Explicit a Priori Inequalities with Applications to Boundary Value Problems (Research notes in mathematics)

V G Sigillito

The Johns Hopkins University

Explicit a priori inequalities with applications to boundary value problems

Pitman Publishing LONDON • SAN FRANCISCO • MELBOURNE

PITMAN PUBLISHING LIMITED 39 Parker Street, London WC2B 5PB PITMAN PUBLISHING CORPORATION 6 Davis Drive, Belmont, California 94002, USA Associated Companies Copp Clark Ltd, Toronto· Fearon Publishers Inc, Belmont, California Pitman Publishing Co. SA (Pty) Ltd, Johannesburg · Pitman Publishing New Zealand Ltd, Wellington · Pitman Publishing Pty Ltd, Melbourne

AMS Subject Classifications: (main) 35A40 (subsidiary) 65N99, 35J25, 35J40, 35K20 Library of Congress Cataloging in Publication Data Sigillito, V. G. 1937Explicit a priori inequalities with applications to boundary value problems. (Research notes in mathematics; 13) I. Boundary value problems. 2. Differential equations, Parabolic. 3. Differential equations, Elliptic. 4. Inequalities (Mathematics) I. Title. II. Series. QA379.S57 515'.35 77-1199 ISBN 0-273-01022-0

© V G Sigillito 1977 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording and/or otherwise without the prior written permission of the publishers. The paperback edition of this book may not be lent, resold, hired out or otherwise disposed of by way of trade in any form of binding or cover other than that in which it is published, without prior consent of the publishers. Reproduced and printed by photolithography in Great Britain at Biddies of Guildford

To Barbara, Robert and Amanda

Contents 1.

Introduction

1

2.

Notation and Some Important Identities and Inequalities

6

3.

Eigenvalue Problems

4.

A Priori Inequalities Applications

I -

Priori Inequalities Applications

II - Second Order Parabolic

5.

~

6.

~

Priori Inequalities

7.

A Priori Inequalities Applications

11

Second Order Elliptic

19

32 III - Pseudoparabolic Applications IV -

45

Fourth Order Elliptic 56

8.

Pointwise Bounds

66

9.

Applications to Eigenvalue Estimation

72

Numerical Examples

80

10.

REFERENCES

99

1 Introduction This research note has several objectives. single volume a number of explicit

~

One is to bring together in a

priori inequalities which are useful in

computing approximate solutions to boundary value problems which arise in such physical applications as heat and mass transfer, potential theory, fluid dynamics, elasticity, and radiation diffusion.

Mathematically this

means that we shall be concerned with elliptic boundary value problems and parabolic and pseudoparabolic initial-boundary value problems. A second objective is to illustrate a method, based on the a priori inequalities, for computing approximate solutions various boundary value problems. a number of numerical examples.

wi~

error bounds for the

This is accomplished by way of examining An important feature of the method is that

the trial functions used in the approximation need not satisfy either the differential equation or the boundary conditions. Another objective is to indicate the techniques used to derive the inequalities.

Thus the interested reader will be able to develop

new~

priori

inequalities for problems of interest if such problems are not covered by the inequalities presented here or by those in the literature. reason we give detailed derivations of most of the inequalities.

For this To develop

each inequality in its most general form would require many details which do not aid understanding but rather tend to obscure the main ideas.

Therefore

we treat special cases of each inequality but in such a manner that exten·· sions to more general cases are evident.

There is at least one occasion

when this approach results in an inequality which is not the best possible 1

one for the specific problem at hand.

When this happens we shall also de-

rive the better inequality. When applied to boundary value problems the usefulness of the

.. ~

priori

inequalities lies in the following areas: 1.

They provide a practical method of

~

posteriori pointwise error estima-

tion for any sufficiently smooth approximate solution.

2.

For linear problems they can be used with a Ritz type procedure to obtain approximate solutions with norm error bounds.

These norm error

bounds can be combined with other inequalities to give pointwise error bounds. Although it was the first area which originally stimulated the research in explicit~

priori inequalities

[26], [30], it is the second area of applica-

tion which we shall emphasize in this volume. THE "METHOD OF EXPLICIT A PRIORI INEQUALITIES" For our purposes an explicit the

~

~

priori inequality is one in which a bound on

integral of an arbitrary sufficiently smooth function u (a term to be

made precise for each specific problem) is given in terms of

~

u which represent data of a specific boundary value problem.

integrals of

In addition

all constants which appear in the inequality must be known explicitly or be computable. it

~

Once the appropriate inequality is in hand the method of explic-

priori inequalities is quite straightforward.

For instance, if we are interested in the Dirichlet problem l!.u = a:au Cl~

+ C!uli

=

f(x1 ,Xg)

in

(1.1)

on then an appropriate 2

B ,

Cl~

~

oB ,

priori inequality is

J .,13 dx

s 0'1

B

J

([).w) 2 dx +

B

0'2

f

(1. 2)

.,13 dS ,

oB

where w is an arbitrary C2 (B) function.

The constants are explicitly known

and are functions of the geometry of the domain B, certain eigenvalues, the coefficients of the differential equation, and possibly, known auxiliary functions. Having this inequality we are now ready to apply the method.

We intro-

n

duce an approximate solution u

a

= ~ a •1• as a linear combination of the k=l k'k

trial functions wk' which, because of the

~

priori nature of (1.2) need not

satisfy either the boundary conditions or the differential equation only be

ca.

If we denote the solution of (1.1) by u, then u- u a

E

but c2(B)

and we can substitute this function into (1.2) to obtain

s

B

(u-u ) 2 dx a =

0'1

s

B

S 0'1

J

B

([).u-[).u ) 2 dx + a

n

0'2

5 (u-ua)

2

dS

oB

(1.3)

n

(f- ~ akM k)2 dx + O'a (g- ~ akwk) 2 dS , k=l k=l oB

f

giving immediately a norm bound on the error of the approximation.

Further,

since f,g and vk' k'-'1, · • • ,n are known functions, the right hand side is a function only of the coefficients a 1 ,···,an.

Denote this function byE and

minimize it with respect to the variables a ,···,a • n

1

This leads to the sys-

tern of n equations in then unknowns a 1 ,···,an:

(1.4)

i=l,···,n

We denote the solution of this system by a*··· a* and thus obtain the 1'

n

approximate solution ua

~

k=l

a*k vk

'

n

which is the best approximation to u

3

in the sense that the bound E (a* •.• a*) on 1'

n

•

Usually more can be said than this. mental solution of

~

J

B

(u-u ) 2 dx is a minimum. a

If we denote by

r

either a funda-

or a parametrix then we can obtain a pointwise bound

on lu(P)-ua (P)I of the form

J

lu(P)-ua(P)I 2 s (

(Mofp)) 2 dx

Kl

+

J

(ofp) 2 dx} • E(a:,a:,···,an) (1.5)

Kl

where P E B, K1 is a region containing P and contained in B such that the distance d(K1 ,B) > 0 and o is a function which is identically 1 in a subregion K2 of K1

,

and belongs to C2 (B).

A RELATIONSHIP TO LEAST SQUARES Our method bears some relationship to the method of least squares.

For

instance, obtaining a least squares approximation to (1.1) involves minimizing the functional

for ua = ~ ak.k' with respect to the ai, exactly as is done in the method of ~

priori inequalities.

lar to the system (1.4).

This procedure leads to a system of equations simiThere are important differences however.

Thus,

contrary to our method, there is no explicit relationship between I and either the norm or pointwise difference u-ua.

A separate analysis is needed

to show that as I - 0 the same is true for u-ua, or some functional of u-ua, for the selected trial functions •k·

Also, with least squares, one has the

additional task of selecting good values for the weights w1 ,w2

,

whereas in

our method the corresponding constants a 1 ,a2 are closely related to the problem at hand and in some cases can be shown to be optimal.

4

OUTLINE OF CHAPTERS This work is divided into chapters as follows.

In Chapter 2 we introduce

notation and develop some important identities.

The next chapter is devoted

to eigenvalues which are important in the development of the equalities.

Chapters 4 through 7 develop the

~

~

priori

in-

priori inequalities applica-

ble to second order elliptic and parabolic problems, third order pseudoparabolic problems and fourth order elliptic problems. pointwise bounds are discussed. of

~

Chapter 9 presents recent work on the use

priori inequalities in eigenvalue estimation.

are given in Chapter 10.

In Chapter 8 the

The numerical examples

2 Notation and some important identities and inequalities In this chapter we present the notation which will be used in the remainder of the book.

we then summarize some well-known identities and inequalities

that are used over and over again in the development of the ities.

~

priori inequal-

Finally we derive three identities which play a central role in the

chapters that follow. NOTATION Although most of the numerical examples will be carried out in two dimensional regions we shall present the inequalities for functions of N variables since this generality is obtained with no additional effort.

We shall use

the convention that 'i denotes partial differentiation with respect to the variable xi so that, for instance, u,i

ou/ox .• 1.

