View of The generalized solution of a nonlinear degenerate parabolic equation and its numerical computation

72 _{B. WOO|)}

U

ìÍÄTHIiT{A'IIC,{.

]ìEVUB I)'ÄNAI?YS}) NUÀ{TIRIQUE

D']' D}1 ,r.IfÉORIÐ DE I]',{PPROXIì{.A.TION

L'TINALYSìì

NUMEIìIQUE ET LA THE0RIE rlE L,appRoxIn[,ITIoNI

illome

lJ,

No

l,

lg8,4,

pp.

73_gB

I{EITERENCTìS

srø. [0, ^1] þar ^dcs 325-3-r3, (1981).

Laþl.acL : --1 þþlica- Sciences clc l'Uni- tntnn. n¿oduli of ord,.iuar I d.i.fferantiat, ()þcr.a.lors,

3, t_15, (1971).

I.Jtc ntod,ttli of sntoolhncss, tr{ath \¡eslril<, $(24),

bin.ati.otts of netúial

224 _ 1250,

Positiue Oþ cyatùt¿

n Institnte _nelhì,

ol j,it11,ar. positiu.c Oþcrators for IIi.glt.cr Ordcr

l lnstltute of ,I.cclrnolosy,

Kanprir (Incìia), of l:uncl,ions oJ a Rcal Variable, ìlacrlillan, J)oler pnl¡lications, Inc., N. r-. (igJS) ftece,rcrl 2S,X.1983.

t)t[alhenntics I)¿þayl¡t¿¿n.t

I'uit, rsilt, ,,t .l rizott,t.

l'ttt:¿t¿, -!ri.,t¡,t,,qi72i

TrfE GENERATJLED sor-,uTroN otr A NoNr,rNriAr{

DEGENERATE PARABOI-.IC EQUATION ANI)

ITS NUIVIERICAI, COMPUTATION

by

]1RVIN SCIfI.)C}ITBR

(Clu j-Napoca)

crassical

Our

h$'à:'i,ï,i,:lì:

cal

so1.t

_done

_b¡'

_{means oI}

;

co

ntinuo h

^r,ve show

that it

cot\/erges

in L,' to the

exact

one

(Theorern 5.2) .

One

dimensional_

problems of in

_[3],

1101. The e_xplicit diffeience

scheme _¡V

_îLr"

euthor in _[11-14.l but under

mor

úa-ely

,p."onr."*, lio

,of "lass

c

^t'å;r!r!:i

attentiorr is devotecl

in

_$

3, to

_{ontoge_}

neo11s bonndary ancl

initial

co

I. The diflorential

problcnr 'l'he problem \\¡e ate concerncd \\,ith can be

v,rittcn

as

(l'1) *:

^¿'¿

^or(,t) |

^o.(x,

^y, t) on

_Q

-a x ^{lo, 7't} (1.2)

u(x,

y,

0)

:

uo@,

),) ^{(x, t,)} = ç

(1.3) u(x, jt,4l,,:

r,r,r(x,

y, t) S:

^d12

x [0,

^7'],

where

o ^e

R2

is a

boundecl, convex domain,

0 < r ( fco. we

co'sicler.

trvo

spatial

variables on11., Tor the sake of siurplicity.

THE CENERALIZED SOLUTION 75

74 ^{ERVIN SCHECFITER}

The

functions occuring

in the

above problem

rvill be

subject

to

the following assumption:

(i)

uo eC(C¿);

q _{= C(S); a} ^e

^C(Q)

) ^{trs,'uþ a 2}

⁰

(A)

(ii) I ^{e ct(R+),}

^{9(w) and p'('u)}

)'0 ^Ior

^ø¿

> 0,

₉₍₀₎

:

0' I)enote

M 2

uo, 1tr, ^&.

We notice

that for thc

examples rve have

in rnind

(such as 9(u)

:

u'',

n <7),

^g'(0)

:0.

IIor,ver.er

this

condition

is not

neccssary

in ou¡

^consi-

derations.

r)[rrrNrlroN.

A

functiorytt'

e

1."(Q),

u > 0

(a.e.),

is

said

to

be a r'veak

sohition o1 (1.1)

-

^(1.3)

^if

^:

(i) It

satislies (1.2), (1.3)

tu a

generalized sense.

(ri)

_e@)

e

L2(0,

T;

ÍIL({))).

(iii) For

^any

f ^e

^{111(Q) such} that /1"

:

6, í).f _ ^âçþt) ôf _ ^{ôç(tt) ôf}

At âr ô'v ôy ^ôY

rf

tlx dv dt

t

has

the

tecluirecl properties

for

^(1'4)' Replacing

it in

(1'5) I^¡e get

I'he

second

integral

can

be written

^as:

¿ 3

\(,,,-r,t,)(,e(u,) - e@))dxctvctt.

l{[ ^Surrrr', - e1",¡ta'fx

a

x fr(,e@,) -

^e(x.z))

^{.it:.@@,) -}

^e@,))

drl?rtn4")- e(',))Ì

^{dxctv dt}

- _+\*{

ti i* ^- ry?1,") ⁿ ti r+' ^- ryl^)":l'.

^c'Iv

" :

: _;

r {(i,+* ^{- +?),,)'* (i} i'v ^- H*)"10. ^o,

(1.1)

\(.

5, -

^S

^{U {(r, y,} r),

^('",

y) ₌

^O}.

Condition (r;) is

to

be interpretecl

in

the fo1lou'ing ^{sense :}

there

exists a sequellce a,,

e

Cn (Ç) such

lhal, a,,+ u

^it:t' L'(Q),

unl"n

tt',

iri

I'z(S) ar'ð. v,,(x,

y, 0)-

^uo@,

t) ⁱⁿ

^Z'z(A).

rHnoRrrll l.l.

^{Under tlt'e} Assuntþtion

(A),

th'e þrobl'ent

(1'1) -

^(1'3)

Jt.as

of

nt.ost on,e ue(ah sol'xt,tion.

Proof..

(see

f9l)

^supposc

^that thcre are tu,o

solütions 'ttr,

u,

^antcl

prolc thät they

òoincicle a.e. 'l'hen b1' ^(1.4), (1 5)

I lt,, - ^{ù?{, -} i.k0,) ^ç(,,))

^'/,

- !,,{r{,,,) - p(",)) ','rfo*rb'

^rtt

:0'

ì I Òl ^c) t (\'

a

'I'he

particular

function

wl'rich

is

positive.

IIence

(,

pq - ^ur)(eþt,) -

^{e(ur)) ctx tly dt}

:

0 J

a

so

that ur -

^{tt,z a.e.}

^on

^Q'

Remark 1.1.

our- interpretation of condition,(z)

^o1

^th.e

Defiuition,

iupli"ì-lir" fãllo'ing

uss"rtinrr: .If .

the

corresp_onding

traces exist

^they

coiiicicle u,ith the d.atã functions. As

it

was proved

in _[7,

Ch.

VII, Th.

2'1]

the

traces exist provided

that

O

is

sufficiently regular'

2. Thc

tliliforcrrce Problem

Consid.et

the rectangular mesh R,, r'vith step

h

) ^{0 in the Ox'}

^Oy

directions ancl

t ) ^{0 in}

^Line

^Ot

d-irection, such

that 0 e Jl'

^l)enotc

O":

:

R,

ll ^O, Q,:

^Q

) ^{Rn and} x¡: ih, y¡ : _ih, tn: /at;

U¡¡(h)

:

[J(x',

j¡,

^t*)

: U(t).

