2. Approximation of Solutions of Continuous Nonlinear PDEs

11

2.

APPROXIMATION OF SOLUTIONS OF CONTINUOUS NONLINEAR PDEs

The nonlinear PDEs considered in this work are of the form

where

n

c Rn is open, f is given, U is the unknown function, while

with F being jointly continuous in all of its variables. For convenience, we shall first consider the particular case of (2.2) when is bounded and T(x,D) is of the polynomial nonlinear form, with

n

continuous coefficients (2.3)

n

D:ij l
T(x,D) = :D +

where we have

x = (t,y) E n,

pij < m and pij

t

19.. I 3 m. 1J

t

E R,

y

E

Dqij Y Rn-l,

pij E D(,

qij E UP-1,

The coefficients in ( 2 . 3 ) are supposed to

satisfy (2.4)

ci E Co(fi)

where for I

E

I = IU

{m}

we denote Ce(fi) = Ce(Rn)In

.

Although the type of nonlinear PDEs in ( 2 . 1 ) , ( 2 , 3 ) covers a particularly large class of evolution type equations of applicative interest, the method of solving nonlinear PDEs through order completion is not restricted to bounded domains n, see the sequel, and the end of Section 5. It is not restricted either to the type of equations with T(x,D) in ( 2 . 3 ) , see end of this Section, as well as Section 8 and the sequel. The only reason that, at this first stage, we study equations of type ( 2 . 1 ) , ( 2 , 3 ) and consider them on bounded domains comes from the fact that Proposition 2 . 1 , and especially Lemma 2.1 below, are so much easier t o prove under these

12

conditions. Therefore, in this way, we can gain a good insight into the essence of the mathematics involved, without being distracted by technicalities. The basic result on the approximation of solutions is the following, and it is important to note that we do not require any kind of monotonicity conditions on the nonlinear PDEs in ( 2 . 1 ) , ( 2 . 3 ) . Proposit ion 2.1

Let fl be bounded and T(x,D) I f f E C0(n) then

given by ( 2 . 3 ) .

v E > o :

(2.5)

3 I'E c fl

closed, nowhere dense, U, E Cm(n\rE)

f - E 5 T(x,D)U,

<

:

f on n\r,

Proof

Let us take a > 0, such that fl c [-a,a]". Let us assume that by continuous extension, we have ci, f E Co([-a,aIn). Given now E > 0 , from Lemma 2 . 1 below we obtain 6 > 0. Finally, let us subdivide [-a,a] in intervals I 1 , . . . ,

5,

n-dimensional interval

such that each

... xlAn

J = IA x 1

with 1 5 A, ,...,An <_ 1

p,

has a diameter not exceeding 6.

If aJ E J is the centre of the above type of n-dimensional interval J, then according to Lemma 2 . 1 , there exists a polynomial f o r x E int J, we have

f(x)

- E

<

T(x,D)P

"J

(x) 5 f(x).

P "J

,

such that,

13

Let us now t a k e

r,

=

([-a,aln

\ u

i n t J)

nn

where t h e union ranges over a l l n-dimensional i n t e r v a l s we d e f i n e U, E Cm(n \ r,) by U = P

J

in (2.6).

If

0nflnint.J

"J

then ( 2 . 5 ) follows.

Remark 2.3 1)

The presence of t h e c l o s e d , nowhere dense s i n g u l a r i t y set

in

Proposition 2 . 1 i s a q u i t e n a t u r a l , minimal and unavoidable type of lack of global r e g u l a r i t y . Indeed, even i n t h e p a r t i c u l a r case when both T(x,D) and f a r e a n a l y t i c , we cannot expect t o have c l a s s i c a l s o l u t i o n s on t h e whole of t h e domain of d e f i n i t i o n a. I n f a c t , even i n t h a t a n a l y t i c c a s e , t h e best general global e x i s t e n c e r e s u l t s t i l l involves a closed, nowhere dense s i n g u l a r i t y s e t , which as i n ( 2 . 7 ) below, can i n f a c t have zero Lebesgue measure, see f o r d e t a i l s Chapter 2 i n Rosinger [6]. 2)

I n view of t h e form of T(x,D) i n ( 2 . 3 ) , it follows from t h e proof of Lemma 2 . 1 t h a t U, can i n f a c t be chosen on each i n t e r v a l J i n ( 2 . 6 ) , as polynomial of degree m i n t h e v a r i a b l e y E IRn-l, see ( 2 . 3 ) .