The summation convention is

also used so that repeated indices in an expression are to be summed from 1 toN and thus, for example, the Laplacian

~u

can be expressed as u,ii"

In

the parabolic cases where a time variable t is present in addition to the spatial variables x =

(~

, X:a,

• • •, xN), repeated Latin indices are to be summed

from 1 to N while repeated Greek indices are to be summed irom 1 to N+l, the N+l-st variable being time. The elliptic problems will be defined in bounded regions B of N-dimensional Euclidean space whose boundary we denote by oB.

No other restrictions are

placed on B except that oB is smooth enough so that the divergence theorem applies in B (requiring oB to be piecewise

CL is sufficient).

On oB we de-

note the outward pointing unit normal by n = (1\ •lla, • • • ,nN) and the normal derivative of u by ou/on = u,ini. 6

The parabolic problems will be defined in a region D = B X (O,T], T < i.e., a space-time right cylinder with base B. the sides

=oB X (O,T]

s

and the top BT =

Dn

'then we define BT (t=T}.

=D n

~,

(t=T},

The so-called parabolic

boundary of D is thus B + S. SOME

IMPOR~NT

IDENTITIES AND INEQUALITIES

For ease of reference we now give a number of well-known theorems, identities and inequalities which will be used in the development of the

~

priori in-

equalities. (A)

The divergence theorem (integration by parts).

···,Nand Sa= Sa(x,t),

I

I

denote smooth vector fields, then

f . . dx ~.~

B

D

~=l,···,N+l

Let fi = fi(x), i=l,

g

~.~

dV

=! s

I

ginida +

gN+ 1dx

BT

-I

B

gN+ldx '

and in a form we shall frequently use

I

B (B)

h f. .dx ~.~

§ oB

h f.n.dS ~ ~

-IB

Green's first identity:

§

u

ov dS

- v

~~)

oB (C)

on

Green's second identity:

I .B

(D)

h,ifidx

(u Av-v Au)dx

§

oB

(u

~:

dS •

The arithmetic-geometric mean inequality (hereafter referred to as the

a-g inequality) with weight

~

> 0: 7

a2

2ab s:a

+a~

This inequality will most frequently be used in the integral form 2

I

u v dx s:

B

I

a if dx +

B

I

a -l v2 dx •

B

The Schwarz inequality:

(E)

I

u v dx s:

(I

ifdx )%

B

B

(!

v2dx)%

and its vector form

I B

f.g.dx s: l.

l.

(IB

N

~

f2 i=l l. dxt

(!

N

~

i=l

g~ dx)% l.

ADDITIONAL IDENTITIES The following three identities are less well-known but are crucial in the development of the inequalities which apply to the second order elliptic and parabolic equations and the third order pseudoparabolic equations. The first identity is due to Payne and Weinberger [30] and contains the Rellich identity [37] as a special case.

The other two identities are

due to the author [39], [441. Let fi denote the i-th component of a piecewise C1 (B) vector field and u a sufficiently smooth function (say piecewise c2(B)),

I

B

i

f u,.

~u

Then starting with

dx, a straightforward application of Green's first identity and

J.

integration by parts results in the identity

(2.1)

+I B

8

(f~.u,.u,.- 2f~.u,.u,.}dx. l.

J

J

J

l.

J

If fi= xi this is just the Rellich identity.

We now use the fact that (2.2)

is orthogonal to n. and hence is a tangent vector. l.

If we normalize

and introduce the notation

(2.4) we obtain from (2.2) and (2.3) that u,.u,. = (au/as) 2 + (au/an) 2 l.

(2. 5)

l.

Using (2.2)-(2.5) in (2.1) gives the first identity

(2.6)

- 2J fiu, 1. B

~u dx+J (f~.u,.u,.- 2f~.u,.u,.}dx B

l.JJ

Jl.J

Analogous identities which will be useful in the derivation of the

~

priori inequalities applicable to parabolic and pseudoparabolic problems are derived in a similar manner starting with

where ut- au/at (we do not use the comma notation for time derivatives), 9

L*u

= Au +

I

s

= A(u-ut) +

ut and L*

[fin.((ou/os) 2

ut.

(ou/on) 2 }

-

These identities are 2fis.(ou/on)(ou/os)]do

-

1

1

2 - 2

I

fXu, L*u dV

+ID

I

N+l u u,.dV f,.

+ 2

D

D

Oi

t

1

1

(f~

(2.7)

u, .u, .- 2d .u,.u, .]dV

0111

I

f Oi u, u dV ott

D

11J

+ 2

I S

f N+l ut ou/on do

f 01 n u,. u,. dx • Oi

1

1

and

I

s

[fin. ((ou/on) 2

-

(ou/on) 2

-

2fis. (ou/on)(ou/os) ]do 1

2

I

D

j.

f 01 u, L* u dV 01

N+l u u, .dV - 2 f,. D 1 t 1

10

}

1

+I (e

01 u,iu,i-

D

+

2 I

D

Oi

2diu,iu,i}dv

f u, 01 (ut-Aut)dV

+

2 SI

(2.8) f N+l ut ou/on do

3 Eigenvalue problems The eigenvalue inequalities discussed in this chapter have wide application to the

~priori

inequalities.

In fact it can be shown that the lowest non-

zero eigenvalues of certain eigenvalue problems are the optimal constants in the a priori inequalities.

Since exact values of the eigenvalues are known

only for special regions, the problem of finding bounds for these eigenvalues is of importance.

We devote the last section of the chapter to this

topic. EIGENVALUE INEQUALITIES The inequalities we are interested in here represent many of the plausible inequalities among the expressions

J ifdS, oB

I

j

u,iu,idx,

oB

B

(~~) 2 dS,

I

(6u) 2 dx,

B

for functions u satisfying various smoothness and auxiliary conditions.

The

inequalities all have the common property that they arise from variational characterizations of various eigenvalues.

I

tf!dx s q- 1

IB

tf!dx s }..-1

I B

tf!dx s ~

I

tf!dx s

B

B

§

ifdS,

~u

0

They are on

B;

(3.1)

oB;

(3.2)

oB

I

u,iu,idx,

I

u,iu,idx,

u

=0

I

u dx

on

B -1

B

s- f oB 1

(3.3)

0;

B (our dS on '

~u

0

on

B,

I

u dx

0;

p.4)

B 11

J tf!dx s o-1 J

(t.u) 2 dx,

u =ou - = 0 on

J

(t.u) 2 dx,

u = 0

(t.u) 2 dx,

rn= 0

B

B

J ifdx

s >..-:a

J tf!dx

s IJo -:a

B

Jlds s p-1

ifdS s p- 2 ifdS s ~- 1

oB

J u,iu,idx,

oB;

f

oB

('Ou)2 dS, on

t.u

J

(t.u) 2 dx,

-= 0

s p-1

J u, .u, .dx

s A-1

~

on

(3.6)

Ju

oB,

dx

0;

(3.7)

B

0.. u dS

0;

(3.8)

i

0

ou on

i

B,

u dS

Ot

(3.9)

on

f

oB,

u dS

0;

(3.10)

oB t.u

(our ds , on

oB

on

oB

B ~

(3.5)

oB

J u, .u, .dx

B

ou

B

oB

f

J B

oB

f

on

oB;

B

B

f

on

on

0

B;

(3.11)

J

(t.u) 2 dx,

ou u =- = 0 on

on

J u,.u,.dxs>..-1 J

(t.u) 2 dx,

u = 0

on

oB

(t.u) 2 dx,

-= 0

on

oB;

(3.14)

u "' 0

on

oB.

(3.15)

B

B

~

~

~

B

~

(3.12)

(3.13)

B

J u,iu,idx B

i (~~r

oB;

s IJo

-1

J B

dS s q-1

o.s

J

ou on

(t.u) 2 dx,

B

The optimal constants in the above inequalities are the reciprocals of the first~-~

eigenvalues of the following problems:

The fixed membrane t.u + >..u

0

=0

on

B,

u

on

B,

on - 0

on

oB;

(3.16)

the free membrane t.u + IJoU = 0 12

ou

on

oB·,

(3.17)

the clamped plate on

0

B,

u

ou

= on =

on

0

(3.18)

oB;

the buckling of the clamped plate 6. 2 u - A b,u

0

on

B,

u =ou - = 0 on

on

(3.19)

oB;

and the Stekloff problems, flu

on

0

B,

6,2u

0

on

B,

62 u

0

on

B,

ou on = p u

on

oB;

ou ob.u =--+ on on

su

0

on

oB;

(3.21)

ou q- = 0 on

on

oB.

(3.22)

-

u = flu

(3.20)

We denote the optimal constants in the inequalities (3.1)-(3.15) by A1

,

~2 ,

That the eigenvalue problems (3.16)-(3.22) do indeed furnish the optimal constants as indicated in (3.1)-(3.15) is easily shown. that

~

J

B

From (3.2) we see

is the minimum of the Rayleigh quotient u,. u,. dx I L

L

J

B

til dx

over all admissible functions u satisfying u

=0

on oB.

