Ockendon J., Howison S., Lacey A., Movchan A. Applied Partial Differential Equations

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158

ELLIPTIC EQUATIONS

Finally, we must specify the tractions, i.e. the stresses that are applied to the

ends of the bar to produce the torsion. The shear stresses on these ends comprise

the two terms

(8w Out

pa(-Y, x, 0) +µa

8x,

8y,0)

and hence the total moment applied about the origin in any cross-section is

M=µaJJDIxz+yz-y8x+x

)dxdy. (5.21)

The fundamental problem of torsion is to solve (5.19) subject to (5.20) and (5.21)

in order to relate M to a and hence determine the 'torsional rigidity'. But many

torsion bars are hollow and we must remember our earlier warning about non-

uniqueness in multiply-connected domains.

5.1.4.4

Plane strain and slow viscous flow

Continuing the previous discussion, suppose we have an infinitely long bar which

is not subject to torsion but whose curved boundary does have tractions applied

to it perpendicular to the axis. We now have what is called plane strain57 with

u = (u(x,y),v(x,y),0), (5.22)

which implies there are both shear and tensile/compressive stresses in the bar.

We now need both terms in the right-hand side of the equations of linear elastic-

ity (4.73), so that

µ 2u+(J1+p) (8z +49X

=0.

It is a simple matter to cross-differentiate and show that both u and v satisfy the

biharmonic equationb8

04th = V4v = 0,

(5.23)

where

20x20y2 +

8y04

= (OZ)2 =

C94x4x

+2004

In practice, it turns out often to be convenient to reformulate this model in

terms of the stresses (see Exercise 5.1). This is very important to gain insight into

57The displacements are non-zero only in the (x, y) plane. The complementary situation in

which the displacement has the form (0,0, w(x, y)) is called antiplane strain, and it is easy to

see that V2w = 0.

"Although the biharmonic equation is fourth order, its solution turns out to be so similar to

that of an elliptic second-order equation that we describe it as elliptic; when it is written as a

first-order system, it is indeed elliptic according to the ideas of §§3.3 and 3.4.

MODELS 159

the kinds of boundary conditions that might be imposed on the biharmonic equa-

tion, and we note that the application of tractions on 8D involves the prescription

of a two-dimensional vector there. This vector is in fact a linear combination of

the first derivatives of u and v, and hence we must be prepared for `twice as much'

boundary data as for Laplace's equation.

Two-dimensional problems of slow viscous flow also lead to the biharmonic

equation. There the assumption of a shear viscosity p in a fully two-dimensional

flow59 gives that the forces generated by the pressure p must balance the shear

forces, in the absence of inertia. Hence, with velocity v = (u, v)T, we can follow

the argument leading to (4.73), with A = 0, to derive the model

V VZV=

24)

P P P

\ V

and, assuming incompressibility as in (5.17),

V v = 0.

(5.25)

Now, following the idea of introducing a potential, as already used several times

in this chapter, we eliminate (5.25) by writing u = O/8y and v = -O0/8x, where

io is a stream function. Then we can take the curl of (5.24) to show that u, v and

,j, all satisfy the biharmonic equation (p only satisfies Laplace's equation, but,

regrettably, in a realistic model boundary conditions can never be prescribed just

for p). Moreover, because the flow is viscous we expect that on an impermeable

boundary v itself is prescribed rather than just v - n, as was the case for an inviscid

fluid; hence we again have two boundary conditions to be satisfied.

We note that, if a slow viscous flow occurs in a porous medium such as a

sugar-lump, sponge or rock, the simplest model is to assume Darcy's law to relate

the velocity and pressure. This means that the slow viscous flow model (5.24)

is abandoned and replaced by Darcy's experimental observation that, on a scale

much bigger than the pore size,

V=-kVP

(5.26)

in the absence of gravity, which is clearly analogous to the Fourier, Fick and

Ohm laws mentioned above; k is a positive constant called the permeability and

it depends only on the geometry of the pores. However, there is a serious question

about what we mean by v in such a complicated configuration and we must be

careful to remember that the fluid does not occupy the whole of space. Nonetheless,

by regarding v as measuring the mass flux, it can still be argued that

0 for an

incompressible fluid, and we end up with Laplace's equation again. It is uncommon

for an increase in the geometric complexity of a problem to be accompanied by a

decrease in its analytical complexity!

Not surprisingly, these examples enable much `technology transfer' between

fluid and solid mechanics.

