NEW ZEALAND JOURNAL OF MATHEMATICSVolume 22 (1993), 43-64

THE ELASTIC TORSION PROBLEM: SOLUTIONS IN CONVEX DOMAINS

G r a n t K e a d y a n d A l e x M c N a b b

(Received April 1991)

Abstract. The torsion function of a plane domain is a function which is zero on the boundary of the domain and whose Laplacian is minus one at every point in the interior of the domain. We survey qualitative properties of the torsion function, and bounds on it and its derivatives, available when the plane domain is convex.

1. IntroductionLet ft be a domain in the plane with a Lipschitz continuous boundary. The

problem of finding a u, twice continuously differentiable in and continuous on the closure of Cl satisfying, for some given positive constant /i,

—A u = fi in f I,

u = 0 on the boundary of f2,

is called the (St Venant) elastic torsion problem. There is no loss of generality in taking /z = 1, and henceforth, this /i = 1 case is refered to as Problem (P). In addition to its importance in elasticity, Problem (P) arises in fluid mechanics: there Problem (P) describes the steady unidirectional flow of a viscous fluid down a pipe of cross-section Q, the pressure gradient along the pipe being constant. Besides arising directly in these applications in mechanics, the torsion function also arises in component parts of larger problems (e.g. [Keal, KK]), as a particular case of harder problems like elastic-plastic torsion problem (e.g. [Fri]), and as a function against which solutions of harder problems are compared (e.g. [PaWe]).

This survey is, in part, a by-product of two other pieces of work by the present authors, (i) The first of these concerns a thorough treatment of the torsion prob­lem in lens-shaped domains, domains which are the intersection of two disks: see [KM1, MK1]). This was a continuation of work, describing just one very special case, which appeared in [MW]. The treatment by the present authors involves explicit exact solutions, numerical computation and theorems on the qualitative behaviour of solutions. The bounds given in this survey were used as checks on the numerics for the solutions in lens-shaped domains, (ii) The second piece of work is a treatment of the same partial differential equation in N space dimensions and with more general homogeneous boundary conditions in [KM2].

Where it is practical to do so in this paper, the order is (0) results concerning the function u, (1) results concerning its gradient, and finally (2) results concerning its second derivatives - particularly ‘concavity’ properties. As there are numerous interconnections this order is not followed strictly.

1991 AMS Mathematics Subject Classification: Primary 35J05, Secondary 73C02.

44 GRANT KEADY and ALEX McNABB

(0 ). The second author has an application of problem (P) arising from work from outside elasticity. This was described in his talk to the 1989 N.Z. Mathematics Col­loquium, an abstract for which is given in the July 1990 issue of this journal. (See also [KM 2 , M K2].) A functional of interest in this application, is the maximum of the torsion function, um, and its location zm,

uizm) = um = max w.

In other contexts, the quantity um also occurs in [KS], equation (1.10), and in [SM], equation (6). We have not found tables of the functional um elsewhere. This is in contrast to the functionals of significance in elasticity, which include the following. The torsional rigidity is

where n denotes the outward normal, and z denotes the coordinate vector for points on the boundary <9ft.

(1). A fail point Zf is a point of ft, necessarily on the boundary of ft, at which the gradient of u has its largest magnitude. Our notation is

pm - max |Vu| = |Vu|(z/). ci

In recent years there has been considerable interest in the following, the wording being adapted from [Pa4]:Question 1. Is it possible to characterize geometrically the boundary point Zf at which |Vw|2 takes its maximum value?Some of the history of this is discussed in [Ka2] who traces the question back to [SV]. Work since [Ka2] is reviewed in Section 6.

(2 ). We now turn to second derivatives. A result from [ML] 1971 is as follows.

Theorem 1.1. (Makar-Limanov). Let u be the torsion function of a convex do­main ft. Then the square root of u, y/u is concave.

A summary of the proof is given in Section 11. Generalisations of the result to higher space dimensions were first proved in [Ken]. Generalisations of the proof techniques to some semilinear elliptic equations are given in [Kea2], A survey of some results related to this is given in [Kal]. Kennington has recent, as yet unpublished, work extending the result to domains on surfaces of constant curvature, e.g. spheres.