The Euler equation

obtained by taking the first variation of this Rayleigh quotient is (3.16). Similarly, (3.17), (3.18), (3.19), (3.20), (3.21), and (3.22) are the Euler equations for the Rayleigh quotients for (3.3), (3.5), (3.12), (3.8), (3.10), and (3.15), respectively. We use Fichera's Principle of Duality [11] to show the equivalence of

(3.1) and (3.15), (3.2) and (3.13), (3.3) and (3.14), (3.4) and (3.10), (3.8) and (3.11).

·For example, to show that (3.4) is equivalent to (3.10), suppose 13

(3.4) holds and we wish to establish (3.10).

Given u satisfying au/on= 0

~ u dS = 0, define v by oB

on oB,

flv = 0

on

B,

ov on = u

on

oB,

J

v dx = 0.

B

Then, by Green's second identity, Schwarz's inequality, and (3.4)

implying (3.10).

Conversely, suppose (3.10) holds and we are given u satis-

§

fying flu = 0 on B,

u dS = 0.

Define v by

oB flv

u

on

B,

ov = 0 on

on

oB,

i

v dS

0

oB

Then, by Green's second identity, Schwarz's inequality, and (3.10),

implying (3.4). That

-1 ~

and

-2 ~2

are the optimal constants in (3.6) and (3.7) follows from

the Euler equations u = flu = 0 ou = oflu = 0 on on

14

on

oB, on

oB,

which are satisfied by the eigenfunctions of (3.2) and (3.3) respectively, and these form complete orthonormal sets for their respective classes of admissible functions. Finally, combining (3.8) and (3.11) yields (3.9) and equality holds for u2 , the eigenfunction of (3.20) associated with p2 , so the optimal constant -2

is p2 • BOUNDS FOR THE EIGENVALUES IN SPECIFIC REGIONS Precise values of X1 simple regions.

,

~~o 2 ,

p2 , t; 2 , q1

,

(\ ,

and

f'..t are known only for certain

For instance for a disc B of radius R we have

Xl

j2 /If'

RS

5. 783/R2

(3.23)

ll-2

y2 /R2

~

3.39/R2 ;

(3.24)

'\

1,4 /R4

RS

103.6/R4 ;

(3.25)

Al

z2 /If'

l'=s

11.446/r

(3.26)

Pa

1/R ;

(3.27)

t;2

5/If

(3.28)

q1

= 2/R

•

(3.29)

In the above, j is the first zero of the zero order Bessel function of the first kind, J 0

,

z is the first zero of J~, y is the first zero of J{ and t

is the first zero of J0 I~- J~I 0 , where I 0 (x) values of X1

, ~2

= J 0 (ix), i = J=I.

Exact

and p2 are also known for rectangular regions.

In the applications of the eigenvalue inequalities to the development of ~priori

inequalities, we do not need to know the precise values of the

above eigenvalues but only lower bounds.

We indicate some useful results 15

along these lines. (A)

The Faber-Krahn inequality [10] which states that the first eigenvalue

of problem (3.16) on B is not smaller than that for the sphere whose N-volume is the same as that of B.

The mathematical statement is

(3.30) Here N denotes the number of dimensions, wN the surface area of the N-dimensional unit sphere, VN the volume of B and j(N- 2 )/ 2 the first zero of the Bessel function J(N- 2 ) 12 • For two dimensional regions Payne and Weinberger [31] have extended the Faber-Krahn inequality as follows: If B lies interior to the wedge of angle TI/a, i.e., 0 Sa S TI/a, for any real a

~

1, then _1

h1

~

n . Ka { 4 a(a+l)

1/ (a+l)

J

(3.31)

where r

2a

sin2 a a da •

(3.32)

Equality holds if and only if B is a circular sector. (B)

Monotony principles:

0t (A)

(C)

16

S

0t (B)

For regions A and B,

,

For a convex N-dimensional region B

A~

B,

(3.33) where D is the diameter of B [32]. (D)

Consider the case where B is star-shaped with respect to some point which

we choose as the origin.

Let r(P) denote the distance from this origin to a

point P E 3B and let h(P) be the distance from the origin to the tangent plane to oB at P.

Then N

(3.34)

~2 ~ --------~~----------

2~ {c:r'· :: + ~J

and

Here the subscripts M and m denote the maximum and minimum values of the associated quantities.

We note that h

= x.n. 1. l.

where x. denotes the i-th coml.

ponent of P and ni denotes the i-th component of n at P. ness of B insures that h

m

> 0.

The star-shaped-

The above results are due to Bramble and

Payne [ 3 1. (E)

The following inequalities are also useful: ~

ql

~

~~

~

~

P2

2

B star-shaped,

rM

+ (1/p)m ,

(3.36)

p denotes the curvature of oB,,

(3.37)

equality holds if and only if B is the N-sphere,

(3.38)

(3.3!1)

Pertinent references for these results are [18] and [33]. bounds see [17], [36].

18

For other useful

4 A priori inequalities I- Second order elliptic applications we now develop explicit

~priori

inequalities which have applications to the

three classical boundary value problems for second order elliptic equations: the Dirichlet, Neumann and Robin problems.

Also included is an

~priori

in-

equality which has applications to the first boundary value problem in the equations of elasticity. The inequalities are given in terms of the Laplacian but the derivation given can be carried over to the general self-adjoint operator Lu (a .. u,.),. where a .. = aJ.i = aiJ.(x) is a symmetric matrix such that ~J ~ J ~J ~ aij~i~j

s a 1 ~i~i• for all x E Band all real

a0 ~i~i

s

~

vectors~= (~ 1 ,···.~N) where

a 0 and a 1 are positive constants. AN INEQUALITY FOR THE DIRICHLET PROBLEM Theorem 4. 1.

Let u be a function with piecewise continuous second deriva-

tives in B, but otherwise arbitrary, then

I

t?dx s

a1

B

I

(Au) 2 dx + a 2

B

f

oB

if dS

where a 1 and a 2 are explicitly determined constants [ 4 ]. Proof:

Introduce an auxiliary function w satisfying Aw

=u

in

B

and w

0

on

oB •

19

Then by Green's identity

I

if dx

=

B

I

u flw dx =

B

§

u ow/on dS +

oB

I

(4.1)

w flu dx

B

and Schwarz's inequality implies that

I

if dx s (

B

§ if dS § oB

(ow/on) 2 dS}% + (

oB

I

if dx

B

I

(flu) 2 dx}%

(4.2)

B

We now bound the integrals involving w on the right-hand side of (4.2) in terms of

I

ifdx.

B

We start with w flw dx

where we have used (3.2). by

(4.3)

Using the Schwarz inequality, dividing both sides

(I ifdx)% and squaring we obtain

the first bound:

B

I

ifdx

:S:

>..; 2 I

B

(4.4)

(flw) 2 dx •

B

Another inequality which we will need later follows from (4.3) and (4.4):

I

w, .w, .dx

B

L

:S:

A. 1-1

L

I

(4.5)

B

f

To obtain a bound on

(ow/on) 2 dS we start with the identity (2.6) of

oB Chapter 2:

§

oB

[fin. ((ou/os) 2

-

(ou/on) 2

}

-

2fisi ou/on•ou/os] dS

L

2 I B

f i u,L.6u dx+

I B

(4.6) i i (f,.u,.u,.2f,.u,.u,.} dx LJJ

JLJ

Putting w into (4.6) we obtain

(4.7) 20

=0

where we have used the fact that ow/os

on oB since w

=0

there.

we choose the vector field fk in such a way that fknk is bounded and has a positive minimum pm on B.

For example, if B is star-shaped with respect

to the origin, one can take fk

= xk.

A

detailed discussion of methods of

constructing appropriate vector fields for more general regions is given in

(4]. we also have that there exists a constant C such that

throughout B if the fi have bounded first derivatives (C is any upper k

. .

i

.

bound for the largest eigenvalue of the matrix f-f,k6LJ+ f,j+ f~i}). Thus we obtain, from (4.7), that

(4.8) ex > 0 where we have used the a-g inequality and

I·IM

absolute value of the enclosed quantity.

We now obtain the desired bound

on

i

denotes the maximum of the

(ow/on) 2 dS by combining (4.5) with (4.8):

oB

i

(ow/on) 2 dS s p~1

oB

(4.9) - K

J

ifdx.

B

This inequality along with (4.4) and (4.2) then gives

(4.10)

21

I

and dividing through by

U8dx, squaring both sides and using the a-g in-

B

equality gives

I

U8dx s 2

B

A; I 2

Thus the coefficients a 1 2 \

f

(au) 2 dx + 2K

B

U8dS •

aB ,

a 2 in the statement of the theorem are

-2

"1

a> 0 • AN ALTERNATIVE APPROACH As mentioned earlier, we are carrying out the derivations for some of the more simple cases but in a manner which immediately extends to more general operators.

If we had not been concerned with techniques which apply to more

general elliptic operators we could have derived an inequality like that of Theorem 4.1 more simply as follows: Decompose u as u

where ah

=h + =0

g

on B, g

=0

on oB so that u

=h

on aB, au

= ag

in B.

