . Moreover, in view of the same ( 4 . 1 6 ) , T is in fact an order isomorphical embedding, hence

Now (5.30) and (5.31) give

Let us denote

then ( 5 . 3 2 ) implies

Our aim is to prove that in (5.34) the opposite inequality _> holds as well. In view of (5.33) and recalling that T is increasing, the inequality (5.32) yields further (5.35)

T(H)

<

inf { T ( G ) I G

P(n)

E

j(!(n),

i(A) g T(C))

.

49

Given now g

E

Co(fi),

from (5.1) we obtain Gg

E

$(n),

such that

therefore (5.35) gives

However, given A

E

Co(fi)

and g E Co(fi),

it follows that

since (5.28) is an order isomorphical embedding. In this way (5.36) yields

Now, it only remains to prove the inequality

This however follows from Lemma 5.2 below. (5.38) finally yield

Therefore (5.34), (5.37) and

which is precisely (5.26), provided that for the sake of simplicity in notation, we identify the order isomorphical embedding (5.28) with an inclusion (5.40)

CO(fi) A I

' jm) , A

50

As a conclusion t o t h e above proof, it i s u s e f u l t o n o t e t h a t f o r A E Eo(fi), we have t h e implication, see (5.26) and (5.33)

Indeed, l e t us take

f E A,

then (5.2) y i e l d s

F E

,$(n),

such t h a t with

t h e convention i n (5.40), we obtain

But

f E A E Eo(fi)

implies


E

$(n),

T(F) =
with A

A

E

T(G)

< T(G),

we have

I

Now we r e c a l l t h a t , according t o (4.16), embedding , thus (5.42) gives

T

Furthermore , (5.33) and (A. 34) imply

which t o g e t h e r with (5.43) and (5.9) r e s u l t i n

Let us now assume t h a t

i s an order isomorphical

51 (5.44)

H

=

$(n)

and l e t us t a k e g E Co(fi)\A.

B = {max {f,g}lf E

We define

.

Then obviously B E Co(fi)

and g E B\A

gives

A c B . # Applying (5.26) t o B ,

T(H)

= A

we obtain K E

$(n)

such t h a t

c B = T(K) #

t h e r e f o r e (4.16) g i v e s

H 5 K i n $(n) # which means t h a t

H c $(fl) #

-

I n t h e proof of Theorem 5 . 2 , we needed t h e following

Lena 5.2 If

A E Co(fi)

then with t h e n o t a t i o n i n ( 5 . 2 8 ) , we have

If E Co(fi),
c

Co(n), A


=

.

The f i r s t equality follows from (A.37) and the f a c t t h a t Actually, we obtain i n addition the relation (5.46)

i(A) =

(v(A))'~

A

i s a cut.

.

We s h a l l now prove t h a t u e inf {
(5.47)

P(fl)

.

For t h a t we note t h a t according t o (A.34), we have inf {
(5.48)

E

C0(n), A

?(fl)

<


while

A

c

< f ] H f E AU

thus (5.48) follows from (A.7).

Now i n view of ( 5 . 4 6 ) , it suffices t o show that

First we note that the inclusion 2 i n (5.49) follows e a s i l y from t h e monotonicity of lp. Indeed, l e t us show that

53 Take

c E cp(Au), c

=

then c

q(b)

and a 5 b,

thus t h e monotonicity of

<

cp(a)

(5.51)

u e

(V(A ) )

It follows t h a t

V a E A

gives

cp

V a E A

cp(b) = c ,

and therefore (5.50).

f o r some b E A'.

p(b)

=

Applying (A.14) t o (5.50), we obtain

2 (cp(A))Ue

*

I n order t o show t h e converse inclusion

we have t o make use of the special s t r u c t u r e of t h e spaces

ff (n) *

Co(fi)

and

Let us take any f E (cp(Au) ) e , then (5.53)

f 5 cp(b),

V b E A'

.

I n order t o prove t h a t indeed implication (5.54)

g E (cp(~))'

f 5 g

f E

in

(v(A))'~,

we have t o obtain t h e

P(n) .

Assume on the contrary t h a t there e x i s t s g E (cp(A))' (5.55)

f

Jg

in

p(n)

.

But f o r t h i s g we w i l l have g E (5.56)

p(h) 5 g

in

P(n), V

P(n), as h E A

.

well as

for which

54

f

Now

and

as elements of

g

equivalence c l a s s e s i n and

b E g.

a , b E C;,(n),

E n,

give a nonvoid, open A

such t h a t

V x E d

(5.57)

a(x) > b ( x ) ,

where d

i s t h e closure of

see (3.4), are

Let us then t a k e r e p r e s e n t a t i v e s

C;d(fl).