Though the best results to date concern convex domains, one can also study the concavity properties of torsion functions on subsets of ft, for domains ft which are not necessarily convex. In this connection, define H and fti as follows. The hessian H is

H (z) = det (D 2u(z)). (1.2)

THE ELASTIC TORSION PROBLEM: SOLUTIONS IN CONVEX DOMAINS 45

The set is defined to be the set on which the second derivative matrix D 2u is negative semidefinite, which, for this problem, is

^1 = {z e | H (z) > 0}. (1.3)

An acceptable phrase with which to describe is ‘concavity set’ . Clearly zm € Q\. As mentioned above, the second author has interests in the location of zm: similar questions concerning fii seem reasonable. In Section 6 we show that, in a convex domain, Zf € Q,\.

For the torsion function of an ellipse Cl = Qi. The question of for which other convex f2, this is true is investigated in [Kosl]: see Section 11 below.

Theorem 1.1 and many other results concerning Problem (P) are proved using Maximum Principles for second order elliptic equations. Our survey emphasises, but is not restricted to, results provable by these techniques.

Numerical information on curvature and other derivatives is relatively difficult to obtain from the usual processes of numerical solution appropriate to arbitrary domains. As a consequence, in calculating with particular examples, there is value in explicit exact solutions such as are often given in older literature. (The most recent English-language survey of exact solutions of which we are aware is [Hi], published half a century ago.) The ability of computer algebra systems to handle involved expressions is of assistance in the re-analysis of the exact solutions. The work in [KM1, MK1] exemplifies the results of such approaches when is a lens-shaped domain.

2. NotationThe notation used both here and in [KM1, MK1, Kea3] is as follows. B(zo, r)

denotes the disk centre zq and radius r. We begin with notation for various geo­metric functionals

A, the area;L, the perimeter;zc — (x c, yc), the centre of gravity or centroid;p, the inradius, and incentres Zi, B(zi,p) C f2;R , the circumradius, and circumcentre z0, Q C B(za,R );Kmax, the maximum value of the curvature over the boundary;ftmim the minimum value of the curvature over the boundary.

We now index where further items associated with Problem (P) are defined. The quantities 5, um, zm, pm, Z f , H and were defined in Section 1. The quantity 8m is defined here

Sm = distance(2m, dfi).

The quantities Pk,Pk = |Vu|2 + kfiu,

are first treated in Section 4, beginning at equation (4.1), and are useful in con­nection with first derivative bounds. The quantities Q(k) are

Q(k) = T + kuH ,

46 GRANT KEADY and ALEX McNABB

whereT — V'y'U'xx ^Ux^yUxy 'U'x'Uyy — ^?(0)-

T first occurs in the proof of Theorem 6.4, and Q(2) in the proof of Theorem 6.5. The systematic treatment of Q(k) in Section 10 is useful in connection with second derivative bounds. As the case k = 2 occurs frequently, we abbreviate Q(2) to Q.

Some further geometric functionals, u r ( p , R) and u r ,a { p , R) are defined in Sec­tion 3.

The torsion function for the disk B(z*,a) is

When the centre and radius are clear from the context, this is abbreviated to u b -

of a plane domain: the letter s denotes arc length around the boundary increasing in the counterclockwise direction. If 2 G dQ., the curvature at this boundary point is denoted k ( z ).

3. Exact Solutions in some Simple DomainsIn this section f i = 1. Simple exact solutions are convenient for checking against

inequalities, and helping in the formulation of conjectures. The solutions given in this section are polynomials and were given in [SV]. (Numerous other exact solutions are available. See [L, Hi). There is some interest in these when the domain Q is easy to describe, and the solutions are elementary transcendental functions. A favorite class of domains in this connection are curvilinear polygons: a curvilinear polygon is defined to be a domain whose boundary is made up entirely of circular arcs and of straight line segments. Explicit formulae for the torsion functions of various curvilinear polygons are available: sectors are treated in [PS], lenses and lunes in [K M 1 , M K1].)

Considering quadratic and cubic polynomials for u one can solve Problem (P) in an ellipse and in an equilateral triangle. One finds the following.