Then

squaring both sides, integrating and using the a-g inequality we have

I

B

I

U8dx s 2I h2 dx + 2 B B

~dx

•

Then from (3.1) and (3.6) we have

I B

22

U8dx s 2

-1

Cb,

§ oB

ua ds

+ 2 A; 2

I B

(au) 2 dx •

THE NEUMANN PROBLEM

= W+

Let w be piecewise c2(B) and define u

Theorem 4.2.

c where

denotes the interior of a sphere of radius 1 1 § ~ dS, S a c B w•N.aN- u"'S a a centered at the origin (we assume 0 E S) and wN is the surface area of Sa. Then c

=-

where a1 and a 2 are coefficients which will be explicitly determined [3 ]. Proof:

The derivation of this inequality is an interesting variation of the

preceding.

It is slightly complicated by the compatibility condition

solutions of the Neumann problem must satisfy.

§

u

which

Notice that by definition

dS = 0.

oB We start with Green's identity

J

§ u ou/on dS

u,iu,idx

oB

B

-J

u t.u

dx

B

and apply the Schwarz inequality

(4.11) p

We now bound

§ pif dS and oB

J B

if dx in terms of

J

B

> 0 •

u, .u, .dx 1

1

= D(u,u).

Denote by Ba the region B - Sa and let fi be a sufficiently smooth vector field defined in Ba·

Then by the divergence theorem we have (4.l:l)

23

An application of the a-g mean inequality applied to the last term on the right of (4.12) yields

(4.13)

+I a B

u, .u, .dx ~

a

~

where a is some positive function in Ba. We assume now that f i

i

and a have been chosen so that

K1 > 0

on

oB ,

- f ~. n.~ s Ka

on

oS a,

p - f n.

~

~

(4.14)

where K1 and K2 are constants (see [3] for details). Using (4.14) with (4.13) we have that (4.15)

where ci is an upper bound for a in Ba·

Now since

§

u dS

0 then by (3.8)

oSa

f

ifdS s Pa-1

oSa

I

u,iu,idx

Sa

=a I

u,iu,idx

(4.16)

Sa

since for the sphere of radius a, p2

= 1/a,

(see (3.27)).

Combining (4.15) and (4.16) it follows that

§

pullds s K:3 • D(u,u)

(4.17)

oB where K:3 = max (a•K2 ,ci), or using (4.14),

§ oB 24

if dS s (K:J/Kd D(u, u) •

(4.18)

Now from the divergence theorem

§

x. n. ifds

oB

1.

dx •

(4.19)

1.

using the a-g mean inequality it follows easily that

J

2rM ,r 4rM u2 dx s - - ;r u2 dS + - 2 - D(u,u) B N oB N

(4.20)

where rM is the maximum distance from the origin to oB.

This inequality

with (4.18) yields

J

ifdx s ~ • D(u,u) ,

(4.21)

B

where

We now obtain the desired inequality by using (4.17) and (4.21) in (4.11) to obtain

or, squaring and using the a-g mean inequality D(u, u) s 2Ks

f

p -l (ou/on) 2 dS + 2~

oB

J

(t.lu) 2 dx .

B

Finally, by (4.21)

J

u2 dx s (2K3 ~ /K1

)

§

(ou/on) 2 dS + 2K:

oB

B

J

(t.lu) 2 dx •

B

THE ROBIN PROBLEM Theorem 4.3.

Let u be a function with piecewise continuous second

derivatives in B, but otherwise arbitrary,

then

25

I

u2 dx

:5

I

a1

B

f

(llu) 2 dx + a 2

B

= a(x)

where a

(ou/on+au) 2 dS

oB

is a positive, piecewise continuous function which is bounded

away from zero on oB and a 1 and a 2 are explicitly determined constants [7 ]. Proof:

Let a function w be defined such that llw

=u

in

on + ,._. ww OW

B 0

on

.,.

oB •

Then we have

I

if dx

B

=

I

u llw dx

- f

w(ou/on+au)dS +

oB

B

I

w llu dx •

(4.22)

B

We now decompose the function u into the sum of two functions h and g which satisfy llh

=0

in

B ,

oh/on + ah

llg

ou/on + au

on

oB ,

= flu

in

og/on + ag

B ,

=0

on

oB •

Now since (4.22) holds for any sufficiently regular function u it holds for u

=h and

from (4.22) it follows that

f

w(oh/on+oh)dS

oB and the Schwarz inequality yields

f oB

a if dS

f a -l (oh/on+ah) 2 dS • oB

(4.23)

If we denote by K a lower bound for the first eigenvalue in the elastically supported membrane problem [29] we obtain

26

I

ifdx ~ K- 1 [D(w,w) +

K

~

§

B

oB

01 ifdSj ~I (flw) 2 dx B

(4.24)

since D(w,w) + 01m ot ifdS

I

D(w,w) +

s

I

ifdx

B

B

where 01

m

G

01 ifdS oB ifdx

is the greatest lower bound for 01 on oB.

Now from (4.24) it follows that

I

~ K-1

§ 01 ifdS oB

(6w) 2 dx •

(4.25)

B

Inserting (4.25) into (4.23) we have

G01-1 (oh/on+ah) 2 ds

•

(4.26)

oB Now take u

I

B

= g.

(6w) 2 dx

Then from

=I

g2dx

B

=I

(~.22)

it follows that

w llg dx •

B

From Schwarz's inequality we have

I

(llw) 2 dx s

B

I

if dx

B

I

(llg) 2

dx

B

and from (4.24) we have

I B

g2dx ~ K- 2

I

(4.27)

(llg) 2 dx

B

Hence we have from (4.26) and (4.27) using the representation u

I B

ifdx ~ 2 (K-1

§ oB

01-1 (ou/on+U01) 2 dS + K- 2

J

=h

+ g

(llu) 2 dx}

B

which is valid for any sufficiently smooth function u. 27

AN INEQUALITY FOR A PROBLEM IN ELASTICITY In this section we derive an

~

priori inequality which has applications to

the first boundary value problem in elasticity

=

Li(u)

u . . . +cr . . . 1,JJ

J,J1

in

B

on

oB

i=l,2,·· · ,N • u =f.

1

(4.28)

In this context u is the displacement vector with components ui which satisfy the system (4.28), a involves the elastic constants A (l-2cr)

-1

and~

(cr

= (Ai

, cr denoting Poisson's ratio), and the F. are proportional to 1

body force.

The

~

)/~

=

the

priori inequality of interest is then given by the follow-

ing Theorem 4.4.

Let u

= (~

·~····,uN)

be an arbitrary vector field with piece-

wise continuous second derivatives in B.

Then

where cr1 and cr2 are explicitly determined constants [ 2]. Proof:

We must first derive an auxiliary inequality.

Let Vi be the ith com-

ponent of a vector function which possesses piecewise continuous second derivatives in Band vanishes on oB.

Then, as in the derivation of (2.6), an

application of Green's first identity and integration by parts results in the identity 2

§

oB

fkv. k[v . . n.+crV . . ni]ds 1,

1,J J

J,J

k + BJ f,k[v . . v . . +crv . . v.. ]dx - 2 J [f,k .v. kv . .+crf,k .v. k~ . . ]dx 1,J 1,J 1,1 J,J B J 1, 1,J 1 1, J,J

28

Since$. vanishes on oB the first and last integrals on the right combine to L

give

I

k [ v. ·V· .+av . . $ . . ] dx + 2 k k ..r f,k [f,.v. kv . . +af,.v. k*· .]dx B L, J L, J L, L J, J B J J, L, J L L, J ,J

Since fi has bounded first derivatives we may easily obtain, for a positive

2

I

B

I

k k k [f,.$. k*· .+af,.v. kV· .]dxf,k[v. ·V· .+av. ·V· .]dx J L, L,J L L, J,J B L,J L,J L,L J,J

s; b

I[$.L,J·V·L,J.+ av.L,L·V J,J . . ]dx

B

where the constant b may be easily obtained,

By the a-g inequality and the

fact that a is positive, we have

§

oB

P[w.

·w. .-+

L. ($) dx L

I 0

J

B

['lt.1., J.w 1., ..J +

avl..

1.·WJ·

,

,

J.

J

]dx

B

.ljr . . dx

1jr.

1., J

1., J

2:

J ljr.ljr.dx

J ljr.ljr.dx

B

by (3.2).

l.

B

l.

l.

l.

Thus the preceding two inequalities give us that (4.30)

Use of this inequality in (4.29) then yields

where we have made the optimal choice for

S.

We are now ready to determine a bound for J u.u.dx. B

l.

To this end we in-

l.

troduce the auxiliary vector x with components xi defined by

i=l,2,•••,N on

0

oB

Then J u.u.dx =J u.L.(X)dx =

B

l.

l.

B

l.

l.

j

oB

u.[x . . n.-+0')( . . n.]d~ +J x.L.udx. l.

1.,

J

J

J' J

l.

B

l.

l.

By Schwarz's inequality for vectors we have J u.u.dx = J L.(X)L.(X)d:x s [ f p- 1 [u.u.+(u.n.) 2 ]dS t[ p[x . .x . . B l. l. B l. l. oB l. l. l. l. oB l.,J l.,J +ax . . x . . dS]~ + 1.,1.