It follows t h a t

p ( n ) = C;,(n)\N,

5

a E f

while ( 5 . 5 5 ) and ( 3 . 5 ) w i l l

a , b E Co(d)

and

n

i n In.

A

We note t h a t we can assume (5.58)

A c Co(fi) #

.

Indeed, i f A = Co(fi) then it i s easy t o see t h a t above i n P ( n ) , hence (A.9) and (A.8) give

p(A)

i s unbounded from

which means t h a t (5.52) w i l l t r i v i a l l y hold. I f on t h e o t h e r hand (5.58) i s v a l i d , then (A.20) gives a constant Y E R, such t h a t (5.59)

h(x)

< Y,

V h E A, x

E

fi

.

Obviously, we can choose Y so t h a t we have i n addition (5.60)

b(x) 5 Y ,

Now we construct

f5.611

V x

E

k E Co(fi)

. such t h a t

for x

k(x) = b(x)

f o r x E A' g A

E

while (5.62)

fi\~

k(x) = Y

k(x) 2 b(x) f o r x E

55

with a suitably chosen nonvoid, open (5.63) Indeed, i f

k

E

A'

5

A.

Then

.

A'

h E A

then (5.59) gives

h(x) 5 I , V x E f l \ ~ while owing t o (5.56) and the f a c t t h a t b E Co(A), we have h(x) 5 b(x), therefore (5.60)

-

V x E

(5.62) imply

Finally, we note t h a t (5.57) and (5.61) give a(.)

> k(x), V x E

A'

which means t h a t

therefore i n view of (5.63), we contradicted t h e assumption t h a t

I n t h i s way (5.52) i n proved, and together with it (5.49) a s well.

Bemark 5.2 I) In order t o appreciate the extent t o which Theorem 5.2 i s more powerful than Theorem 5.1, we only have t o see how much Co(h) i s l a r g e r than C 0 ( h ) . One answer t o t h i s question i s given i n (7.67). This i n p a r t i c u l a r means t h a t C 0 ( n ) contains a l l bounded functions f : n + R which a r e discontinuous on closed, nowhere dense subsets l' c n.

56

In view of Theorem 9.2, one may ask whether the result in Theorem 5.2 is not in fact a direct consequence of Theorem 5.1, in which case the above rather lengthy proof of Theorem 5.2 would be superfluous. The fact, however, is that with the notations in Sections 4 and 9, we have, see in particular (4.6) and (9.1) 2)

T:X

I Y

u-v

where V is the N equivalence class of T(x,D)u, this way it follows that, see ( 4 . 4 ) , (4.5), (9.4)

see ( 3 . 4 ) , (4.6). In

Therefore Theorem 5 . 1 does not prove (9.25), but only the lesser inclusion T(X,)

1

co(n)

while obviously we have

Remark 5.3 It is rather surprising how easy it is to extend the existence and uniqueness result in Theorem 5 . 1 to unbounded domains fl 5 Rn. This i s a simple and direct effect of the fact that the spaces of functions o r generalized functions

57

are flabby sheaves, see Subsection 7 . 2 . Therefore they can be easily and naturally extended to, that is, embedded into the respective spaces corresponding to any larger open set which contains fl. This advantage of easy and natural embedding o r extension is in sharp contradistinction to the case of the spaces Co(fl) and P ' ( f l ) which are not flabby sheaves, see Kaneko. In this way the spaces of generalized functions

used in this work enjoy from the start an advantage over the space of distributions P ' ( f l ) . Before presenting Corollary 5 . 1 , we need Lemma 5.3

Given fl C_ Rn open, together with a closed nowhere dense subset the following relations hold for every t! E Iii

l'

c

fl,

Proof

The relation ( 5 . 6 4 ) is an immediate consequence of ( 3 . 1 ) , as well as of the fact that a finite union of closed, nowhere dense sets is again closed, nowhere dense. The relation (5.65) follows from (5.64) and ( 3 . 3 ) , ( 3 . 4 ) . Relation (5.66) is a direct consequence of ( 5 . 6 4 ) , (5.65) and (4.1) - (4.6).

Now (5.67) will result from (5.65) and (A.12), if we note that in the order relation 3 in ( 3 . 5 ) the size of the closed, nowhere dense set I' is not important. Finally, (5.68) results from (5.66) and (A.12), based on a

similar remark on the order relation ST in ( 4 . 7 ) .