59Rather than the quasi-one-dimensional flows such as the paint flow of §1.1.

160 ELLIPTIC EQUATIONS

5.1.5

Acoustics

There is no better illustration of the delicacy of elliptic differential equations than

the vitally important science of acoustics, i.e. small-amplitude wave propagation

in fluids or solids.80 We will only consider the former here, and then only in the

so-called frequency domain where we assume the sound field is being generated by

a loudspeaker operating at just one frequency w. Then, in the simplest case, when

we write 0 = 34 (u(x)e-"dl) in the wave equation 820/8t2 = 4V20, we find

(V2 + k2)u = 0,

(5.27)

where k = w/ao; (5.27) is known as Helmholtz' equation.81 The seemingly similar

model

(V2

- k2)u = 0,

(5.28)

called the modified Helmholtz equation, has absolutely nothing to do with acoustics,

as will soon become evident from a study of its mathematical properties. However,

(5.27) and (5.28) can be related in the context of chemical reactions because we

have seen that, if the heat source is proportional to the temperature, (5.10) reduces

to (5.27) or (5.28) when the reaction is exothermic or endothermic, respectively.

There is also another clear physical reason to expect the properties of solu-

tions to (5.27) and (5.28) to differ widely. Suppose we have any of the boundary

conditions (5.11)-(5.13) but with zero forcing. With these homogeneous boundary

conditions, u = 0 is a solution of the problem, but is it the only one? We would

expect an acoustic resonator, modelled by (5.27), to be able to vibrate of its own

accord; in other words, we would expect the existence of eigenfunctions, and prob-

ably many of them. We will soon see that a trivial application of Green's theorem

shows that, whereas (5.28) admits only the zero solution, there is no such unique-

ness result for (5.27). Of course, our experience suggests that non-zero solutions

only exist at certain 'eigenfrequencies' w = w;, but this is not always true. Our

investigation into this problem will inevitably involve a little of the vast theory of

eigenfunctions of elliptic differential operators, but only the rudiments are needed

later in this chapter.

One other crucial remark needs to be made concerning acoustics in the fre-

quency domain, and that concerns the model in cases where the physical domain

of interest extends to infinity. Since the model describes wave motion, we expect

it to be able to distinguish between situations in which waves are `incoming' from

those in which they are `outgoing'; this suggests that we need more precise condi-

tions at infinity for Helmholtz' equation than we would for Laplace's equation in

an infinite physical domain.

801n solids we usually obtain vector elliptic equations, which are closely analogous to Maxwell's

equations in the frequency domain.

61The complex time dependence of 0 means that we need to take complex combinations of

real solutions of Helmholtz' equation. The choice of e-i't rather than ei'

is made in order to

simplify later formulae.

MODELS 161

5.1.6

Aerofoil theory and fracture

We conclude this section with two examples which highlight the global importance

of the specification of any local singular behaviour that may occur in the solution

of elliptic equations. A trivial illustration of the sort of thing that can happen is

the Dirichlet problem for Laplace's equation

V2u=0

in the open sector

0<r<1, 0<9<2

in plane polar coordinates, with u = 0 on the boundary. If we insist that u is

bounded at the corners of the domain, the only solution is u = 0. However, it is

easy to verify that

u a

(r2

sin 29

satisfies all the conditions except for boundedness at the origin. Thus a singularity

at one point of the boundary makes its presence felt everywhere. Such singularities

could occur in any of the aforementioned situations, but we just mention two

striking examples which perhaps have more everyday technological importance

than any others in this book.

5.1.6.1

Aerofoils

In §5.1.4.1 above, we blandly asserted that the boundary condition on, say, an

impermeable obstacle placed in an irrotational streaming flow, such as in a wind

tunnel, is the Neumann condition for a velocity potential satisfying Laplace's equa-

tion. Moreover, it is well known that a properly designed aerofoil has a sharp trail-

ing edge, i.e. OD must not be smooth if the aerofoil is to work well. The details

of the fluid mechanics of this situation are too intricate to enter into here [29]

but one simple scenario can be plausibly laid out. Suppose that a two-dimensional

aerofoil with surfaces y = ff(x), 0 < x < c, is so thin and so nearly aligned with

the flow that it is always close to the line y = 0, 0 < x < c, and that the free

stream velocity in the x direction is U (see Fig. 5.2). Then, as in the linearised

Fig. 5.2 Flow past an aerofoil.