3)

I n view of t h e f a c t t h a t

rC

t

alone, being independent of

i s obtained as a r e c t a n g u l a r g r i d

generated by a f i n i t e number of hyperplanes, we obviously have (2.7) 4)

mes

(rE)

=

0.

By a similar construction, one can obtain a version of ( 2 . 5 ) , i n which

14

f

(2.8)

s T(x,D)U,

3 f+

rE,

on n \

E

Leua 2.1

Given f E C o ( n ) ,

then

V€>O: 3 6 > 0 : v X o ~ n :

polynomial in x

3 pX

(2.9)

E

IR"

:

0

Remark 2.2 As seen in the proof of Lemma 2 . 1 , we can take 6 = min

{T(E),

C,

1)

where r is the modulus of continuity of f and C > 0 is suitably chosen, depending on the coefficients ci in ( 2 . 3 ) . Furthermore, we can take Px (x) = a (t xO

0

where

x

=

-

to)

m

(t,y), xo = (to,yo)

E

IR",

see ( 2 . 3 ) , while

ax

E IR

is

0

suitably chosen. Proof of Leua 2.1

Since f i s uniformly continuous on fi, 6 > 0, such that 1

f o r given E > 0, we can take

15

For given xo

(to,yo)

=

Px (x) 0

=

a(t

E

n, to)

-

l e t us take m

,

with x

( t , y ) E Rn

=

.

Then ( 2 . 3 ) yields T(xo,D)Px (xo) = am! 0

n,

while f o r x = ( t , y ) E

we obtain

We note t h a t

Let us now take a = (f(xo) - ~/2)/m! C > 0

It follows t h a t t h e r e e x i s t s and

the

of we have

the

structure

x = ( t , y ) E fl,

IT(x,D)Px (x)

2

It - tol

<

62

products

<

am![

<

1, such that

*C

in

T(x,D),

C ( t - tol, for

-

0

Let us now take 0 < 6

depending only on

E It - t o ( 5 ;I

.

llfll,

m,

such

It - tol

<

I.

,

that,

llcillm for

16

Then, we can t a k e

.

6 = min (6 ,6 } 1

n,

Given x,xo E

2

we have

T(x,D)Px (x) - f ( x )

=

0

(T(x,D)P

xO

But i n case

(x) - am!) + (am!

am! - f ( x0) = E

- ; I

s

-

f ( x o ) ) + (f(xo) - f ( x ) ) .

it follows t h a t

IIx - xoll 5 6,

Since

-

f

f(xo) - f ( x )

s

E

; I

t h i s completes t h e proof of ( 2 . 9 ) .

Now l e t us r e t u r n t o the general case of t h e continuous nonlinear PDEs i n ( 2 . 1 ) , ( 2 . 2 ) , and consider them on an a r b i t r a r y , possibly unbounded domain

n.

For x

E

n,

l e t us denote

t h a t i s , t h e range i n IR of F(x, ...). Since F is j o i n t l y continuous i n a l l of its arguments, it follows t h a t Rx must be a bounded or halfbounded i n t e r v a l i n IR, or p o s s i b l y , t h e whole of iR i t s e l f . This l a t t e r c a s e , which i s e a s i e r t o deal with, see (2.11) below, happens with a l l of t h e l i n e a r PDEs, a s well as with most of t h e nonlinear PDEs of a p p l i c a t i v e i n t e r e s t . I n p a r t i c u l a r , it obviously happens with t h e type of n o n l i n e a r p a r t i a l d i f f e r e n t i a l operators i n ( 2 . 3 ) .