30

J, J

t BJ x.x.dx J L.(u) L.(u)dx}~. l. l. B l. l.

since x.

L

=0

on

oB

we may use (4.30) and (4.31) and the a-g mean inequality

with Wi replaced by xi to obtain

+ 2Ai 2

J

Li(u) Li(u) dx.

B

31

5 A priori inequalities II- Second order parabolic applications The a priori inequalities presented in this chapter have applications to the Dirichlet, Neumann and Robin problems for second order parabolic equations. Although they are derived in terms of the heat operator Lu

= ~u

- ut, the

derivations can be carried over to the general second order operator Lu (aiju,i),j- c(x)ut' where c(x) > 0, aij

= aij(x,t)

that aS-S· sa LJ .. s.s. sal S·S· for all (x,t) 0 L L L L L L (s1 ,···,sN) and

&o

=

is a symmetric matrix such

E D and all real vectors s =

and a 1 are positive constants.

Inequalities applicable

to semi-linear problems are also given. The approach used in the derivation of the inequality for the parabolic Dirichlet problem has some similarly to that of the corresponding elliptic case but for the other problems new techniques are needed. AN INEQUALITY FOR THE DIRICHLET PROBLEM Theorem 5.1.

Let u be a function with piecewise continuous second deriva-

tives in x, and piecewise continuous first derivatives in t throughout D, but otherwise arbitrary.

I

u2dv s

0'1

D

where

(Lu) 2 dV +

D

0'1 ,

Proof:

I

Then

0'2

and

0'3

0'2

I B

0'3

I

u2dcr

S

are explicitly determined constants [39].

Introduce the auxiliary function w which satisfies the boundary value

problem

32

u2dx +

L*w

=~w + wt

w

0

=u

in on

DUB

using the divergence theorem and Green's second identity we can write

I

UA dV =

D

I

u L*w dV =

D

I

u ow/on da +

S

I

w Lu dV -

D

I

u w dx •

B

The vector form of the Schwarz inequality then yields

I

UAdV s

(I

D

•( I

I

ifda

S

(ow/on) 2 da )l.i +

S

ifdx

(I B

+ A~ 1

B

I

(5.1)

(Lu) 2 dV )l.i

D

our object now is to bound (5.2) and

I

s

(5.3)

(ow/on) 2 da

in terms of

I

u2 dV.

D

To obtain the bound on (5.2) we use the fact that

-I

if dx

B

=

I

oif lot dV

D

and recalling that w

2

I

w[L*w - w 6 w] dV

D

=0

on S we obtain

-I

w L* w dV •

(5.4)

D

This equation will be useful very shortly; it also yields (using (3.2) integrated over [O,T]) the important inequality

I

D

w, .w, .dV s A1-1 ~

~

I D

(L*w) 2 dV •

(5.5)

33

The desired bound on (5.2) is now obtained as follows:

I

wl!dx +

B

11. 1

I

wl!dv s

I

I

wl!dx + 2

B

D

D

w, .w, .dV ~

~

A. 1

I

wl!dv

D

(5.6)

s

-1 A. 1

I

(L*w) 2 dV =

-1 A.1

I

tf3 dV

D

D

where we have used (5.4) and the a-g inequality. To obtain the bound on (5.3) we use Equation (2.7) recalling that ow/os ow/ot = 0 on S and

r

f

J

N+l

nN+l w,iw'idx = 0 since w = 0 on S U BT:

BT

(5.7)

Now choose fN+l ~ 0 in D U B, then the last term in (5.7) is nonpositive and we drop it.

Further, the term next to it contains the term

I fN+ 1 (w t ) 2 dV s 0 and we use the a-g and Schwarz inequalities on the other D integrals containing wt to cancel out this term. We thus obtain the inequal-

-2

ity

where K

and where C is an upper bound on the largest eigenvalue of the matrix ·~ j i -tt,~6ij- f,i- f,j}.

a positive minimum pm on S we have

34

t

t

If we now chose the f , t=l,•••,N such that f nt has

I

(ow/on) 2 dcr s K

S

I

(5.8)

u2 dV

D

where K

p

-1

m

-

K.

The inequalities (5.6) and (5.8) thus yield the inequality of the theorem where 2

-a

:>.. 1

2 K •

,

A SEMI-LINEAR CASE

we now establish an inequality for a function which satisfies the semilinear equation Lu Theorem 5.2. piecewise

CL

= f(x,t,u).

Let u and in t.

Lipschitz constant M.

I

'lf 2 dV s a 1

D

be functions defined in D, piecewise

ca

in

X

and

Furthermore we assume that u satisfies the semi-linear

= f(x,t,u)

equation Lu

~

I

and that f satisfies a Lipschitz condition in u with Let W(x;t) = u(x, t) -

'lf 2 dx + cx2

B

I

'lf 2 dcr + a 3

S

I

~(x,

t).

Then

:r2dv

D

where F(x,t) and cx1

,

Proof:

f(x,

t,~)

- Lql

cx2 , cx3 are explicitly determined constants [39].

Introduce the function v = V eb(T-t)

in D where b is a positive constant. L'lf

(Lv-bv) e-b(T-t)

Then

(5.9)

35

and we seek a bound on

I

=I

e2b(T-t) ~adv

D

if!dv •

(5.10)

D

To this end introduce the function w which satisfies the boundary value problem L*w - bw

w

=v

0

in

DU B

on

BTU S •

Now much of the remainder of the derivation proceeds in a manner parallel to that of the preceding theorem and so we will just give the highlights.

Pro-

ceeding then as in Theorem 5.1, we obtain the inequality

j

vSdv s:

{£ if!da [

(ow/on) 2 da

• ( / if!dx + (b+Ad-

I

and thus we need bounds on

1

t

£

+ ([ ifdx + (b+A 1 )

£

ifdv

y~ (5.11)

(Lv-bv) 2

if dx + (b+A 1 )

B

dvt

I if dV D

and

I

(ow/on) 2 da.

S

The first bound is obtained from the easily derived identity

I if dx + 2 I w, . w, . dV + 2 b I if dV B

D

1

1

2

D

I

w(L*w-bw) dV •

(5.12)

D

Using (5.12) and (3.2) along with the a-g inequality we obtain

I ifdx +

(b+Ad

B Furthe~

I D

36

I ifdv s:

(b+A 1

D

) -1

I

v2 dV •

(5.13)

D

(5.12) also yields the useful inequality w, i w, i dV

s; ( 4b) - 1

I D

v2 dV •

(5.14)

To obtain the bound on

+ 2

- b

N+l J f,. ~

D

J

J

s

(ow/on) 2 do we modify (5.7) in an obvious way

J

w w,.dV- 2 f exw, w dV t ~ D ext

~exifdV - b

D

m

B

~

~

fN+l ifdx

B

Now choosing fN+l ~ 0 in p

J

- J f N+lw, .w, .dx

Dand

the ft such that ftnt has a positive minimum

on S we have

from which follows the inequality

J

(ow/on) 2 do s K

S

J

v2dv

(5.15)

D

where

(5.16)

where again C is any upper bound on the largest eigenvalue of the matrix

-{~ex

6 ij- di-

f! j} •

From the Lipschitz condition satisfied by f we have (.:>.17) 37

so that by (5.9) we have

J

J (Lv-bv) 2 dV = e 2 b(T-t) (Lw) 2 dV D D

J

s 2M2

f

e 2 b(T-t) w2 dV + 2

D

e 2 b(T-t) rdV

S

and this inequality along with (5.13) and (5.15) in (5.11) gives

(£

v2 dV )

%s ( K

!

v2 da

)%

+ [ 0 we obtain the inequality

s 2 K- 2 (KJ v2da + (h+X 1 )-1

D

c,d

S

J

v2dx + (h+Xd- 2

B

I

e 2 b(T-t)rdv}.

D

AN INEQUALITY FOR THE NEUMANN PROBLEM Theorem 5.3.

Let u be a function with piecewise

c1 in t, but otherwise arbitrary.

J

if dV s

Cl'1

D where

38

Cl'1 ,

J

D Cl'2

and

Cl'3

(Lu) 2 dV +

Cl'2

C2 in x, and piecewise

Then

f

B

u2 dx +

Cl'3

J (~u/~n) 2 da S

are explicitly determined constants [19 ].

proof:

f + g + h where

Decompose u as u Lf f

= 0, = u,

of/on

= Lu,

Lg g

= o,

Lh

= 0,

=0

h = 0

og/on =

o,

oh/on

ou/on

in

D ,

on

B ,

on

s

and successively substitute f, g and h into the identity

I

ifdx

Bt

=I

t

2[[

ifdx -

B

t

w L w dx dT - 2

0 BT

IS

w,iw, i dx d-r

B-r (5.18)

t

+ 2

I w ow/on I oB,.

ds d-r

Putting f into (5.18) yields

and integrating from t

J f2dx

dt s 'r

D

J

0 to t

T gives (5.19)

J!dx •

B

Putting g into (5.18) yields

J g2 dx

t

s -2

Bt

J J 0

g Lu dx d,-

B,-

t

s ex

J Jg

2

0

Br

t

dx d'T" +ex - 1

J J (Lu) 0

2

dx d'T"

B,-

by the arithmetic-geometric mean inequality for arbitrary positive ex.