Ptr.t

U¡(tt)

- u(h\ u(L

¹⁾

ancl sirnilar

ly ^(J;,

^(I-,

lor the

back'i,ara

dltt"r"o"es in thc

space variables.

The forrvard dilferences ^rvi1l be clenotcd-

Ly U-, U,,

^(I ,' d.x d.y dt

!

a a

-l-

i

C)

Í(x ^{v, t)} :

o4@,

y)f(x, y,

0) dxdy

lq@,)

- ^e(u,)ldr

76 ^ER\/IN SCIIECHTEIì

As

usual

L,,U(k)

:

U

;(lr) +

^U _r\(h.)'

To the differential problem

^(1.1)

-

^(1.3)

^we

^associate

^the

^follo$,ing

difference scheme:

(2.1) ^{u¡Uù: a,e(u(À))} +

^q'(tr)

^on

^Qn

(2.2)

U(0)

:

uot,

(2.3)

^Ulrn

-

^ur(x,

^Y, lrc) t¿

^1,

^2,

^.

'. ^/i : li]

I-Iere

u¡¡,:

^uçlao

and' 1,, is

the

set of points (xo, y¡)

€

R,, such ^{t:ha19 aL} least one

ot the

tour neighbouls

|

^(x¡+r,

_),j), ^{(x¡-t, y¡),}

^(xo,

^{y¡+t), (xoy¡-r) lies outside () '}

^O,,

: -rl \

n

- È¿/¡ . r/r,

Q,: {(*, ),, t); (*, y) -

^Q,,,

t:

t1,

t2, ...,

^t,,}.

The functions

u,

and 1,!'(r¡ are defined as follows:

ur(x,

y, t):

^or,r(x*,

1",

^t)

r*,,lreLe

(**,t"") ^e

^dQ

is the

nearest

point to (t,,

_1t)

(or

otre

of thcnr but the

same

on all

levels) ^.

TrrrioRliì{

2.I. IÍ

a.ssurnþtion

(A)

holds ctttd (J

is

a solutioto of lhc dif.fc' rctcce þroblem (2.1)

-

^{(2.3), tloen}

0<u3x[0,

where

A[o:0+7-)M.

Proof

;

We show

that

U(h')

=

⁽¹

+ ^kr)II

^for

^/r . ^1'

^')'

-;

^'

., 1i.

J.ssurrre

tlre contår)..'lhen

there exists a

triplet of

^jndices

(ut, tt,

h),

h>

^1so

^that

L|,,,,,(h)

> (i f- ^hllVI

^anð.Uo¡(h

-

¹⁾

= (1 I

^(/¿

- ^{\)r)Lt ior all}

^(i,

^j);U',,(h)

beíng maximtlm on Oo,

By the definition oI

^l'1'1,

(xn,,

Ju, rA) e O,, ancl

L,,(J,,,,,

<

0.

tstr.t

U,,,,,(h)

:

U,,,'(h

- ^{t) -l}

;t-\"'7(U""'(it'))

+

^o(n)

¡t:rd

LT,,,,,(k)

r

⁽¹

+

^ltr)X4,

u'hich

ieads

to a

conttadictiorr.

5 rllll cENEIìALtzrD SOLUTION ₇₇

No*,

cousid.er

the following linear

honogeneous algebraic system:

V¿¡

: rù,,(a¿¡V¡¡) (*t,

y¡)

=

^Q,,

(2.4)

^I/¡¡ly

n: 0

^a¿

^>

^0'

r,¡ì¡r,\

^2.1

. Iìor

^{curu¡, giaem a.¡¡, the systent,} (2.4) ød,mifs

only

the triuial'

sohtliovt,.

Proof

; II, on the ^contrary, for an

_.(x,*,

y) _{= %,.}

".g. V,,, 10,

^this

cotrple cá1

be

chosen

so that

L,o(a,,uVu*)

ì ^{0, but this}

contradicts (2.4) .

A siuitar

argurnertt applics r'vhen 2,,,,

¡

^0'

Consider

nol',

the linear system:

Vvtj(k)

:

Z¡¡(/t)

)-

t\',(o"¡¡(lt)Wü(h))

+ m¡¡(/e) ^(x,,

^y¡)

=

^{In

(2.5)

^I|/¡¡(lt)

:

Z,¡(k)

ott

^I',, a¡¡

2 0,

7' ^'.

{ln-' ft'

coRoI,L-\I{Y 2.1.

lìor u

^giuttrt'

Z ^and

^h

: l, 2,

' '

', I{

^{Jixecl' (2'5)}

has ct ttniqwe soltttiott ^.

This is

^an inunecliate consecluence

oI

Lemma ^2.1.

A¡other-. conseqllellce

oI the

sa¡re Irem¡ra refers

to the

systeÛl ^:

l4r,¡(k).-

^I(¡¡(lt

-- l) Ì

tÀ,(øiift)W¡¡Qt))

)-

^m.¡¡(h)

^on

^O,

(2

6)

I'I/tj(tt)

:.

t¿z(xt, -)'¡,

tt) (xr,

_-t'¡)

= ^1,,, lt: 1, 2,

_'

"' Ii'

lV4(O)

-

^rr,,(r¡,

-i1) ^on

^Oo'

rrcr-e

I,'iQu-, R, I'

= 0,

_ç(0)

:0, I =

^Ct(R*), eil _T'¡r(h)

"(/'D-

^V¡¡(t¡\

⁺

^O

e'(0) ^Vrj(h):0

r,ri-\L-1t.\

2:2.

Stt'þþost: co'nditi'on

('1)

hotrls and,

V:Q,,-, Iì is

g'iveto such

tltot V 2 0

^a'tod. tz

r.r

^rL,.

^'fhen

^{(2.6) høs}

^a

^{xtniquc sol'Lttion}

l,l' :

Qn-

^IN

sttclt tlt,øt

II/ ^>

0,

Prottf

; It is rcadily

scen

frorn corollary

^2.1

that there is a

unique solution

for the

svsterr.

II I7

takes negative ^va1t1es, there

exists.t

^couplc

(nt.,

n)

such tha.t

trl ,,,,,(lt)

< 0,

Ln(o,,,trY,,,,,(h))

>

0

a:ncl 11r,,,,,(h 1)

>

^0,

but this

is impossible accorcling

to

l-hc ecluation.

'lìr

:

lernma

is

provecl.

4

a¡¡(lt)

-

I

78

F'or

a

fixecl

å e _{1, 2, ..., Ii},

(2.6) delines on operator:

G(/a) : R"n

* Ri, Il(D *

IA&), s

.:

card f)r,.

r,riìnrÀ

2.3.

Under tlte assu,m.þtion,

(A)

thc oþerølor

G(k)

hns

ø

^rt.nirlue

Jixed þoint .for

^nn¡t

^k - l,2, ..., ^I{.

Ì?roof

:

^{Since 14/}

^>

^0,

llw ^(þ.)ll,

- h,l ^lr,i(h)l -

^/t,

I

^Vr,,¡n¡

:

Q" Qt'

: lfÐ(W,¡(1,

_()n

-

¹⁾

+

r\,^(a¡¡(lt)Vvtj(ÌèD

I

^,t,¡(t¡).