The letter n will be used to denote outward normal directions at the boundary

V3a420

9

(p, R) (6, a) if b < a l v/3’ \/39Urr4 p

{x 2 + y2 < p2}

In both cases zm, the centroid, the circum-centre and the in-centre coincide. In both cases Zf is at the intersection of the incircle and the boundary. (See Section6 for references to the recent examples that this is not a general property.)

THE ELASTIC TORSION PROBLEM: SOLUTIONS IN CONVEX DOMAINS 47

Define

uR(a, b) =

UR,A{a,b) =

1

^ J ) ’A 2ab

(3.1)

(3.2)47r a2 + b2

These definitions are arranged so that for any ellipse,

Um = U r ( R , p ) = U r , a { R , P )•

The reason for these definitions is given in Section 8.For the equilateral triangle, when we substitute for p and R in terms of a we find

2a2 - 2UR 15 > 5tt

a2y/ 3 a*= while um =

The simple generalisation of the equilateral triangle solution to solve a corre­sponding problem with homogeneous Robin boundary conditions is given in [KM2, MK2],

(The equilateral triangle is the extreme case of a family of domains, f^ , bounded by x = 0 and a family of hyperbolae. The torsion function for flc is given in Milne- Thomson [M Tl].)

4. Sperb ’s P* FunctionsPartly to give checks against known identities when p = 0, we consider solutions

of—A u = /i > 0. ( D M )

The left-hand equality in the following is the definition, the other equalities being easily checked:

1 0 uxPk = |Vu|2 + kpu — — 0 1 Uy

'U'X Uy —kpu

l D 2uB M uyux uy kpu

(4.1)

We have

Pk =Am1 - ̂ if k / 1, k / 0,

—w2A(log(u)) if k = 1

When Q is a disk, Pi is constant. When Slis a strip, P2 is constant. Integrating equation (4.1) we have,

(1 + k)S = [ Pk, (4.2)Jfi

from which various bounds for S have been found, after finding bounds on Pk-

We have

AP0 = A(|Vu|2) = 2 (u2xx + 2 u2xy + u2yy),

APi — (UXx /y) “t- 4Uxy

48 GRANT KEADY and ALEX McNABB

Thus, as both Po and Pi are subharmonic, the maximum of Po and of Pi is achieved on the boundary. (Sperb [Sp], p. 74, generalises this, for appropriate Pfc, to higher dimensions.)

The quantity P2 = —4w3/ 2 A (y/u) , satisfies,

VP2 div— = 0,

\vu\z

so that the maximum of P2 is necessarily attained either on the boundary of the domain fi or at a point where |Vw| = 0.

For solutions of the torsion problem,

§ = - 2 | V ^ k,

everywhere on the boundary of f I. See [Sp, p. 76].Using the Hopf form of the maximum principle it was shown, for plane convex Cl,

by Payne [Pa2], that the maximum of P2 cannot occur on the boundary of Cl. The proof is effected with the results of the two preceding paragraphs, and establishes the following.

Theorem 4.1. For the torsion function u of a convex domain Cl,

P2 = |Vit|2 + 2/iu < 2fj,um in Cl. (4.3)

(For generalisations, see Sperb [Sp].)

5. Bounds on umIn this subsection p, = 1 and Cl is convex. Sperb [Sp] integrated inequality (4.3)

to give,Um < = ^distance(zm, dCl)2 < i p2, (5.1)

where p is the in-radius of Cl.We remark that u r (R, p) < p2/ 2. Both of these inequalities are sharp for long

thin domains.Let k denote the curvature of the boundary of Q, and let Kmax denote the maxi­

mum, and « min denote the minimum values of k. Payne and Philippin [PP] prove, for problem (P),

< Um < • (5 -2 )^m ax min

See also [Kosl]. The inequalities become identities for a disk. (For a lens, though, they do not tell us any more than monotonicity under domain inclusion.)

6. First Derivative BoundsIn this subsection fi = 1. The function P0 = |Vu|2 is subharmonic so achieves

its maximum on the boundary of Cl. Sperb [Sp], p. 85, gives numerous inequalities for pm. Since, in the plane case, P\ takes its maximum on the boundary of Cl

(6.1)

THE ELASTIC TORSION PROBLEM: SOLUTIONS IN CONVEX DOMAINS 49

Using equation (4.2)

(6 -2)

For these, equality holds for a disk.Inequality (4.3) gives, for convex Cl,

P m < 2 um < p2, (6 .3 )

which becomes an equality when Cl is a strip. Various improvements, in which equality is attained when Cl is a disk, are available if one allows further functionals of Cl to occur in them: see end of Section 8.