Multi-

plying by e -ext and rearranging gives s ex -1 e-ext

Jt J (Lu) 0

2

dx dT

B,-

Integration with respect to t from 0 to T and multiplication by e

exT

then 39

gives

I

T

g2 dx dt .,;;

a-1

D

t

I

eaT

I I

e-at

0 T

= a- 2 I

(Lu) 2 dx dT dt

0 Bt

I

(ea(T-t) - 1)

0

(Lu) 2 dx dt

Bt

I

s a - 2 (eaT_ 1)

(Lu) 2 dx dt ,

D

Setting a

where integration by parts was used.

I

g2 dx dt s; a- 2 (ea -1)

r I

D

= aT -1

gives

(Lu) 2 dx dt

D

rI

s; 1. 544138653

(5.20)

(Lu) 2 dx dt ,

D

with the optimal choice of a

= 1.59362.

Finally, putting h into (5.18) yields

I

h2 dx

=-

t

2

I I

0

BT

h, . h, . dx dT + 2 1

BT

1

t

I I

0 ST

t

s; - 2

I I

0

h, .h, .dx dT +a - 1 1

~

ou

h ~ ds dT

1

on

J I{~u)2 ds dT

0 S

on

(5.21)

T

Suppose B is star shaped with respect to a point which we take to be the

I ST

40

ha dx s d -1

I ST

h2 x.n.ds-_ d-1 1

1

I B

T

(h2 x.),. dx 1

1

putting this into (5.21) gives

Jh

2

dx

~

t

a- 1

J J (~~r ds

dT + a(nd-1 +

t a r2d-

t

2 )

0 ST

BT

2

dx dT

(5.22)

0 BT

a(nd- 1 +ta r 2 d- 2 ).

Define K(a)

J Jh

Multiplying (5.22) by e-K(a)t andre-

arranging gives

Integrating with respect to t from 0 to T and multiplying by eaT yields

J h2 dx

dt ~ a- 1 eK(a)T

D

t

J

t

e-K(a)t

0 =

(aK(a)) -1

J

J J 0 ST

(eK(a)(T-t)- 1)

0 ~

2

(~u) ds dT dt n

J (out on

ds dt

st

J

2 (aK(a)) -1 (e K(a)T - 1) S (ou on I\ ds dt •

For given region and T, minimize (aK(a))

-1

• (e

K(a)T

. - 1) wLth respect to a,

to obtain

(5.23) Combining (5.19), (5.20), and (5.23) along with the a-g inequality yields the desired inequality with a1

=

4.633r,

3T,

3[c2 (R,T)] 2

•

THE ROBIN PROBLEM

Theorem 5.4.

Let u be piecewise ca in x, piecewise cl in t throughout D,

and otherwise arbitrary.

Then

41

where

~

=

~(x,t)

is positive on S and a 1

,

a2

,

a 3 are explicitly determined

constants [41]. Proof:

Using the divergence theorem and Green's second identity we can

write J u2 dV

- J (T-t),t u2 dv

D

D

Tf

lfdx + 2 J (T-t) u (~u-Lu} dV

B

D

Tf

ifdx + 2 J (T-t) u (ou/on + Su) dcr

(5.24)

s

B

- 2 J (T-t) u,iu,idV - 2 J (T-t) ~ifdcr

s

D

- 2 J (T-t) u Lu dV D

An application of the a-g inequality yields the inequality J u2 dV

~ T J ifdx +

D

B

t

J (T-t) 13-1

(ou/on+~u) 2 dcr-2J (T-t)u Lu dV

S

D

where we have also dropped the negative third term in (5.24).

A second

application of the a-g inequality then gives J ifdv

~

(1-a)- 1

(T J u2 dx +.!. J (T-t) B

4

(T-t) 2

(Lu) 2 dV}

D

J

+ a-1

~- 1 (ou/on+~u) 2 dcr

S

D

for positive a < 1. Starting with (5.24) we can obtain the following result for a semi-linear equation: Theorem 5.5.

Let v

and~

be functions defined in D, piecewise C2 with

respect to x and piecewise ~ with respect to t. 42

Let v satisfy the

semi-linear equation Lv = f(x,t,v) , where f satisfies the Lipschitz condition

uniformly in the set (x,t) E D, -

where

a

~

< u
Ida on t t

S

- 2 J e(T-t) u,. (u+u ) , .dV - 2 ~

D

~

t

J

D

e(t-t) u Lu dV

(6.21)

2 J e 0 on oB and use the Schwarz in-

equality along with (7.9), (7.10) and (7.11) to obtain

60

§

(t.w) 2 dS ~ 2p -l

oB

m

We still need to bound integrals involving the function

f

(ov /on) 2 dS =

oB

§

(ou/on-o~/on) 2 dS :s: 2

oB

§

~·

We have

(ou/on) 2 dS

oB (7 .21)

A bound on

§ (o~/on) 2 dS

is obtained using the identity (2.6):

oB

:s:

c

§ ~ o~/on

dS

oB

I ..

~ ~,.dx = § ~ o~/on dS since 6~ = 0. 1 1 B oB Now using weighed a-g inequalities with weight a = 2 p:1 , S = 4 p:1 we obtain

where we have used the fact that

t

m (7 .22)

Finally, we note that

y ifdS

Jcf by Theorem 3.1, where

(7.23)

oB

B

a2

is given by a 2 on page 22.

The inequality of the theorem is now obtained using (7.2), (7.4), (7.20)-

(7.23) and the a-g mean inequality:

61

INEQUALITIES FOR OTHER BOUNDARY VALUE PROBLEMS When we attempt

t~

treat other boundary value problems for the biharmonic

operator we immediately run into difficulties.

First, explicit

~

priori

inequalities, as we have come to think of them, have not been developed for some of the problems.

Second, for those that have, rather lengthy and

involved derivations are required.

For these reasons we shall summarize the

known results and refer the interested reader to the appropriate literature for the details. Consider then the second boundary value problem for elastic plates: f

in

B,

=g

on

oB,

M(u) = h

on

oB,

u

(7.24)

where f,g and h are prescribed data and M(u) is proportional to the normal moment, i.e., M(u)

2 6u - (1-a) ( -o u

1 -ou) . +p on

os 2

In the above equation a denotes Poisson's ratio and p denotes the radius of curvature on oB.

priori inequalities (in our sense of the term)

Explicit~

which have applicability to this problem have not appeared iP the literature to our knowledge.

However, Bramble and Payne [ 5] have

given the follow-

ing result: For any biharmonic function V with pi.ecewise fourth derivatives in a domain B whose boundary has bounded curvature, the following ~ priori bound holds for any p-th order derivative of V at a point 0 in B:

lv(P) (0) l 2 s K1

f

v2dS+Ka

oB where the constants K1 determined.

62

f (oV/os) 2 dS+Ka f (o2 V/os2 ) 2 dS+i 3 the fundamental solution is not square

integrable and a weighted Schwarz inequality must be used in the last term of (8.4) giving the pointwise bound

I u(~) Ia

s Kd~)

J

if dx + K2 (c:¥) (~)

B

J

r -c:¥ ([).u) 2 dx

(8.6)

B

with K1 as before and

K~c:¥)(~) = 2

rc:¥(~f) 2 dx

J

(8.7)

sp' Of course, c:¥ is chosen so that ~c:¥) is finite and r-c:Y(h.u)a is integrable. Using (8.4) or (8.6) a pointwise bound for the difference between an approximation u* and the solution w of, f

in

B ,

=g

on

aB ,

[).w =

w

for instance, the Dirichlet problem

is obtained using the inequality of Theorem 4.1 with u

I w(~) -u*(~) 12 s K1 (~) (c:¥1

J

(f-[).u*)a dx + c:Ya

B

+ K2 (~)

J (f-[).u*)a d'c B

68

= w-

u*:

§ (g-u*) 2 dS} aB

(8.8) •

FOURTH ORDER ELLIPTIC PROBLEMS Drawing on the results of the previous section we quickly sketch the approach to this case.

For the biharmonic operator A2 the fundamental solution is

[ 16]:

for all odd N and for even N > 4, and

(-l}(N-2)/2 23nN/2 (2- ~)!

r N-4 log r

See [16] and [22] for information on parametrices for more

for even N s 4.

general fourth order equations.

The fundamental solution has the property

that u(t;;)

= lim p-0

Thus, Green's second identity yields u(t;;)

I

(uH-1)fA2 u)dx

sp'-sp

I u(t;;) 12 s Kdt;;)

I

ifdx + Ka (t;;)

B

I

(A2 u) 2 dx •

(8.9)

B

Having (8.9) it is now clear how to obtain pointwise bounds in biharmonic boundary value problems for which an appropriate

~

priori inequality is known.