I-Ierrce,

if

u,e

prrt

),

-

^rllt2,

ltI|l(tùll, < ttrlttr,,ln

_ll

-

¹⁾

⁺

2^tù>:,e94/¡¡(tt))

J,

^¡tp

I

^a¡(/r),

u ll', f) _u

bccatrse I,V

: V on

I-r.

l)enote

p(,1)

: ttlW,6 * t¡ |

2tt ).nt,(ÒQ)

|

^{r ùtrn(Q)}

and

S;"(,I)

: _{{ø e R-, x} 2 0, llrll, !

^p(/r)}.

'l'hus

G(À)

:RÌ' -"

^{So (/¿)}

atrcl lìror'ver's theorern is applicable.

lhe

unicluuess follou's ^flon-L f,emrna 2.2 Rcnrark 2.1. The fixecl

points U(ñ), l¿:1, 2, ..., 1(,

satisf\':

(2.7) U¡(t) -:

A^9(U(A))

+ &(h) ^o11

Q,,

(2.8)

tJ(0)

-

^xrot,

(2

9)

^Ul¡.,,

:

xt,,.

3.

llÍesh-hurcl.it¡ns

'l'o

begirr

rvith,

u,e

rccall the formula of partial

snnrruatiotr Su.ppose

U, V arc

r'ectors

with

components

U,, V,, þ=i=q þ, ^q =2, þ<q,

h l, ^{¡tlu))n 1-} u,tv,t

^c

_¡IÌ,

h:þ lr

7 ^THE GENERALIZED SoLUrloN _7l)

In

^order

^to

cleiine cxtensions

oi

rnesh-functiotr, g'e introduce

the "cells"

;

o¡¡: _{(x,,r,) ^e pz

_ilt,

{ ^x 1:(i t

^{\)lL, jh,}

^<1, <(j + ^l)k}

Q1(/t)

-

^<,t¡i

X lln,

^(å

-l-

^1)r[.

Ðenote

a), : _{U or,r,}

_Q,,

: _l)

^q,,.

o--c l) ^,l _¡¡.(J

\Ve obscrve

that

O,, is

the

rectangular donrain generatcd

bythe

mesh-poìnts also detroted b1'

1¡,. Ihis is the

greatest such clomain contained'

in Q'

Q,,

has

a

similar

propelty. fn the

same ^\Ya)¡

the

smallest

rectangular

domain

containing f) respectivcly Q

^are:

Ai,: U a¡j, 0,,: U

_{tr,r0QlO ttijîQ+o} ^in,.

lu thc

secluei

l e shall

use

three types oI

finite-clemeut

interpolauts

oI nresh-Iunctions.

ø) (0)-interpolation: Given

V,:

_Qu-o R,

rts

(0)-interpola1r- fr,, is cleTjncd

in the

rectangular clotlaitr _Qn as Ïoliorl,s ^: i,,l,t,,tt)

-

^V.,¡(k)

for any

_Aii(lt)

çç,,.

Ò)

(l)-interpolation: Ihis

assigns

Ío

Vn

a

continuous

function

Ví,

deli'

rred on _Q,,, such

that on

each cell q¡¡(h)

it is the

T,agrange interpolate _{_of} clcgrce one

in

each

variablc, of the

rralues ^o1'

Z ^on the

vertices. Clearil',

I/;,

lnas integrable

first order

generalized derivatives.

c)

Mixed interpolation:

z1r¡

is

^defined

on the

rectangular domain

Ç, in the folloling

manner :

on

each rl¡¡(lt)

ít

is linear in

y

and

I

ancl cons-

iant in x.

V¡¡(x.,,

y, t) is an interpolation polinomial

orr

the

face ₁₁ <

f

_-t,

3 ),j+t,

^l¡o

< t < t¡,¡1.

Arralogously

one

clefines

Vp¡ lor

constant ^¡/.

Clearly, all the

^¿r-bove extensions

can be

adapted. when _Q,,

is

changed"

irrto _Ç,,- or f),,.

In

lvhat follo¡¡,s extensions defined on _Q,,

or Q, ^will

^atltoma-

tically be

prolonged

for ¡ e lIG, ^?'l

^by

V'(t):it(t):V(I{r), ^{¡ e} lIG, Tl.

Thus tlre qualitics

ol V'

^anð,

i ^wlll

^remain unchangecl.

It is

easil1. seen

that:

(g.2) Y : V)u¡ ^{on an!}

^Q,¡(k)

ancl

analogousl1,

for

y.

ERVIN SCI{ECI]TER 6

(3.1)

^lø _l¡:þtl-1

l,

^{(l/,)r, u n}

BO IIRVIN SCI-IECHTEIì ô

llecause

lve

are

mainly

concerncd i,vith

the situation'"vhen

l¿,

c*0

u'e shall deal

with

(generalized) sequences

of

mesh functions. Neverteless, we shall speak

of

a

function V (or Zr)

d.efined

on

Ro'

Next,

u'e

iecall

an

important

theorem due

to fadyzhenskaia [5]'

-lrIEoRBì{

3.I.

Suþþose tløat:

(i)

Tkere

is a

constønt C ind'eþendent of lt' and

I

^{sucln tl't'at}

for

^{V :}

^Qon -R:

(3.3) "h'ÐVz

_0à

^<

^C,

(iù V is

d.efinecl on tlte nr.esk-þoints outside _Qo so th'at (3,3) kol'cls on the zahol,e Ro.

'flrcn,

if

^one of the sequence

{i}, ^{V'}, {Vo}, {Vp¡}

^{isueøkl,y conuergent}

in

L'z(Q) uhen /t,,

t*0,

^{th.e søtne-is}

truefor

tlte other tlt'ree sequences of exten- sxons.

'I'he properties we are formulating below

will play an important

rôle

in the following

sections.

(P)

:

The couple (Q,

ur) is

said

to

have

the property (P) il

there exists a

fnrrction

f :Q- ^R

^sucir

^that:

(ù "fl": ut

^S

:

ôO

x [0, r]

(i.i)

Í ^e

^C(Ç)

^with

bounded d.erivatives

up to

^th.e second order

in r,

_1r

and Tirst order in

/.

(P,) : The couple

((),

ør) is said to have thc property (P,)

iÎ

there exists

a

nresh-fu.nction

J

^: Qo-"

R

such

that

(ù .fn(h)lro: %,

h

: l, 2,

^{. .}

., K

(iù

"f¡(h),

"f"(k),fr}ù,f,;(Þ),"f,7(Ìù areboundeclonfl, h: l,2, ..., ^K.

the

dillerences are

taken

on

points of the

mesh _Q,,

on

r,vhich

they

rnake sense. ^LLz

was

defined

for condition

(2.3).

Remark 3.1.

If ^(P)

holds,

the

restriction oI

f ^to

^{Liue mesh Ç,, satislies}

condition (ii) of (P,)

ancl

f(h)|",:1(2 ^)-'rn, lt': r' 2' " '' I{'

^lr,,l

^<Clt'

C

independent

of h, t.

Thus (Pr) is "7tear\y"

satisfied.

IÌemark

3.2.

I:f ø, is

independent

ol x, y,

tb,en

ur-

u,L

aîd

botln

(P) ancl (P,)

hold.

'Ihis is a

situation very often occuring in practice. Man¡,' authors have studied the problem of the conditions under which (P) holds.