162 ELLIPTIC EQUATIONS

models (3.11) and (4.70), 0 is almost equal to Ux and so we can write ¢ = Ux + 4)

to obtain

V20

= 0, 1V41-, 0 as JxJ -* oo, (5.29)

but we will see later that we must consider the decay rate at infinity more carefully

than we have in (5.29) if we are to explain the theory of flight. Lastly, we can

replace the exact condition of tangential flow, namely 84/8y = f (x) 8¢/8x on

the aerofoil y = ft(x), by

=Uff(x) ony=±0,0<x<c. (5.30)

We are thus led to a boundary value problem for a harmonic function in a multiply-

connected domain; we might regard the discontinuity that inevitably occurs across

the segment 0 < x < c of the x axis as a line of singularities, and in particular

we may expect especially violent singularities at the `leading' and `trailing' edges

(0, 0) and (0, c). One problem for elliptic theory is to decide what to do about

these singularities, which will also be crucial to our understanding of the theory

of flight.

5.1.6.2

Brittle fracture

Our final example concerns the modelling of cracks in perfectly elastic solids. A

realistic configuration is illustrated in Fig. 5.3(a) for a crack y = 0, -c < x < c,

in a material in plane strain under tension in the y direction. However, we will

confine ourselves to the easier cracks that can occur in antiplane strain with a

displacement field u = (0, 0, w (x, y)) where w is a harmonic function. There is no

traction on the crack surfaces, so that

8 -+0 asy-i0,-c<x<c

(5.31)

from above or below, so that the problem can still be represented as in Fig. 5.3(a).

Also, assuming symmetry,

w = 0 on y = 0, IzI > c, (5.32)

where the material is pristine. Finally, we suppose the crack is subject to a uniform

shearing at large distances, modelled by

w -ry aslyl -goo,

(5.33)

where ,r is a constant.

The symmetry about y = 0 means that we need only solve the problem in

a half-plane, in contrast to the aerofoil model (5.30), where we need to find a

harmonic function in the doubly-connected region exterior to the aerofoil (unless,

of course, the aerofoil is symmetric and f+ = -f_).

The boundary conditions (5.31) and (5.32) are an example of what is called a

mixed boundary problem, the data being part Neumann and part Dirichlet. This

WELL-POSED BOUNDARY DATA 163

(a)

(b)

Fig. 5.3 (a) Brittle fracture in pristine material; (b) partly opened pre-existing sack.

switch engenders possible singularities at the tips of the crack which we can readily

see by separating the variables in local polar coordinates (r, 0), say near z = c,

y = 0. The function rA sin AB is a local 'eigensolution' as long as

where n is an integer, because it is only for these values of A that w = 0 on 0 = 0

and 8w/80 = 0 on 0 = a. As in aerofoil theory, we shall see that this 'singular

behaviour' has far-reaching implications for the predictions of the mathematical

models. Even at this early stage one might, for example, enquire as to the difference

between models for a crack and for a 'contact' region. The latter could arise if the

two slabs in Fig. 5.3(b), each containing a slight indentation in -c < x < c, had

been brought together; how would the mathematics distinguish this situation from

that in Fig. 5.3(a), which involves a crack surrounded by pristine material?

With all these examples as motivation, we now embark on as general an account

of the theory of elliptic partial differential equations as we are able to give in a

book of this type. We will clearly have much to say about Laplace's equation and

the effects of singularities in the data, and we will confine most of our comments to

two-dimensional problems. However, a great deal of our discussion generalises quite

easily to higher dimensions and sometimes to higher-order and vector equations

as well.

5.2 Well-posed boundary data

5.2.1 The Laplace and Poisson equations

We could begin our discussion by repeating the procedures of Chapter 4 and

writing Laplace's equation in characteristic variables z = z + iy and i = x - iy as

02u+8su482u

8x2 8y2

8z 02

noting that u is the sum of a function of z and a function of z. Then the solutions

satisfying analytic Cauchy data, say, u(x, 0) and 8u/8y (x, 0) given, can be written

down from d'Alembert's formula (4.38) as

164 ELLIPTIC EQUATIONS

u(x, y) =

(u(z, 0) + u(f, 0)) +

(5.34)

but we already know from our discussion of the Cauchy-Riemann system and

analytic continuation that this recipe is ill-posed. This is a relief because we can see

from the models in §5.1 that it would be a disaster if well-posedness for Laplace's

equation demanded analytic Cauchy data.