The quantities Pk are not the only significant functions involving no higher than first derivatives. With ub denoting the torsion function of any disk, the functions

S/u -'S/ub — 2u and log(|V(u — ub)\2), (6.4)

are both harmonic. Inequalities involving the former are given in [Pa3].We now investigate Zf and other critical points of Pk on the boundary of Cl.

The proof of the Lemma below depends on using the Hopf form of the Maximum Principle on the subharmonic function Pi, and is given in [Sp], p. 86.

Lemma 6.1. Let k ( z ) denote the curvature of the boundary of Cl at z e dCl. Suppose that the boundary of Cl is C 3 in a neighbourhood of Z f . Then

K(Zf)pm < (6.5)

Equality occurs only if Cl is a disc.

Rem ark. The inequalityk (z )\\7u (z )\ < i ,

holds, not just at Zf, but at any point 2 6 dCl where

^ w - o , £ ( , ) > *

the same conclusions hold with 2 replacing Z f .

Attempts to answer Question 1 of Section 1, that is, to describe the location of fail points Zf have a long history. The history, back to St Venant [SV], is described in [Ka2]. Developments and extensions are given in [Swl, Sw2] and [R l, R2]: Numerical computation is a convenient and easy way of obtaining data which can be used in formulating conjectures. (The NAG routine D03EAF is a particularly appropriate tool for locating fail points and for investigating other aspects of the behaviour of Vw on the boundary.) There are many open questions, some of which we now list. Define

<Pi(z) = \z - Zi\2 and tpm(z) = \z - zm \2. (6 .6)

At this stage it may be worth looking for results in special classes of domains and we suggest that convex domains which have a single axis of symmetry may be a suitable class to investigate. (Amongst the motivations for treating this case are possible applications in problems like those discussed in [KK, K eal].)

50 GRANT KEADY and ALEX McNABB

Question SV1. For which convex and singly-symmetric domains Cl do the fail points occur at a point of intersection of an in-circle, a largest inscribed circle, with the boundary of the domain? Equivalently, when is

mm <pi{z) = ip&f)? (6.7)zEoil

Certainly this is not true in general. The first counterexample we found was for a slender isoceles triangle. (The initial computational evidence came from using NAG routine D03EAF. The semi-infinite strip is analytically easier because a Fourier series expansion for the torsion function is available.)

Though Theorem 1 of [Kos2] is wrong (see [Kea3, Sw2]), it suggests to us the following variant of the preceding question.

Question Kosl. For which convex and singly-symmetric domains Cl do the fail points satisfy

min (pm{z) = <Pm(zf )7 (6.8)zEoil

When the convex domain has two axes of symmetry the two preceding questions coincide. The case of two axes of symmetry is treated in [Ka2, Swl, Sw2, R l, R2]. While, undoubtedly, even here there is more to discover, the case of a single axis of symmetry is also important, and there far less is known.

For lens domains it is shown in [KM1] that:(i) |Vii|2 increases as one moves along each of the circular arcs, moving away

from the axis of symmetry;(ii) Zf is on the axis of symmetry on the arc of lesser curvature;

(iii) equations (6.7) and (6.8) are satisfied.These results have parallels in the results in [Ka2] where convex domains with two perpendicular axes of symmetry are studied. See Theorem 6.3 below. This leads to the problem of how to generalise these two sets of results, for example the following.

Problem S. What further geometric properties will ensure that, for convex and singly-symmetric domains ft, (ii) the fail points will occur on the axis of symmetry, and (iii) that equations (6.7) and (6.8) are satisfied?

Some of Kawohl’s results do not depend on symmetry at all. Recall that arclength s around dCl increases in the counterclockwise direction. Let Zq 6 dCl and 11+(zo) be the half-plane bounded by a normal line through zq and containing points 2 € dCl with s(z) — s(zo) > 0 and arbitrarily small. If D is a set, denote by D r the set obtained by reflecting it in d n + (z0)- If

(ft \n+(20))r c ci nn+(z0),

we say that zq has the positive reflection property. If the inclusion is in the reverse sense, we would say that Zq has the negative reflection property. Maximum Prin­ciple reflection techniques along the lines of those discussed in Section 12 establish the following. See [Ka2, Fri].