In

connection

with

parabolic problems

condition (P)

appears

in Friedman [4,

^Ch.

III,

$ 4l

(Class C21-") and fadyzhenskaia

[6]

^{(C1ass O2,1).}

TTIE CENERÄLIZED SOLUTION B1

/r. First'ordor

differenees

fn this

section

we are

concerned

rvith first

orcler diÏferences

of

the discrete

solution U

and

their

boundedness.

First

here

is

some

notation:

oro

: _{(r,,

y¡)

= oo,

_@¿+t, ^{y¡) and (x¿,}

_{}i+r) =}

^c),,1

Qi :

_{(xu,

_{),¡, tt)} = ^Rni(xt, yi) =

^{Oo+, Ìt,}

:

L,

2, ..., Iq'

f¡-:1(xr,

y¡)

=Qni (x¡-t, yi) = f,) _i l¿-: _{(*,,

^y¡)

=1,; ^{(x¡ r,}

^{1t¡)e Ilu..}

+1,

_and

-1,

_have sirnilar meanings regarding y.

In ^the

^sequel

^{we shall}

^{alu'ays consider}

^{the solution U} of (2.I) -

^(2.3)

extencled over

the

mesh-points

of

_Qi,,

in the

following rvay:

(J(xu,

)'¡,

^tt)

:

tl(x¿',

!'r, tt), ^(xr,

^y¡)

= fii'"O,

where

(Ì",

y';'¡

e

dO

is the

nearest

point to

^(xt, ),¡).

'r'rlnor{E\r 4.1. Suþþosc thøt

(i) U

i,s t\t'e sol'ution o"f

(21) -

^(2.3)

(ii)

(O,

ur)

has ^tloe

þroþerty

^(P,,) ('íi'i) Cond,ition

(A)

^/t'old's

(ia) u, is

Liþschitz continuous.

Tlt,en tlt,cre cxists n. consttut't

C

indeþendent of h,,

r

^{sz+ch tlLat}

,hrD _@,(J), I

^q,(U),)

^<C

ot+

Proof

:

^Condition

^(li)

^ensures

^the

^existence

^of a fuuction V: Q,-ffi

strch

that

Vlru

:

r't',

with

bounded differences

up to the

scconcl

older in x, y

^and.

first

order

in L We take V : U

outside _Ç,,.

Put

14/

: U - ^VI

Multiplying both

sides

of

(2.1)

by

dr,zW(h) and summing uP,

s'e

get

(4.1) tlÈlw(tì)uí(h) :h,Dw(h)l\,,,e(u(h)) ! ^a.(n)l

^h

: r,2, ..., I{.

o.

lhe

leTt-hand. side can

be

transfortnecl using

the

iclentit-v ^: a(n

- t,),

^{-t, ltr.z}

-

^hz

I

@

-

^b)t)

into

(4.2)

I

!t,,D(u,(tl

': (t

- ^u,(h -

¹⁾

^{+ +ui(h))} - ^tJrzf

(). tzllr) Lr¡(ft

-

¹⁾

-'h t

6 - L'artaìr'sc numérir¡ue et la théorie dc I approxirnation

- Tone 13, No 1. 1984.

B2 ERVIN SCHECFITER

Since I4l

: 0

on

0ï...q.

^and,

W.: 0

on

ll ; W,: 0

on

-1,,

14.3) "h,ÐW(tr)L,,p(U(h,)) :

.crûÐW1n¡t^,pru(h))

:

f)à

-"l,,rrÐ

_of lw"(h)

e,pØ.)) +

^w,(h)

e,pft,))1.

Frrrther' ^.rve _have

- ^{t,,Dt/, ç.(u)} : h,Dv,;(u) * ,F ^{v;ç(u) _ hÐv,ç(u)}

A

^sirnilar

identity is valid

for

70

Ð', ^r,(u)

Replacing,

(42),(!_!),

(4.4)

in

(4.1).319 snmming up

for h: t, 2, ..., I{, we get after

applying

ouie

tròre'(3.1) :

,,u Ð _Çti ^{e" p,(u)} i

^L/,

e,(L\) . ^.,hrÐ _e¡

_to,

_{ulq(t) +} ,hf

_{ã: r}

_D v;ç(u) +

r.i

-r D

_i'¡

^vrvlu) -Ð

_I.t

v;ç(u)-D,

_-I.¡

v¡p(u)) +.,n\ lwlø *

i ^¡/t2D tv,lL ¡

^¡z

_Ð {u{r<)v(K) _ _v(0)u(00.

e¡

Q¡ o¡

Tnki'g i'to acco'.! (pr)

and

'rheoren 2.r it

follou.s

that there

exists a

constant C inclcpenclent

of the

mesh sizes such

that

"h'Ð ^{Q. ç,(u)} + ^{u, q,(o)) <}

^c.

aÍ

Finallv

u'e noticc t.hat,p,(u),z

s.q'(Mùr?"(u)u"anc1 that le"e)l

^{is bounded}

on

_Qi,*

'

01,,

so our cstilnaîc is'tiuËl

^'

rìemark

4.1

. .*-is.readily

_seen

_frorn tire proof of the abo'e

theorem

1la1 in _ProPcrty (1'r)-it is

cnough

to

r,,1,por" insteacr

ol lhe

boundcdrress oI

thc

_sccord orc]"i differcnccs,

tfiot

or À^'in thc

¿i air"ät,.

lror.nì.

Il-cmark 4.2. 'Ithcorcm

4.I

holds rvhen (r.,r)

is

replaced by

- ^(ir:'¡

^(Q,

^ur)

^iras

^{the property}

^(p).

rrrdeccl, let,s

takc in this

case

I/ : _-flo^ ^,lhcn _{the right}

hancl side

of

(4.3)

is tti

l.¡e rcplacecl by.

-.r'Ð fii,(h)e"(u(h)) ) tv,(tr)e"(u(tDl -,hÐwVt)e"pUrD _rt - -

^¡tË

^Ð -I''

^r,r,1z,;

^{e,gUù) +} ,hÐ!V(Ìt),e,p(ttD r), +,nÐwø1e,,(u(h)).

^r'

11 ^TFIT GENLRAL]ZED SOLUI'ION

tt ,1

Recalling

now that

according

to

r{emark 3.1,

there

exists

a

constant C

independenL

of /t,

such

that:

lw(h)lSCtr, ^on l,

lW"(h,)l

< C on r'h, lw,(k)l < C on t;;,

we have

lor

C independent

of

^/2,

itl)

g4w,1tt)e"V(tt))l

+ lw(h)lle"(uØ))t) < c

T_

tt))

g4w,1tr)e,(u(h))l

+ lw(h)l lç,(u(h))l) < c.

Thus,

the prool of

'I'heorem

4.1

can proceed unchauged. Refore passing

to the next

theorem rve introduce

the

follorving

notation:

M :

nlax _{'i

,

1y¡:(xu, y¡)

=

^{Qn} ;}

nx:

min

{i;

11,¡:(x,, y¡)

e

d),,}

and similarly N,

rr,

for

jr.

For a

given

n

S

j

= N, A[(j):

max

{i; ^(*,, !) =

^{Q^}}

Tlre

nreaning

ol m(j), N(i), n(i) is

^clear.

l'r-inonnì{ 4.2. Snþþose tlt,øt

(A)

ltold,s &nd.