Let us therefore reconsider Poisson's equation in the form

Vu = f (x, y).

(5.35)

Motivated by the models in §5.1, we begin by considering (5.35) in a closed

bounded region D but with only a single boundary condition rather than the

Cauchy data needed for (5.34). All possible linear conditions are encompassed by

the Robin condition, which we will write in the form

au +0T =g on OD,

(5.36)

where a, 0 and g are given functions of position, and we recall that 8/8n denotes

the outward normal derivative; we expect a and j9 to be of the same sign from

(5.13). Our immediate worry that we may have allowed the solution too much

freedom by reducing the number of boundary conditions is unfounded because of

the following theorem.

Uniqueness theorem Suppose u satisfies V2u = f in a bounded domain D with

smooth boundary 8D, with an + 0 8u/49n = g on OD. Then, if a solution exists,

it is unique provided that a never vanishes and f3/a > 0.

An elementary proof uses Green's theorem over D in the form

ffV.(UVU)dxdY=f U8n

ds, (5.37)

where U is the difference between any two solutions of (5.35) and (5.36). Since a

never vanishes,

(OU)

J JD

IVU12

dxdy = -

d8; (5.38)

considering the signs of the two sides of this equation, we we that both sides must

vanish and thus IVUI = 0 almost everywhere in D. This means that U is constant

almost everywhere and hence zero from the boundary condition.62 Although we

have stated it in two dimensions, the theorem holds in any number of dimensions,

with the same proof.

62We cannot exclude the possibility that U is non-zero on a set of measure zero, but then

U would probably not satisfy Laplace's equation on such a set and we will not deal with such

`pathologies' in this book.

WELL-POSED BOUNDARY DATA 165

As a corollary, we note that, when a = 0,

IVU12

dxdy = 0,

so that u (if it exists) is non-unique to within addition of a constant. Thus, for

example, the solution to the torsion problem of §5.1.4.3 is only determined to

within an additive constant in w, which just corresponds to moving the bar bodily

along its axis. It is also possible to give specific examples to show that, if /3 96 0

and a/,6 changes sign on OD, or if o//3 < 0, uniqueness can no longer be assured

(see Exercise 5.4). We recall that the latter case corresponds to heat flow from

cold to hot in (5.13). Also, it is clear from Green's theorem that, when a = 0, no

solution exists unless, as in (5.14),

lID

(5.39)

The uniqueness theorem and its corollary have been stated assuming that OD

and the data are as smooth as necessary. We must also remember the example at

the beginning of §5.1.6, where, even though Dirichlet data were prescribed at all

points of 8D except one, the existence of a singularity at that point resulted in a

solution that was non-zero in the interior of D. This can happen whether or not

8D is smooth, and it will cause us much trouble later in this chapter.

These results will come as no surprise to experts in two-point boundary value

problems with Robin boundary conditions for ordinary differential equations, but

here, in addition to the singularity problem we have mentioned, we have to worry

about the possibility that D is not simply connected. Suppose, for example, we

adopt the following commonly used device to solve the torsion problem (5.19)-

(5.21). There is often a computational and theoretical advantage in working with

a Dirichlet problem rather than the Neumann problem for to, so we introduce its

harmonic conjugate // for which a simple calculation reveals the Dirichlet problem

V2'=0,

where the zero-stress condition (5.20) requires that

ty -

(x2 + y2) =constant on 8D,

the constant being arbitrary when D is simply connected. Things are even eas-

ier computationally if we can take the constant to be zero and write 63

5 (x2 + y2) + 0, so that

V2+G=-2

with

tp = 0

on 8D.

(5.40)

Now, if D is an annulus, as is often the case in practice, ' is a different constant

on each component of the boundary of the annulus, but what is the difference

63 is called the Prandd stress function.

166 ELLIPTIC EQUATIONS

between these constants? We certainly do not have a well-posed Dirichlet problem

without this knowledge. The answer lies in the fact that, in transforming from the

physical variable w to a stress function st', we have introduced an indeterminacy

and we must be sure that when we eventually retrieve w from ii, by solving the

Cauchy-Riemann equations, we have a physically acceptable displacement. In this

case the w we retrieve is multi-valued unless

JOD t9

da = -

J D

On ds

is zero around any circuit in the annulus that cannot be shrunk to zero. Hence

is uniquely defined by the condition that

aids+I a

(5.41)

taken around any such circuit (the first term on the left-hand side is in any case

a constant independent of the circuit).