58 The extension of Theorem 5.1 t o unbounded domains ll

C

IRn

is given i n

Corollary 5.1 If

n E IRn

i s an a r b i t r a r y open s e t , then

f(q(n)) 3 c&(n)

(5.69)

t h a t i s , for every

f

E

c:d(fi),

there e x i s t s a unique

F E

,id@),

such

that

T(F) =
(5.70)

,ko(n)

which means t h a t F i s the unique generalized solution of t h e continuous nonlinear PDE i n (2.1), (2.2)

Proof Let us take

f

E

Cid(n)

and a closed nowhere dense subset

i'

c n,

that

Further, l e t us take a closed nowhere dense subset

where

n,,

where

BY

c n\r, such t h a t

a r e p a i r wise d i s j o i n t , bounded open s e t s , which s a t i s f y

BY c n\r,

(5.74)

C

Y E

DI

is t h e closure of

llY i n Rn.

such

59

Obviously, we can also assume that (5.75)

mes (X) = 0

however, this is not necessary in the rest of the proof. (5.75), it follows that (5.76)

f

I fiv

E C O ( f i u ) , u E IN

Regardless of

.

In this way, by applying Theorem 5.1 to each

nu,

with

u E N,

the

relation (5.76) will give (5.77)

F~

E

$(nu),

v

E

IN

such that (5.78)

T(Fv) =
q(nv),u

E

IN

.

Now (5.72) - (5.74), (5.76), (5.77) together with (5.68) and (7.86), (7.87) will yield (5.79)

F

E

q(n\(r

u z))

which in view of (5.67) as well as (4.6) and (4.15), satisfies (5.70).

Now based on Corollary 5.1, we are in the position to use the abstract result in Theorem 9.2 and obtain the corresponding ultimate s t r e n g t h e n i n g of the existence and uniqueness results presented so far in Theorems 5.1 and 5.2, as well as in Corollary 5.1 itself. Indeed, for arbitrary, possibly unbounded domains n C [R", as well as f o r general continuous nonlinear PDEs in (2.l) , (2.2), we obtain

60

Theorem 5.3

For any open s e t (5.80)

n

< IR",

T(q(n))

we have =

P(n

I n o t h e r words, f o r every

A E

P(n)

t h e r e e x i s t s a unique

F E

&(n),

such t h a t

(5.81)

T(F)

=

A i n P(n)

In particular

i s an order isomorphism, see (4.16).

Proof I n view of Theorem 9.2 and pct 2) i n Remark 5.2, it s u f f i c e s t o prove t h a t

However, t h i s i s a d i r e c t consequence of (5.69), as well as of (3.4) and

(4.6).

w

Remark 5.4 1) I n view of (3.13), it is obvious t h a t Theorem 5.3 implies both Theorems 5.1 and 5.2. The c r i t i c a l ingredient, however, which allows t h e strengthening of Theorem 5.1 t o Corollary 5.1, and t h e r e f o r e , t o Theorem 5.3, i s given i n Lemma 5.3. This l a t t e r property i s obviously r e l a t e d t o t h e flabbiness of t h e f i v e respective sheaves, t h e r e f o r e , it is not connected with t h e a b s t r a c t r e s u l t i n Theorem 9.2.

61

In the particular case of continuous nonlinear PDEs in ( 2 . 1 ) , ( 2 . 3 ) , it is easy to see that Theorem 5 . 3 can be further extended to the case when the coefficients are no longer continuous on fi, and instead they only satisfy the condition

2)

A further similar extension of Theorem 5 . 3 can be obtained in the case of the general nonlinear PDEs in ( 2 . 1 ) and ( 2 . 2 ) , if instead of the continuity of F in all its variables, we shall only require for instance that

where X c n is closed, nowhere dense and may depend on F, while I E IN is the number of the variables DEU(x) in F, with p E IN, IpI 5 m. Obviously the nonlinear PDEs in ( 2 . 1 ) , ( 2 . 3 ) which satisfy (5.84) are a particular case of the nonlinear PDEs in ( 2 . 1 ) and ( 2 . 2 ) satisfying ( 5 . 8 5 ) . It is quite obvious, however, that singularities of F which are still more general than in (5.85) could be accommodated in existence and uniqueness results such as that in Theorem 5 . 3 . 3)

It is important to note that we have as well P - r e g u l a r i t y type

results concerning the existence of generalized solutions provided by Theorems 5 . 1 , 5 . 2 , Corollary 5 . 1 and Theorem 5 . 3 above. Indeed, in view of ( 2 . 1 3 ) , we can assume in (2.15) and (2.22) that (5.86)

uc E P(n\r, .)

In fact, we can assume that (5.87)

Uc

analytic on

since according to ( 2 . 1 3 ) , Uc is a polynomial in a neighbourhood of every point in n\I',. In this way ( 5 . 1 ) can be strengthened as follows

62 (5.88)

T($(fl))

2 Co(fi)

in other words, in (5.2) we can assume that the generalized solution F of the continuous nonlinear PDEs in (2.1), (2.2) or (5.3) is p - r e g u l a r in the sense of (5.89)

F E $(n)

.