U

^,is tlt.c solutio.n of _þroblciri,

Q.l) -

^(2.3). Assu,rte tl,tnt ot,r,is L,iþscloitz con,tittt,tous

,in x, y, t. Lat

con- dt:tioto

(P) or

(P,,) hold..'fhen th,e.re

exits n

constatot C indeþend.ent of lt,, ^s,ucl¿

lh,ct.t

,''o* ll a uj,ll ^{< c.}

¡o, r , ll

i.t _lll

^..1o¡

Prortf

:

\,\rs h¿vç:

to

shou.

that

i1¡ ^u'1r,

^:y,

^t)(x,

^{y)dx rt1,l}

.

.,1,1,11,,"(n),

,

rr

I

ior

arny

\l c

^Cí

(f))

and,

t c _{[0, 7'.].}

Reca11

that L/' u,as

extended tc,

Õì x to,

^11.

Iror

_{(ø, 1,,}

t)

^{e= g¡¡(lt)}

10

t''(x. t,.1) -. _-L

LU,¡(h) L¡¡e(x,

j,,

^t)

., \-_, ), ,t .lr"_,

n'lrere

tlre

sum

is

exte¡rded or¡er-

the

vcrtices

ol

^clr¡(h) and

for P e

^Il,,

ft,

f\

^{q¿¡(k) :}

,,,,,

(tr\ - ^t I lor. P -

^(x¡'

^!¡,

^tt,)

' lO for ^{P +}

^(x,,

)'¡,

^tt,)

thc

basic fur.ictiorrs

L¡¡¡

arc

linear rn x, ¡t,

^t.

,8-1

ÌIence on

q¡¡(lz),

k :0,

I,

nSo r'r'e harre

ERVIN SCI-IECIITEIì

I2

¿i

u'(.r, v, ù :

) ^l{*,*, -

^x)(y¡+,

-

y)(u¡),tr,71 _1(/r)

1

1- ^(,x¡+,

- ^x)(1,-

),¡)(U¡)¡+r,¡(tù

I

^(x

- ^x,)(y¡t,

^_1,) (J)¡,¡1_1(Ìe)

| +

^(.v

- ^x,)(! - ^ylp)4ft)).

13

In

vieu, of

the

definitio¡r t.I l,¡j

a¡d

reclling

that

(,,

=

^{Cí (O) :}

t,t,¡t, t@u))

s c,

'rax {r+r, l; I

^I

^l,ll ^,

^any

^{i, j}

ancl

*fr1 {ldrtit-r,¡1,

1d,,,1',,¡i}

<Crtt'-"- _I#I

Using l'heorem 4.1

r-1 ^11(j)-2

*Ð ,,t^,,Ip.(uo(A))

^('tn¡).,l

< cn'rax

{r+r, lfll, l#l}

ü.)

,K-1

\,i,

^{tu;,t*, .v, t)) þ(x, y) rtx ,t_t,}

-

"l ^l, {u;,{*,r,,

^¿))

ü(r,

.v)

t u

^tt¡,

I

of \ ^r¿¿

(4.5)

_a-

_o,Ð (L,,e(J,¡(tl)

^_t

n,¡(k))\!

110,,U, s)ctr d.s

j

+ h,D

I',, d,,

Ílere Jor

(x,,

),j,

^t¡)

e

{l¡,

d¡¡(r,

s):

(1

- ^/)(t -,)ü(", f

^rtt,

^y¡ f

^s/¿)

*

1-

r(I -

^r)

^{ü(r, t}

^{_F r/t.,}

^y¡ I

^sh)

|

^rsþ(x¡_1 1_

rh,

_{;v¡_r}

*

^s/4

*

J-

⁽¹

- ^r)sþ(x, I

^{rlt., .;t,_r}

l-

^r/r).

t_!,!:,:)'ii

,t!), - ^[-,

^sorneof

theterms,ltrreright-hanclsidearezero.Rettrr- ntrg to _{il'ay U}

u'as extencled beyoncl (4.5) arcl

ta$1s into

accorurt f),,, u,c get the

r,iplchitz"orrlirr.rity

:

or urandäe

(,1.q

1.,1. ,] ^tut,r*, v,

^t) þ(x, v) ^{ct:v dv}

í,4.7) h

<

Ch max lþi

f)

Norv,

(4.8)

Sirnilar

consicler¿rtions

take

place Tor

hD

q,;((1,¡(/t))

ct¿¡: ,,'b'

^,:H.', ,e,;(J¡¡(tr))

^d,,

:

d_l N(j)_2 N_l

: -

^Jt, j -t i -n(jl

^{D n} D.ç"(Uø(/r))

(,t,1, 1-

hD

j=t s,((ju¡¡-,,¡(tr)) ct¡¡1¡_1,¡

_ -

^g "(U,,(jt,j(/r)) ^J,,t¡t,¡).

h'Ð ç,¡(u aft\

^d¡.

,O

:

Ir; ^{u e L'(0,}

^T

^{; Bo), 4 = L'(0, l' t}

^Br)1,

0 < ¹ < fcc,

^{.þ, ,l}

>

^1.

^I1

^{lve endow}

l( with the

norm

itatry,ç6,r t

"¡

nll rill,"*,,,",,,

, )r(o) ₀₀

! !

)t¡(tt)

tt,\\0,,,,,

s) dr ds

D

_l' ut

U .s d,r _ds

{

^Crh

urar

_{l,.f l}

o

So

if

u'e

takc

^jntc¡ accoLtnt

the

continuous imbedding ã3(A)

C

^Cl(O)"

estinates

⁽⁴

6) -

^(4.8)

^{prove our}

^theorem.

5.

Conlelgonce

oI tho

tliscrcte solution

l'his

section colrtajus the main result

ol

the paper Tornulated i-n'-[heo-

retn

=-.2.

We

also gir,re

here

sorne

lunctional analytic results l'hich

^rve,

neecl

in our

proofs.

Consider

three Banach

spaces

F6,

81,

Br, satis{ying the

lollorving algebraic

and

topological inclusions:

BoCBCB,.,

llo,

B L reflexive.

I'et

lØ becomes

a

Ban¿rch space

(obviorsly

W

CL'(0, T;

^B)).

r,Dr,IÌ\,IA 5.7. Sltþþose tJte imbedd,ing Bo

( Bt

^øs comþøct ond tJtql

I _'<

1þ,

^q

<o:.

^{Tlten, V¡r}

ç ^L'(0, T; B) is

also cor'u,þact.

For the þroof

^sec

ì,ions _i8l.

ôr) ^{ERVIN SCHECHTEI{} 74

r,Eì{r,rA 5.2. Søt'þþose

,l €

Ð(O

x [0, T[).

^Tlte'n

for

^lt,,

r

sufJ'iciently (õÇ,

-7p.(u) i, -

^7p,(u)Ç,

^+ãil)

^{ctx cty ctt 'l-}

* _[;,@, ^y)

^Ç@,

^y,

^{o) d.:v}

_'ty :

o.

l)

,Herc U is a solution of (2.1)

-:

^(2.3).

Proof

.'