17

In the case of systems of linear o r nonlinear PDEs, such as for instance dealt with in Section 8, the range set Rx will obviously be a subset of a corresponding multidimensional Euclidean space. Given x E n, it is obvious that a n e c e s s a r y condition for the existence of a c l a s s i c a l solution of ( 2 . 1 ) , ( 2 . 2 ) in a neighbourhood of x is that (2.11)

f(x) E Rx.

We shall assume that, instead of ( 2 . 1 1 ) , we have satisfied the somewhat stronger condition (2.12)

f(x)

E

int Rx, x E ll

.

Obviously, when

then both conditions ( 2 . 1 1 ) and ( 2 . 1 2 ) are satisfied. Therefore, the results in the rest of this Section apply to the particular nonlinear PDEs in ( 2 . 1 ) , ( 2 . 3 ) as well. The corresponding general version of Lemma 2 . 1 above, this time valid for continuous nonlinear PDEs in ( 2 . 1 ) , ( 2 . 2 ) , on possibly unbounded domains n, is presented now. Lena 2.2

Given f E Co(n),

then

vx,~n, 0 0 : (2.13)

3 6 > 0,

vx€n: IIX -

P polynomial

xolI 5 6

* f(x)

18

Proof Let us take xo E

tP

n.

Then, f o r

6

with p E lNn,

E R,

> 0 small enough, (2.12) yields IpI

<

m

such t h a t F ( x o , t o , . . . , t ,,... ) = f ( x o ) - 2 c

Let us take a polynomial P i n x E Rn,

tp,

D!p(xo) =

P E

or”,

*

which s a t i s f i e s the condition

P I s m .

In t h i s case, obviously, we obtain (2.14)

T(xo,D)P(x0) - f(xo) =

-

Since both T(x,D)P(x) and f ( x ) (2.13) follows easily from (2.14).

are continuous i n

x E 0,

property 0

The important general approximation r e s u l t , corresponding t o Proposition 2.1 above, and valid f o r nonlinear p a r t i a l d i f f e r e n t i a l operators of type (2.2), on possibly unbounded domains 0 , is given i n Proposition 2.2 If

f

E

(2.15)

C0(n),

v € > o : 3 I’€ c ll closed, nowhere dense, U6 f -

Proof Let

then

6

5 T(x,D)UE s f

on “\re

E

Cm(n\re) :

19

where, for u E IN,

with

the compact s e t s Ku

a r e n-dimensional i n t e r v a l s

au = (au,lY--,au,n), bU = ( b y , l ' . . , b u , n ) E Rn, a u , l I bu,l,...Ya v,n 5 bu,n We also assume t h a t Ky, with u E IN, a r e l o c a l l y f i n i t e , t h a t is

.

(2.17)

v x ~ n : 3 x E Vx g n, {v E IN

I

Vx

neighbourhood of

Ku n Vx #

d} i s

finite

We a l s o assume t h a t the i n t e r i o r s of disjoint.

We note t h a t such

with

Ky,

x :

with

Ku,

u E DI,

u E IN,

a r e pairwise

e x i s t , see for instance

Forster.

Let us take now 6 > 0 given a r b i t r a r i l y but fixed. Let u s take u E IN and apply Lemma 2 . 2 t o each xo E K,,. Then we obtain 6 > 0 and P XO

polynomial i n x E Rn,

Since Ku

such that

is compact, it follows t h a t 3 6 > 0 :

V xo E Ku :

(2.18)

3 P

XO

polynomial i n x E Rn :

v x ~ n :

IIx

- x0ll

5 6

* f ( x ) - < T(x,D)P 6

XO

(x)

f(x)

XO

20

Now, i n view of ( 2 . 1 8 ) , within the compact

Ku, we can repeat t h e argument

c Ku

i n the proof of Proposition 2 . 1 and obtain generated by a f i n i t e number of hyperplanes.

a rectangular g r i d

Further, we can define

such t h a t

<

f - c

KU\rv,€ .