Similar c"-regularity versions can be obtained f o r Theorem 5.2, Corollary 5.1 and Theorem 5.3. These p-regularity results on generalized solutions will be important in the study of the group invariance properties of the smooth versions of general nonlinear PDEs in (2.1), (2.2), see Section 18. 4) We should like to give a few first indications about the meaning of the generalized solutions (5.90)

F E

q(n)

obtained in Theorems 5.1 - 5.3 and Corollary 5.1. For that purpose, it is useful to give a meaning to the spaces of generalized functions $(R) themselves. This latter task can be dealt with, provided that here we refer in advance to the two main results in Section 7. Indeed, the nature of the spaces will be clarified along two directions. First, it

q(n) will be shown that q(fl) can be simply related to the space

Yes (fl)

of

Second, by using the sheaf real valued measurable functions on fl. structures involved, we can equally simply relate to various

q(fl)

classical spaces of real valued functions on fl.

In order to relate

q(fl) to the set

les (fl)

of real valued measurable

functions on fl, let us recall (7.11), according to which we have the order isomorphical embedding

63 where P(n) is defined i n (7.6). Therefore, with t h e n o t a t i o n i n (7.26), and i n view of (7.25), we obtain t h e order isomorphical embedding

Let us t u r n now t o t h e r e l a t i o n between

$(n)*

and various c l a s s i c a l

spaces of r e a l valued functions on n. The main r e s u l t i n t h i s regard i s obtained i n (7.172) according t o which we have t h e following commutative diagram of sheaf morphisms

P(A)A

Cd(A)

A

1

& $(A)*

$(A)

I

.i. i n j

T inj

inJc:d(A) A P(A)

Co(A)

$)(A)*

f o r every nonvoid open subset A g n. Here t h e mappings a and 0 a r e t h e canonical quotient mappings defined by (4.4) and (3.4), r e s p e c t i v e l y . F u r t h e r , according t o (7.173), a l l t h e spaces i n (5.92), except f o r and C'(A), are flabby sheaves, see (7.92). We can n o t e now t h a t , with t h e exception of

$(A),

* P(A) and

Cm(~)

$)(A)*,

a l l t h e o t h e r spaces a r e classical spaces of real valued f u n c t i o n s on 0. Indeed, i n view of (7.40), t h i s holds f o r P(A)as w e l l . On t h e o t h e r hand, i n view of (4.4) and (3.4), a and B a r e s u r j e c t i v e , while i n view * of (A.23) i n t h e Appendix, $(A) and P(A)a r e o r d e r dense i n $(A) and

P(A)*r e s p e c t i v e l y .

(5.93) i n (5.92)

P(A)

c:d(A)

I n t h i s way t h e sheaf morphisms

A

$(A)

g i v e a d e f i n i t i o n of t h e spaces

c l a s s i c a l spaces

P(A).Indeed,

inJ$ ( A ) * $(A)*

i n terms of t h e

C d ( ~ ) , and t h e r e f o r e , i n terms of t h e c l a s s i c a l spaces

as mentioned i n Remark 7.5, t h e spaces C:d(~)

as t h e smallest flabby sheaves which contain t h e spaces Cm(h).

can be seen

64

Finally, the commutativity of the sheaf morphisms in (5.92) shows that the mappings

are natural extensions of the classical mappings (5.95)

As mentioned at the beginning of this Section, the existence results in Theorems 5 . 1 , 5 . 2 , 5 . 3 and Corollary 5.1 - obtained on the basis of the ’pull-back’ partial orders (4.7) - can significantly be eztended by the use of the more general non ’pull- back’ partial orders ( 1 3 . 1 1 ) . Here we mention several of the convenient features of such extensions. 5)

First, in the case of non ’pull-back’ partial orders, it is no longer necessary to make the strong identification in (4.3) which, as we have seen, will automatically lead to the injectivity of the mappings ( 4 . 5 ) and (4.16). Therefore, the non ’pull-back’ partial orders (13.11) will unlike in this Section - lead to existence results which need not automatically be uniqueness results as well, see related comments in Section 6 . Further, as seen in Theorem 13.1, and in particular, in Corollary 1 3 . 1 , nontrivial and significant extensions of the existence result in Theorem 5 . 1 can be obtained for a variety of non ’pull-back’ and so called admissible partial orders which can be associated with the same given nonlinear partial differential operator T(x,D) , see (13.35). Finally, the nontriviality and significance of the non ’pull-back’ partial orders is illustrated through their applications in Section 18.