Recall

that,

e.g., ^r1.,, is

the

(0)-extension

of

(tþ,)¡7(/e). -tllultiplying each cc¡ràtion

of

(2.1)

bythe

corresponding

valtes

^of {., i,veget after sumlna-

tion

^:

h

r tù

| Ð lur(¿) -

^{L,^}

^e(u(h)) -

^{o(n) I .¡(r,)}

:

'r.

rreuce

if t

is

t"o""tr"

,-a11

that

^,1,(1q

:

0, h,7

15.2) ,h,D D _o:o ^tu,(¿)

^,1,,(,1)

-

ç,((l(/t)),1,"(/r)

- e,(t/(ir))

^,1,,(/,)

-

o;l_

-

^o(h)

^ü(¿)l

^-t-

tûÇuo,l,(0) :0.

,\ou' if /t, t

are so amali

that

supp

{i,

^supp

ü,,

suPP,.l.',

çO, X ^t0,

^7'[,

{5.2}

becotres identical

to

^(5.1).

r,Dln{A

5.3. (I,ions l9l)

^Let

D

bc ø bounded' d'oma"in iu' W" ctn'cl I'L¡,

lt'

^e

'-: I-"(D), þ > l, j : l, 2, .

^{S¿r,þþose that:}

i,) llu¡ll¡=Cj:1,2,

¡.cith, tJ'tc consLnnt C ind'eþendent oJ

j.

1't'¡ -t 't¡ ø.e' ^{Oft' I) '}

xú

-7

^'r.t, 'aenkly in, L"(D) swall

{5.1)

iii)

','!'Jt,cu,

r,rìrn,rA 5.4. Sr't'l>þosc

that

,D

C R"

'ís

ø

bounded' d'ontøin' an'd th'at ^tke

::;t.(lu!:¿'tce

{"¡} CC(D), j : l, 2, ...

^h,as t}t'e Jol'l'ozui'ng þroþerlíes:

i,') -; ^lu¡lSC i-1,2,

1ii) u¡-t.

^{1,1, &..e. on D.}

(iii)

ihere'exists

q> |

such tlt,at w

= L"(D),Then

^the seql'Lencc.contøirt's

,,.

trrbstqutrt,

tonurrgrlnt'in

L(D) _Jor

any

þ'fl, {øl'

art'd tlt'e

limit

tt' e

,n I-'(D).

15 ^rtslE GENERALIZED soLUrIoN 87

Proof:Thepreviouslemrnaensllfesllnalu¡-->.ol,r,veaklyfuL,(D|.

Because ót _1;¡

ttr"t"

is a subsequence _{ø¿}

C

{u¡) ^such

that Lt¡-'L!,

weaklY

In L'(D),

fo ^nð' u e À L'(D).Since there

exists

a

constant C,

þ>- |

in ^that

llrrlí,;

< ^C, it

^follows

^that

^tt'

_{= L'(.D)'}

accordinf

to

Egorov's theorem

thcre

exrsts a slllall

m ^D

^{such lirrut}

^{ur-' ø}

^uniformlY

^on D\Do'

Thus for þ

^>-

l:

a

\ ", - ^rt,' d'x:

\ ", - u'

ctx

*

\ ^lt, -

^{1't'' dx'}

-D D,/ DU ^DO

Since

U,",- ^{u'a*)'t' =} ()"",

rlþ

f ^.)

dx ^II rlþ

p d

,tt Do

the

^{sequence and}

the

absolute

continuity of

the

u

in'

i'1n¡. In

^{what follorvs a}

^llar

^{over t1-re sub-}

,:"i;:i

subsequence' ('i) Con'd,ition

(A)

lr,ol'd's

(ii) u, is

Liþsctt'itz continuous

in øll

^aøriøbl'es

(iii) U is

tlte solution of þroblem (2.1)

-

^(2'3)

(iv)

(ct,t, Q) køs tke þroþerty ^(P,,)

or

(P).

Tlten

tlwre

^{cxists a} subsequence

{k, ,} < _{k, ,}

and. a f'tttr,ction

u = L'(Q),

ôu ?' _{o L,(o\ vtch}

_lJ¿at

;

(il

_k,,(UrD'

_-¡

^{I) in, L'(Q)}

for ^q e _[1,

^co _I

(jil

_@JU¡,)r'

-' 'u:, ^@,(ui))' n

ôJruectltl')'

i'n

^L'z(Q

(jjì _e(Û,,)+-ü

a..e. crucl

u 2 0 a'e'

on _Q'

ProoJ'; Ileca11

first that u

has been defined ^o11

Qi

and

the

extcnsion

u,

^a.s,o

ón t0, ^{11. From}

Theorem 4.1 ^and.

(ü)

we have

the', ^that

^:

(5.3)

^'c/ù

²

^(e,(u)z

t

e,@)2)

<C

'

Consecluently

taking iuto account theorem

^2'1

, ^lor a cotrstant Cr

^we

get ^: (5.4)

I

0;

(q(U)')' * (q"(U)')' I (ç,((l)')')

^{d'x dy dt}

<

^Cr

88

alld

^also

(s.5)

((p(U)')'

-l-

(ç.(t/)r'r)' I

(ç"(U)e¡)z)dx dy dt

<

^Cr

Qt:'

Here _Qif

is the

rectangular

domain

generated

by the

mesh-points

of

the

set

denoted

in the

same way.

Fronr

(5.5) \\¡e get

for

a subseqtlence

of

indeces

{k, r}:

q((l¡)'- a, e,(Ul,)(r)* ^ur,

eu(UT,)ot- ^az,

weakll' Á

^Lz(Q).

^Now,

^accotding

^to

^Theorem

^3.l:

,p"(LI;)'- ut,

gu(Uí)'

n rr,

rveakly

fu

^Lz(Q).

On

the other

hand,

(s.6)

(ç"(U))r,r

: þ#

rvhich iuiplics

(ç,(u))r,r

:'+

âu

òl

77 ^THT GENERÀLIZED SOLU'IION ^Qi)

Mot'corcr

if llterc

cxists ø cott,stant øo

2 0

such

lltøt ç(x) >

^x

for x ):

^xo'

lh,ett,

h,)

^U;

+

u,

i'n

^L'(Q)

for

^ønY

I ^<

1,

< ^-lrc.

Proof;

Suppose

u is the lunction of ^'Iheorem 5.1 and u:9-t(!)^

Clearly, "(i), (l;)-hold. Since

¹⁾

é I-'(Q), tt ₌ L'(Q)

and

bv femma

^5'3

Uu-

^u weakly

Á

L'(Q), which proves

(i,ii). 'lhe

solution

U of Q)) -

^(2'3)

uoiirfi",

^(5.1)

lor { = ø(o x _[0, rf) and

ft.,

t sulîiciently

^small.

Si'cc

ËRVIN SCITECHTER 16

u

(iií)

U;

-r

w.¿ucttÌtly

in L"(Q), I <.þ <co

^{a.n,d. n..e}

. lt, r-,0

i@, l,0)-'ù(r, ^{y, O),} ''iTor'rly in strictly í'terior

domains,

\\'c

stc

itbt"-(S.Í¡ iL'.i. iake'into

account

(ù, (iù,

tlnat

u is a

solution

ol

(1.'l).

Finally iÎ

q@)

2 I ^for

^v,

2

^N¡,

then

¹⁹ _'@)

l.x,

^so

^that qt(x)<xIxo,Torx20'

l'his

implics

that

n-t191L'¡)')-'

^'t¿

in L"(Q) lot

a:n1'

þ e _[1,

-f co[.