SECOND ORDER PARABOLIC PROBLEMS To obtain pointwise bounds in problems involving the heat operator L 3/3t, we need the fundamental solution

r*

of the formal adjoint

=A-

L*= A+~ot: 69

-xi) - ~~i 4(1"-t) }' 2

1

r *·,x,t;~,'T)

exp {

(2jil)N (T-t)N/2

,. > t •

The function r* has the following properties [151: (a)

L*f * (x,t;~,T)

(b)

lim t-"T

I

=0

for each fixed

r * (x,t;~,'T)

f(~)d~

B.r

(~,T)

'

= f(x) '

for every continuous function f

in

Bt •

Di, Da, of D as follows:

Now define two subregions

t

t

< T} ,

< T}

where r 1 < r 2 and r 2 is such that Da is entirely in D. Introduce a C2

~(x,

t)

function~

-{

=ID

as follows:

(x,t) E

1

~

OST\Sl

(x, t) E Da-~

0

(x, t) E D-Da

Then for any C2 function u u(P)

= ~(x,t)

u(x, t), we have, for any P

[uL*(T\f*) - T\fL*u]dV

where we have used Green's second identity for L: vLu - uL*v

(vu,.) 1

'i

- (uv,.) 1

'i

- a(uv)/at

Setting * L* (T\f)

70

= f *L*T\ +

* 2 f,iT\'i s

H(x,t,~,T)

(~;r)

E

~

,

we obtain u(P)

= I [uH-~r*Lu]dV

•

D

An application of the Schwarz and a-g inequalities then gives the bound lu(P)I 2 s Kt{P)

I

tfdV + KJa)(P)

D

I

r-a(Lu) 2 dV

D

where and

K2 (P)

For problems in only one space dimension we can take a = 0 since r* is square-integrable in that case.

See [39] for results on the selection of

a so that KJa) is finite.

71

9 Applications to eigenvalue estimation In this chapter we present a method which gives improvable upper and lower bounds for eigenvalues of self-adjoint elliptic operators given only rough preliminary estimates for them.

The method applies to the classical membrane

and plate eigenvalue problems as well as to Steklov eigenvalue problems. uses some of the

explicit~

It

priori inequalities given in previous chapters,

and, as is usually the case when applying methods based on

~

priori inequal-

ities, the trial functions used in the procedure need not satisfy either the eigenvalue equation or boundary conditions. Our approach here deviates somewhat from that of previous chapters in that attention will be given to particular problems only after the method has been first developed in a general setting.

Further, in order to make

the chapter self-contained, the illustrative numerical examples are included along with the theory of the method. AN A Pa3 TERIORI INEQUALITY Central to the method is below.

the~

posteriori inequality given in Theorem 9.1

This inequality is related to one given by Moler and Payne [24]

(see also [14], [25]) but differs in the important aspect that it can be combined with

~

priori inequalities to permit the estimation of eigenvalues

in terms of quadratic functionals of "arbitrary" test functions. refer to this combined method as the method of a

posteriori-~

We shall

priori inequal-

ities. We proceed now to develop the tor with domain 72

~(A)

~

posteriori inequality.

Let A be an opera-

which is dense in the separable Hilbert space H.

Let

A be symmetric, so that (u,Av)

(Au,v) ,

u,v E

~(A)

and let A have pure point spectrum tAi} with corresponding orthonormal eigenvectors [ui} which are complete in H.

Let A* be an extension of A, so that

~(A)

~(A).

~(A*)

c

The

~

H

with A*u = Au for u E

posteriori inequality is given in the following theorem.

Theorem 9.1. u*

c

Assume the above hypotheses.

For any number A*' and any

E ~(A*), suppose there exists a function w satisfying (9.1)

Then \lw\\ s-\lu*\1

=e

(9.2)

and thus if e < 1 there exists an eigenvalue Ak of A satisfying

(9.3)

Proof:

By symmetry and (9.1)

so that

or },i-A*

~

73

Now since (A.} has no finite limit point, there exists some k such that l.

= min lA .-A*I/IA .1 l. l. i

and consequently,_ for this k, IAk-A*I I (u*' ui) I s l<w,u.)l l.

IAkl

for all i, and thus

...

...

IAk -A) 2

L; I ( u*' u i) 12 s ~ l<w,u.)l 2 l. i=l i=l

lA k 12

(9.4)

We now use the completeness of the (u.} which implies that \\w\1 2 =~1(w,u.)l 2 , l.

\lw\1 2

l.

l.

=~l(u*,ui)l 2 = \lu*\1 2 , in (9.4) to obtain 1Ak-A*I2 --~~--

1Akl2

\lw\12 s ----llu*ll 2

(9.5)

thus proving the theorem. It is not desirable to have to actually obtain the function w which appears in the theorem and it is at this point that we introduce an appropriate

~

priori inequality which estimates w in terms of u*.

AN EXAMPLE -

lliE FIXED MEMBRANE PROBLEM

Let B be a bounded region of Euclidean N-space with boundary oB. Hilbert space H be on B.

n2 (B),

Let the

the space of functions which are square integrable

Let A be the negative Laplacian -6 with domain

entiable functions vanishing on oB.

~(A)

the twice differ-

Thus A* will be -6 with domain

~(A*)

the functions which are just twice differentiable on B. This gives the classical fixed membrane problem -6u =Au 74

on

B,

u = 0

on

oB ,

(9.6)

whose eigenvalues and eigenvectors satisfy the hypotheses of Theorem 9.1. ~

The appropriate 1\wl\2 =

priori inequality to use is that given on page 22

J '-' dx

s 2A1-2

B

J

-1 (llw) 2 dx + 2q1

for wE ~(A*).

§

'-'ds

(9.7)

oB

B

Combining this with (9.5) gives 2A 1-2

IAk -A*I2

J

-1 (6 u*+A u*) 2 dx + 2ql

B

s

9 u;ds

oB

JB

lA k 12

(9.8)

u!dx

Now the right side of (9.8) is a ratio of quadratic forms in the arbitrary twice-differentiable function u*.

Thus we can let u* be a linear com-

bination of test functions, say

(9.9) and minimize the right side of (9.8) with respect to the coefficients ak as in the Rayleigh-Ritz method.

This leads to the relative matrix eigenvalue

problem

0 '

(9.10)

where

and

Now let e be the smallest eigenvalue of (9.10), then

75

or, if e < 1, (9. 11)

giving upper and lower bounds for the fixed membrane eigenvalue Ak which is closest to A*·

OTHER EIGENVALUE PROBLEMS The free membrane eigenvalue problem is 6u + IJ.U = 0

ou on

0

on

with eigenvalues 0 the

~

in

B '

oB ,

= 1.11
. 1-2

aa

2>- -1

aa

pm

I

ifdx + a 3

B

The constants a 1

I

ifda

(10.7)

S

,

a2

,

a 3 are given by

1

-1

K

where pm and K are as defined on page 34. Since (10.6) satisfies (10.3), the first term on the right hand side of (10.7) drops out.

Thus we need only compute the explicit constants a 2 and

aa. The value for a 2 is easily obtained since for B, >- 1 = 1 and thus a 2 = 2. The expression for a 3 also reduces to a simple expression if we take f 1 and f 2

=ft

= 1.

Then in

K,

=x,

p. 34 we have 85

c =1 '

I

fN+l fN+l 'i

'i

fN+l

I

0

M

so that

and then

Finally, setting u = w- ua, where w denotes the solution of (10.3)-(10.5), we have that n

T

J J -n 0

N

(w- ~ a cp ) 2 dxdt s: 2 n=l n n

J

N

(cos x -

L; a cp (x,O) ) 2 dx

n=l

n n

This expression is now minimized with respect to the an, as explained in the introduction. The results of our calculations are given in Table 4 where we have also included pointwise bound calculations as discussed in Chapter 8, and for comparison, the values of the exact solution given by w(x,t) = e

-t

Several observations concerning these calculations are of interest:

86

cos x.

X

t

Approximate Solution

Error Bounds on Approxi.