This calculation reveals a common bugbear of working with `potentials' such as

' or is they are often helpful theoretically and computationally but their arbitrari-

ness (or so-called `gauge invariance') calls for constant vigilance. This example, and

the vortex problem in §5.1.4.1, illustrate the message that great care must always

be taken when considering uniqueness for elliptic problems in multiply-connected

domains.

5.2.2

More general elliptic equations

We have already seen in our discussion of the Helmholtz equation in the introduc-

tion that questions of existence and uniqueness can depend very sensitively on the

coefficients of elliptic equations. Hence we will not enumerate statements about

equations other than the Laplace and Poisson equations, except to say that the

ideas above can be tried but they may or may not work. As an example, we cite

the biharmonic equation V4u = 0 with `Dirichlet data' in which u and Ou/On are

prescribed on a dosed boundary 8D. The problem of uniqueness can be resolved

by considering the difference w = u - v, for which

0 =

fJwOawdxdy

= ff

(V. (wV(V2w)) - Vw V(V2 w)) dxdy

ffa

(V. (V2wVw) - (V2w)2) dxdy = ff (V2w)2 dxdy,

each divergence integrating to zero by Green's theorem. Hence V2w = 0 and,

because of the boundary data, w = 0.

The use of Green's theorem to prove uniqueness is only at all easy when the

differential equation is in divergence form. This makes uniqueness theory difficult

for vector equations, and for higher-order equations other than the biharmonic

equation. Also, problems of mixed elliptic and hyperbolic type pose their own

THE MAXIMUM PRINCIPLE 167

peculiar challenges for existence and uniqueness.64 Fortunately, there is one other

tool that we can use in many practical situations.

5.3

The maximum principle

It is almost obvious that, if Laplace's equation holds in a closed domain D, the

maxima and minima of u occur on 8D; one would never see a membrane, as

modelled in §5.1.4.2, with a `bump' in it, unless, of course, a pressure difference

was applied across it. This is certainly true if we exclude points at which 82u/8x2

and 82u/8y2 both vanish because, if one is positive and the other negative in the

interior of D, at any critical point where 8u/8x= 8u/8y = 0, we have

82u82y

'"U

y)2

8x2 822

< 0

and so the critical point must be a saddle. To make this argument rigorous we note

that, if an auxiliary function V satisfies Poisson's equation (5.35) with f > 0, then

certainly one of 821'/8x2 or 82V/8y2 is positive and so V cannot have a maximum

in the interior of D. It may have a minimum or saddle in the interior, but any

maximum is on 8D. Hence, if V2u = 0 and we write V = u + Ere/4 (where e > 0,

r2 = x2 + y2 and (0, 0) is assumed to be in D), V2 V = e > 0 and so any maximum

of V is on 8D. Hence

2 2

u+E4

where k 1v

lbiu are the maximum values of V and u, respectively, on 8D and

R is the largest distance from the origin to 8D. Letting E - 0 in u < Afu +

e(R2 - r2)/4, we see that any maximum of u also occurs on 8D (and, similarly,

any minimum).

The maximum principle enables us to rederive part of the uniqueness theorem

in §5.2 and also to do much more, as we now see.

Uniqueness

For Dirichlet data for Poisson's equation, the difference between two solutions

is a harmonic function vanishing on the boundary. The fact that it attains its

maximum and minimum there means it is zero.

Comparison theorem

Consider two Poisson equations V2u, = f (x, y), i = 1, 2, with the same right-hand

side f and smooth Dirichlet data g; = g, (x, y), such that 91 < g2 pointwise. Then

ul(x,y) < u2(x,y) in D because u2 - ul attains its positive minimum on 8D.

Similarly, if 91 = 92 but the Poisson equations have different right-hand sides ft

and f2 with f, > f2 pointwise, then ul (x, y) < u2 (x, y).

s"Note that our accumulated knowledge of hyperbolic and elliptic equations can be used to

help to understand mixed equations such as (3.13) which are elliptic and hyperbolic on either

side of a `sonic line', in the case that data is given on a curve I' that crosses this line. Clearly we

would have an overdetermined problem if we prescribed Cauchy data on the 'hyperbolic part' of

I' if that data enabled us to find both u and its normal derivative on the sonic line.