{i. Initial a¡tl

hountlar¡i contlitions. Liniqnncss

oI

aproximal'ing sequcrrúc"

LrtIr{A ^6.1. (I'ions

_kSl)

_ry X is ¿ ^{L'(0, I'.;}

^X'¡.'.

a¡¡alZ t;10, 7';'x), t 1'þ < ¡,n,

^changing

^it

õn. ø set oJ rt'easure zero

of ]0,

^{7-[) ls}

coNsrieu¡NcÞ 6.1. ^Suþþose

u

is the solut'ion constructed,'in Tlteot'¿ttt 5'1

l-lten

u(0)

: ltlr:o

nt'a'hcs sense.

f rrdcecl

iI

^n<'

takc in

^(1.a)

J@,

y, t) : _{f'(x, t)fr(t)}

u,ith /, ^e

^Ð(o),

^{Jz = Ø(0,} î) it

^becomes:

l(, ,o#), r,) - ^- _U+', ¡ù,i:)-U+, _{¡,), T;\*}

+ ^((a,J,), f,): ((Lq(")' Í,), f') +

^((a'

-f")' Í')'

This

inrplies

that

equation (1.1)

is

^also

^fuÏilled in the

sense

oI

^clis-

tributions.

^Because

ôe^fu),ôe!)=Lr(Q)

òx ôt)

ù*

t)u

âx

ancl also

(jj). 'lhls

completes

the

proof ^.

Xcxt,"lêt

us take

in

T,emma-íJ,

þ: q:2,

^B0

- ^f1'(O), B: ^L'\t)), r

^>-

| and finite, 81- H4(A). Then by

Thcorem

4.2

anð'

(5'6),

rp(U;)'

ís prccompact in

L2(0,

'1'; L,(Al. Then a

subsecFreuce (clenoted

in

the

tutrì"

.n,oyi

oI 9(J¡)',

converges Lo

u =

^Lz(Q)

in I-'(Q).'l'his

enables us to aply

lemma

^{5.4 rvhich. proves (i) .}

this

entails, passi¡g

if

neccssat¡'

to

alro-

thet

snbsecltlence,

(jjj). 'lhe prool is

completed.

Next,

ir,e introãirce

a neit

continnous extension

o[ the

discretc lnnc-

tion [/

definecl on

the

mesh points of Qr,. Namel5'

u; : ç-'(ç(ur)')-

This is a continuous interpolate

of U

or.er the rectangular clomain Qr, ivhich belongs

to

Hr(Ql,).

,rlrrlc,r.Ðlr'3,2.

tln¿r,

the hyþotltcse s

of

Th,corert,5,l , tlt,t're cxisls rt

func-

t'ion,

u - L'(Q), u >0

^{ø.e., uitl't'}

ôq(u) ôr

^{i)q(u) ây}

-

^Lr(Q)

such. I.lt,øt

for ^a

^sul.tøble subsequence

of

ind'ices

{h, t}:

(i) 9(U;)'- e(u) in

L"(Q)

for

^nny

þ e ll, ^cnl

(1i)

e,(U¡)'- ry ^, p,(U¡)'-, 'l:,u

^ueahly

itt'

^L'(Q)

90

\\¡e ha\¡e

4:

_ryçn¡

i ^ø =

^L2(0,

T;

^H-tQD.

ancl l,emma 6.1 can be applied.

with X :

(¡1-1(O)'

--- _ÍriË

fãllowing

lemma^ii a

particular'

"as" oi îh"ot"ttt ^{2.1 hom}

^r,ions

17,

Vol. IIl.

L''

"n**ro', 6.2. Let

.f = Lr(o,

"; ^Ht(o))

^and. suþþos.e^.Ç2

to

be sufficientl'y ragtilø.i. Then the ^trøce

fl,

exists ønd belongs to H.'Pp(2.)'

' _Nlorrorrr

tke mø'þiiug:

t, * ul,,

^H''o(Q)

*¡1ttz'o(S)

^'is contitt'uott's' IIere

¡7rrz,o(S)

: Lz(0, T);

H1t2(ôe))

C

^¿,(S).

'r'HÐoREM6.:l.IfQissr'tfficienttyregul'ørnttdcond'ition'sofTh'eoremS'l

h,ol,rl, tka.n

u

_þossesses ø trøce

ul*

Proof

:

Since 9(ø)

=

^H''o(Q),

^by

T,emma 6'2' ,p(u)1,

=

¡¡trz,o(S)

C¿'(S).

on tlre

other hand.

q-t(x) < x I

^{øo so}

^that

ø1" also

exists

and.

the

trace operator

-'-- is

continuous.

t*'Doonnn

6'2'

søt'þþose tloat

Q

'is sufficiert'tl¡''

"rt-'n''

^{Assurnn thct't t't'}

is ^tlte

^soluti,on

tontirrñíttt in

^{Theorent' 5'2 øntt ih'at}

^-tltt

conditions

of

tkis tlteorent, are

fulfillecl.

^Then'

uls

:

xat

øn,d

Ull, n

%lr, uniJ'ormlY on ^S.

piooJ

:

^According

to ^'lheorem 5.2 there cxists a

seqtlerlce {U,,}

C C ÍU¡j,

^n,

:

1,

2,

. . .

,

^(Ju

CC(Q)

^such

^that

(J,,-'" 1'(,

in L'(Q), P e

_[1, ^oo[.

At the

same time there is another ^seqtlence

V,=

^{C' (Ç) such}

^that

Proof

; ^By

Consecluence 6.1, øl¿:6 exists

ald

belongs

to

Z'z(O)' Using

the

same seçFlence as

ln the proof of the

previous theorem rve get ^: (Jn(x,

t,

^{0) --> oto@,}

^Y)

^{on Q.}

The proof is

complete.

A better

und.erstaãclinq

of

how (1.3) is

fulfilled

and a more precise charac- terizatiotl

of the regulãrity

^of

O'is liven in

^Theorem

^6.4. But first

we for- mulate:

rrrìrrìrÀ 6.3. Suþþosr

o q

R2

is

^{ct. bound,ecl} dont,øin such thøt _.its fy.2n'

iltr ilili

diuictcdinh.¡lnlte ^õwmber

of

arcs whose tangents ntah,e witlø eitker

Ox or Olt

a.n øngle greøtev thø'n_ø þositiae ^constønt.

Then

for

^{an1, fwnction}

^V: Qr,nR

^ønd,

øly

sufficientl'y sma'll

r <0,

thcre c:vist constanis

A

^q.nd

B,

indeþendcnt

of lt, t,

^such

^tkttt:

rtûf,T/,(h)

=

ArzrtøzÐ w?,Øl 1- ^v?(h))

)- ^Brhr D

^z(¿)

s,,h ^s'Jt

19 ^TFIE GENEI{ALIZED SOLUTION 91

(6.2) Here

S,r: Qill ^S,,

^S,

: _{M ^e

_Q', d(X't, S) S ^z},

S¿: ll¡ x _{1, ^2, "',K)'

'lhe þroof is ^given

h _l2l.