Exact Solution -t u =e COS X

0 .2n .4n .6n

0.2 0.2 0.2 0.2 0.2

0.8187307 0.6623671 0.2530017 -0.2530017 -0.6623671

0.0000583 0.0000583 0.0000583 0.0000583 0.0000583

0.8187307 0.6623671 0.2530017 -0.2530017 -0.6623671

0 .2n .4n .6n

0.6 0.6 0.6 0.6 0.6

o. 5488116 0.4439980 0.1695921 -0.1695921 -0.4439980

0.0000110 0.0000110 0.0000110 0.0000110 0.0000110

0.5488116 0.4439980 0.1695921 -0.1695921 -0.4439980

0 .2n .4n .6n

1.0 1.0 1.0 1.0 1.0

0.3678794 0.2976207 0.1136810 -0.1136810 -0.2976207

0.0000072 0.0000072 0.0000072 0.0000072 0.0000142

0.3678794 0.2976207 0.1136810 -0.1136810 -0.2976207

0 .2n .4n .6n

1.5 1.5 1.5 1.5 1.5

0.2231302 0.1805161 0.6895101(-1) -0.6895100(-1) -0.1805160

0.0000123 0.0000123 0.0000123 0.0000154 0.0000407

0.2231302 0.1805161 0.6895101(-1) -0.6895101(-1) -0.1805161

0 .2n .4n .6n

2.0 2.0 2.0 2.0 2.0

0.1353353 0.1094885 0.4182087(-1) -0.4182100(-1) -0.1094886

0.0000127 0.0000127 0.0000145 0.0000228 0.0000620

0.1353353 0.1094885 0.4182090(-1) -0.4182090(-1) -0.1094885

0 .2n .4n .6n

2.5 2.5 2.5 2.5 2.5

0.8208500(-1) 0.6640812(-1) 0.2536559(-1) -0.2536571(-1) -0.6640818(-1)

0.0000253 0.0000253 0.0000374 0.0000590 0.0001556

0.8208500(-1) 0.6640816(-1) 0.2536566(-1) -0.2536566(-1) -0.6640816(-1)

0 .2n .4n .6n

3.0 3.0 3.0 3.0 3.0

0.4978704(-1) 0.4027856(-1) 0.1538495(-1) -0.1538505(-1) -0.4027803(-1)

0.0000758 0.0000934 0.0001381 0.0002178 0.0005743

0.4978707(-1) 0.4027859(-1) 0.1538505(-1) -0.1538505(-1) -0.4027859(-1)

0 .2n .4n .6n

4.0 4.0 4.0 4.0 4.0

0.1831575(-1) 0.1481759(-1) o. 5659724( -2) -o. 5659134( -2) -0.1481537(-1)

0.0004035 0.0004972 0.0007355 0.0011600 0.0030588

0.1831564(-l) 0.1481766(-1) 0.5659844(-2) -0.5659844(-2) -0.1481766(-1)

.an

.an

.an

.an

.an

.an

.an

.an

TABLE 4* *Approximate values for negative x are obtained by reflecting those given across the t-axis since the solution is symmetric with respect to the t-axis. 87

(A)

The approximate values are very close to the actual values; in many

cases approximate and actual values agree to seven significant figures. Agreement between approximate and actual values is always much better than the error bounds indicate.

Thus although the error bounds are generally

good when looked at as a percentage of the approximating value, they are pessimistic when compared to actual errors. (B)

As is evident from the discussion in Chapter 8, the pointwise error

bounds worsen as the boundary is approached (K1 (P) and K2 (P) become unbounded as P approaches the boundary). values of t.

This is noticeable at the larger

Notice, however, that the approximations themselves are not

so severely affected. (C)

For t

~

3.0 the error bounds, and to a much smaller degree the

approximations, become progressively worse.

This happens because as time

increases more trial functions are needed to construct the approximating function if a given error is to be maintained.

In our calculations we

restricted N to be less than or equal to 10. With the exception of the values obtained for t = 0.2 and 0.4, all results in Table 4 were obtained using ten trial functions.

At t

functions were employed while nine were used for t = 0.4.

This was neces-

0.2, eight

sary because the system formed in applying the minimization pcocedure tended to be slightly ill-conditioned, especially for the smaller values of t. Thus for the first two values of t, calculations employing ten trial functions yielded worse results than calculations employing fewer functions. This tendency toward ill-conditioned systems persisted throughout the calculations although it was not as pronounced at the larger t values.

Because

of this tendency, and also to avoid loss of significance due to the subtraction of two nearly equal numbers when mean square errors on the boundary 88

were computed, all calculations were done using double precision arithmetic. It is extremely important that highly accurate methods are used to determine the a n 's since the error bounds, but not the approximations, are quite sensitive to errors in these constants. One last obvious observation concerns the selectivity of the method. That is, an approximate value can be calculated at a few points without the need to perform calculations at many additional points in which one has no interest.

Thus if one is interested in having an approximation at the

point (x1 , t 1 ) , the approximation can be computed iUBIIediate ly without "building up" the solution through a succession of calculations from t

=0

to

SOME EXPERIMENTS IN TRIAL FUNCTION SELECTION From the previous examples it is evident that the accuracy of the approximation will be highly dependent upon how closely the set of trial functions can approximate prescribed data in the norm induced by the ity.

The situation is usually this:

~

priori inequal-

we have selected our trial functions

from a complete set R of functions, i.e., products of sines and cosines, products of powers of x and y, polynomials which satisfy the differential equation, etc.

From this set of trial functions some, or even most, may

not be useful in approximating the prescribed data.

It would be very de-

sirable if once the minimization process has been done we could in some way use the results thus obtained to

select

those functions from our set of

trial functions which are of most use in the approximation, add more terms of that type, discard those which appear to be of no use and rerun the procedure.

Some indication of the importance of the trial functions can be

derived from an analysis of the relative values of the coefficients of each trial function and their effect on the computed norm error bounds.

89

We now discuss the results of some experiments along these lines via the treatment of a biharmonic boundary value problem. the computations is that given in Theorem 7.1.

The inequality used in

Consider the problem

4 cos x sin y

(10.8)

on the unit square with u and ou/on chosen on the boundary so that the solution of the boundary value problem is u(x,y) = cos x sin y.

We mention that

from the boundary conditions and (10.8) a natural first guess for the trial functions would be the products cos (~} sin

ti}.

These would have given ex-

cellent results since the actual solution is included in this set.

To bet-

ter illustrate our ideas, we consider instead trial functions of the form m n

x y

as providing a "general" set of trial functions which lead to a rea-

sonable problem to analyze and program.

In addition, this group of trial

functions contains many functions which are not in the power series expansion of the solution and hence would not be expected to be useful in an approximate solution.

Also, notice that except for the first few terms,

none of these functions satisfy either the differential equation or the boundary conditions.

With these trial functions the approximating function

becomes N N m-1 n-1 ~ a x y n=l m=l m+N(n-1) ~

Case 1.

We choseN= 3 in (10.9), thus obtaining an approximating func-

tion with 9 terms. pansion of u.

Of these 9 terms, only 2 appear in the power series ex-

The minimization procedure yielded the function (all coeffi-

cients have been rounded to 2 significant figures) .012 + .0027x + .0012x2 T l.Oy + .047xy - .53x2y - .22y2 90

(10.9)

-

.094xy2

+ .19x2y2 •

Case 2. tions 1, x,

This was the same as Case 1 except the first three trial func-

xa

were deleted because they did not appear to be important

judging by the values of their coefficients.

This case resulted in the

following approximating function: ua (x,y) = LOy + .05lxy - .53i8y - .22y2 - .096xy2 + .19i8y2 , with very little change resulting from the previous case as would be expected if the deleted trial functions were not important.

Nothing more

would have been gained here except for the fact that we also started computing the norm error terms and found that 98 percent of the total error was due to the error in approximating

~ 2 w,

while the error due to the ap-

proximation of the three boundary terms contributed the remaining 2 percent. This suggested that we add to the approximating function more of the terms of the expansion of Case 3.

~2 w,

i.e., cos x sin y.

The six trial functions of Case 2 were used plus x4 y, x8 y, y3

,

u (x,y) = LOy + 0.28xy - .53i8y + .0025y2 - .03lxy2 + .00033i8y2 a ( 10.10)

+ .049x4 y + .0044x6 y- .16y3 + .12i8y3

-

.025x4 y3 + .0006lx6 y3

with a decrease in the norm error bound of two orders of magnitude, while the contribution to the norm error due to the approximation of to 13 percent of the total error.

~2 w

dropped

But just as important is what happened

to the coefficients of xy, y2, xy2 and X8y2 (which do not appear in the power series expansion of the solution) from Case 3 to Case 2, as compared to the coefficients of y and i8y (which do appear in the series expansion of the solution).

Whereas the coefficients of the latter did not change

(at least to the two figures reported here), the former ones decreased in

91

absolute value by at least a factor of one-half and some by several orders of magnitude.

This suggested that these terms are not important to the ap-

proximation and should be deleted. Case 4.

Doing this results in Case 4.

This is the previous case with xy, y2, xy2 and ,(3y2 deleted.

This resulted in

At this point it is clear what additional trial functions should be added if a better approximation is desired. Although this example is in some sense a special case, the results of the experiment lend a good deal of support to our hope that the results of the minimization process, via analysis of the relative values of the coefficients of the trial functions and the norm error bounds, can help in selecting the appropriate trial functions from a large number of candidates.

We

now turn to more detailed numerical examples.

A BIHARMONIC EXAMPLE As our last example we consider some biharmonic boundary value problems defined on three regions:

a square, a triangle and the

quadran~

of a circle.

The first two regions were chosen for their simplicity, the last because it represents a region which presents some difficulty.

For each region we

chose the boundary values so that the test problem had as its solution u(x,y) = ex + cos y.

As in the previous section the appropriate inequality

to use is that given in Theorem 7.1:

f B

92

u2dxdy:s:a1 f(t~2 ufdxdy+a2 fu2dS+a3 g(ou/onfdS+a4 f (ou/os) 2 dS B oB oB oB

As the approximation function we used (10.12) Again we point out that except for xn, yn, i=l,2,3, none of the trial functions satisfy the differential equation or the boundary values, and extraneous terms are present via odd powers of y. (A)

The square.

Consider the biharmonic boundary value problem B = ((x,y)IO<x