T(x,D)UU,€ 5 f on

But i n view of (2.17) it follows t h a t ( 2.20)

rc

=

u rU,€

c fl and l'

U€M

= fl\ U

i n t Ku

a r e closed and nowhere

VEM

dense. Moreover ( 2 . 1 6 ) , (2.19) and (2.20) w i l l yield

which s a t i s f i e s ( 2 . 1 5 ) .

H

Benark 2.3 1)

As i n ( 2 . 7 ) , i n t h e general case treated i n Proposition 2 . 2 , we again have

( 2.21)

mes

(re)

=

o

since according t o ( 2 . 2 0 ) ,

i s a countable union of rectangular

g r i d s , each generated by a f i n i t e number of hyperplanes. 2)

As i n ( 2 . 8 ) , by a similar argument, we can obtain a version of ( 2 . 1 5 ) , i n which

21 ( 2 .22) 3)

f

<

T(x,D)Ut <_ f

t

c

on fl\rc.

It i s easy t o n o t e , by following t h e argument i n t h e proof of Lemma 2 . 2 , t h a t ( 2 . 1 1 ) implies t h e property

(2.23)

V x E n : 3 U E c"(fl) : T(x,D)U(x) = f ( x )

However, a d i r e c t a p p l i c a t i o n of Lemma 2 . 2 i t s e l f o f f e r s t h e s t r o n g e r property V X E n :

( 2 .24)

3 6 > 0 : V A c fl n B(x,6), 3 u E P(n) :

A finite

T(Y,D)U(Y) = f ( Y ) ,

:

Y E A

as well as a proof of it which can have an i n t e r e s t i n i t s e l f . Indeed, given x E fl and c > 0 , we obtain from ( 2 . 1 3 ) ( 2 .25)

U- E c"(fl)

such t h a t

By a similar argument, we can f i n d ( 2 .27)

U+ E Cm(fl)

f o r which

Now, f o r A E [0,1]

l e t us define t h e c o n v e x c o m b i n a t i o n

22 (2.29)

uA =

(1 - n)u- + AU, E P(n)

and t h e continuous function p :

[o,q

n

+

RI

Then ( 2 . 2 6 ) , ( 2 . 2 8 ) give

p results i n

Hence the continuity of

(2.30)

V y 3 A

E Il E

n B(x,6)

:

[0,1] :

T(Y,D)UJ(Y) = f(Y) since the above equality j u s t means t h a t p(A,y) = 0. Let us now take any f i n i t e s e t A c n f l B(x,6) and apply (2.30) t o each a E A , obtaining thus UA E c"(n) which s a t i s f i e s ( 2 . 1 ) a t a . But a s A a

i s f i n i t e , we can consider on n a p a r t i t i o n of u n i t y given by aa E c"(n), with a E A , which s a t i s f i e s aa = 1 on a neighbourhood of (2.31)

0

<

a,

3 1 on n

Ca,=l aEA

on il

Let us then define U (2.32)

a

U = C aa* UAa aEA

E

c"(il) by

..

The r e l a t i o n s (2.29) - (2.32) w i l l now give ( 2 . 2 4 ) .

23

The i n t e r e s t i n ( 2 . 3 2 ) i s i n the f a c t t h e function U which it defines, and which s a t i s f i e s ( 2 . 2 4 ) , i s always a convex combination of t h e same two functions U- and U+ i n (2.25) - ( 2 . 2 8 ) . Finally we note t h a t i n a similar way, one can obtain t h e following stronger version of ( 2 . 2 4 ) , namely V A (2.33)

3

u

c

fl,

A d i s c r e t e i n fl :

E P(n)

T(x,D)U(x)

=

f(x),

x E A