THDorìEt'r 6.4. Suþþose

Q

satr,sfies conditions

of

Lemnta

6'3'

^Assutne

thøt tt,

is

th.e soløítion, obta,ined.

in

T'h,corent,6.2. Let

r = cQ) ^be a

^Liþscløitz

contin'trou,s fun,ctiott. ^{sot'clt, that :}

l Z0

^tt'nd

Jl":tt','

Tlt.c¡'t (6.3)

that

(6.4)

ERVIN ^SCLIECHTER 1B

(6,1)

since accorcling

to

Theoren 6.1 the trace ^operator

is

contil1t1ot1s:

V,l"-"

^ør,1"

in

I'z(S)

which

^implies

in vierv

o1 (6.1) ^:

(Jul,* _ul, in

¿'(S).

on the

other hand

u,

being

fipschitz continuoÍs

and

by the way

^{u2 was} corrstructed, Uul"

*

ü1, urriformly.

,r.FrrloREM e',á.

¿.t'*'ore

tha.t cond,itions

"J

Tkeorern 5.'2 ^are foilfilletl' 'fÌøen øtl¡:s exists ^and,

(i)

wl,:o

: 110 x ₌

{)

(ùi)

Uf

(x,1,,

0)

*

tto@,

y)

uniJ'or'mly on ^Q.

t-(

(,"(rt\

-

<off))2 d't' d1,

fl,¡* 0

¿.s

¡-o

⁰

,. ] tvt-l Y\Jtt -"'

s,

Proof.. As in the prool

of

Theorem

6.2

\ve use

the

seqtlence

lu"\i:r.

Recall

tltat

^(J,,

_{= C(Ø}

^and

U,,-,tt. in L(Q)' y,=@'.

_.

l,et {Í*}

^be the'ðótt"spotrding' ^secluencè

of

discretizations

of

f .

.f¿rtrì1'tnat

(6.3) is

troi ttu".-lheir

^there exists a constant C

> 0

^such

r-

\øfù -

^{eU))' dx ctl, ttt}

^{s c,}

S,

for a

sequence

oI

numbers

r',

convelging

to

0.

According

to

^{(6.2) :}

-ctù\ _(p(u,) -

^,?(f,,))'

^s Ar" rtÊ;'^, t(p(u") -

q(f,,))",

+

s-rr, ^S rh

-1-

ç(u,) -q(r"D?l ], ^Brhp {e{u") -

^,p(l;))'.

lt 1

max lU,,

- ^Vrl <:

^,

-11_a

7L

i

²

ç i

EIìVIN SCHECI{TER 21

tel

i10l

tl ^l l

lt2l

tls l [14]

TLIE CENERALIZED SOI,UTION 93

o9

lìnt

_lzr,

- f,l ^{< I-r,}

^so

^tltat

2û

'-Ð

_{t' sth}

^(*(u,) - ^q(Í,))'s

^Dt',

Ð

indepcirdent

or

h,,

r, t' ^(atd

^n).

Conscclucntly

for

_?

<

^/,

+ \ t'etu")' -

e(-f")') rtx ^{tt'Y rtt}

3 I{,t'

sr\sp

/(,

harring

the

same properties as 1(.

Norv l<:tting p

- ^0,

^{s,e gct}

O l e i rr i k, O. 4., I{ a l a s h n i k o v, ^.A.. S., ^C h o * I' i - l i t, .Equ.ation' of the,on- station.ary fil.tration tyþe, Izv. AN SSSR Set. IIat. ¿í (S) 461-469 ^-(1958, iu l{nssian)

I{ rr s a rr o v, ^{lVI. A,., I-he} stucly of lhe d.iJferen.ce method for ^{t}u: sç¡y¿¡on of a} þørabolic crluøliott uítlt. degenerecy, fz-v. AN Azerb. SSR, Ser. Iriz-'I'eh i IIat. no. 8,27-Bl

( 1s81) ^.

S c h e c 1r t e r, D., ^'fhe conuergence of the cxþlicit tliffcrcnce aþþroxim.øtion to solue the Cauclty-Dirichlet þrcblem for non-lineat'þaro.bol'ic cquations, Mathernatica, vol. Pl,

67-8r

(1979)

S c lr e c ht er, Ê., On u,lion bolic eq¿øtio,., ¡.ev.

il'Analyse Nurn e cle ll, 89-97 (1979)

Sclrelrchter,8., 1? cyical t.þaraboliccqua,ti,ott.,

Stndia Uuiv. Il XXV

Sclrechter, D., Sottte þroþert'ics of a n.ot'tlitt,car þørøboli.c differcttae s¿,l¿¿rz¿, Iìer..

cl'Arrall'sc Num. ct de la Théorie de l'Approxirnrtion l. 10, no. '2, 22õ -282 (lg9l)

lìeceiYed 5.III.1983,

IìLrcultûtce, de À'[at¿nt,aticã U nì ^L¡ c r s ilateq, ß abe; - B o1,1, a.i

slr. I(ogcílniceanu Ny.l 3100 Clu.j-Naþoca

1

,

(p ( U,)

' -

^9(f ,,)')" ^d x d.t, dt

!

_{I{ ,"}

Fina111' Tot /t,,

a--,0

^(r't,

-'

^co),

' [ (,, /)'lt'

^{r/t, rll <-}

^Iirr,

'i'

rvhich contradicts (6.3).

Remark _6.1.

The

existellce

of the

lunctiorr

f ^is

^ensured

^by

^condition

(P)

. This condition call i)e

\\'eakerled b1'supPosing

that zi, is _fipsclritz

continuous

as it

\\'as

proved in

1.1].

THEoRTìI{

6.5.

The wholc sequen.ce

U;

tentls

Jor lt,

^-'-,01¿o llte ^otuique solulion

u,

þroaiclcd, ^tJtcLt cotoditiort.s oJ 7-hcoret,tt, 6.'2 arc

flrlfilletl.

This is immediatcly seen Trorn tl-re rluiclculrcss oi the

limit of the

sec¡ren- cc

,

^1t.

IìIlIrDRl.r)NCDS

[1] ^{A I o r]. s s o} l, O., Dxlcn.sion, of functiotts satisflti.ttg Liþschil': ct¡ld,il.ions, Arkiv lör ]latc- rnatik, Il6, nr 28 551-561 (1967).

f2l CorrLarrt, R., Irrierlricl.Ls, l{., Lcrvl-, ÌI., ^Übtr dir fartitllcn Di/,',ren cttglti- ch.ungen. d¿r tn.alhttnalíschen Physl.h, Ilath. Arrn 11.100, 32--74 (1928)

[3] ^{D c s c 1o rr} x, _J., On tllc e(lxt.atiolx of ]]oussintsq. 7-oþics in, ttuntct,tcrl anall,sis, III, 81-102. Acaclcnric Press (1976).

['1] ^{It r i e d ru a} n, t\ , Pnrlial dtfJcrential er1'ttalìous ol _þarah, lic Íyþe, Plentice-Il¿rll, Jlrc.

(19(ì4)

f5]I,acl yz.hetskaia, O A, Botnt.iløry-ualue þr';l,l¡lnsoJ íhct¡ta.lltctnatical Physics, Nauka, l{oscol' (197't, irt Rrrssiarr).

[6] I,acl yz,hen,skaia, O.,{., So1o,rnikoi., \¡. t!, Ur¿ltscva N.ñ., ^Li.near

ønd cuasilinear equaliotts of lhc þatabolir ^11,75a, Nauka, lloscol' (1967, in Russian-

[7] I,ions, J. I,., ]Iagenes, IJ., ^{Prc,!èmes aux límites n.on} honrogencs et aþþIicati.ons"

\/ol. I, II, Dnuocl (1968).

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l)unocl (1969)