THE ELASTIC TORSION PROBLEM: SOLUTIONS IN CONVEX DOMAINS 51

Theorem 6.2. If dfl is C 1 near zq G dfl and zq has the positive reflection property, then

d u , . „ , d / du . . x

s :w < 0 andWe now return to the case with symmetry.

Theorem 6.3. (Kawohl). Suppose that the convex set Q is symmetric about both the x-axis and the y-axis. Let T — dfl fl {x > 0,y > 0} be C 1. Suppose also that the curvature k increases along T as x increases. Then

(i) every point of T satisfies the positive reflection property, so |Vw|2 decreases as one moves along it, moving away from the axis of symmetry;

(ii) Zf is on the axis of symmetry;(iii) equations (6.7) and (6.8) are satisfied.

Kawohl in a private communication pointed out that, for any lens which is sym­metric about {x = 0}, the right-hand part of the upper bounding circular arc has the positive reflection property. Thus, it may be that a reasonable approach to Problem (S) will aim to establish items corresponding to (i) in Theorem 6.3. Various items related to the geometry of this are given in [Kea3].

There are many possibly simpler questions in the area.

Question K os2. Let tt be convex and symmetric about {x = 0}. Let Y -(x) denote the lower part of the boundary of fl. Suppose

distance (zm, (0, r _ ( 0))) — min pm(z).zEdfl

Does |Vw| have a local maximum over dfl at (0, Y1(0)) ?

In [Kos2] it is claimed in the alleged ‘proof’ of his wrong ‘Theorem 1’ that the answer is ‘yes’ . This ‘yes’ may well be proved at some stage in the future. See [Kea3] for an analysis of the errors in [Kos2].

For convex Q, Zf and fii are related as follows.

Theorem 6.4. Let u be any solution of the equation —Au — 1 in some region containing a domain D. Suppose that un < 0 on the boundary of D, where n denotes the outward normal to the boundary of D. Let Zf(D) denote the location of the maximum of |Vu|2 over the closure of D. Suppose that the boundary of D is C 3 near Zf(D). Then

Zf(D) G fix « = » 0 > Unn(zf(D)) > -1 . (6.9)

Consider solutions of Problem (P ). Suppose that the boundary of fl i? C3 near Zf. If the curvature of the boundary of fl at Zf is nonnegative (that is, locally at Zf the boundary is convex), then Zf G f2i.

P roof. The implication to the right in (6.9) is trivial as Zf G Oi requires both that Unn < 0 and that uss < 0 and, using the differential equation, the result follows.

52 GRANT KEADY and ALEX McNABB

We now turn to the implication to the left. The function P0 is subharmonic so that, using the definition of Z f ( D ),

9Pq dP2(ununs + ususs) = ~ 5̂ ~dn ^ ̂ at Z f ( D ) .

The latter means that ununn + usuns > 0 at Z f ( D ) , and the hypothesis on the sign of un gives u^unn < —unusuns at Z f ( D ) . Since

we have that

from which

Next

uss -t- unn — 1,

Uns = — (1+Wnn) at Z f ( D ), (6.10)Un

'̂ ‘nV'nn ^ (1 "I" Wnn) at Z f ( D ) . (6.11)

H — Unn'Uss ^ns)

= - {u2n + u2s)unn - u2s)(l + unn) at Z f ( D ) .Un

Inequality (6.11) and the hypotheses on unn{zf(D)>j in (6.9) then establish that all three terms in the last expression for H (zf(D )) are nonnegative and hence prove the implication to the left.

The last sentence is easier to establish (with or without the preceding more general argument). The Dirichlet boundary conditions establish that us — 0. Also uns = 0 at Zf so that

H — “I- unn) at Zf.

We remark that using us = 0 in inequality (6.11) gives unn(zf) < 0, so that it is not necessary to hypothesise it.

The curvature at a point in the boundary of Q, the level curve u(x,y) = 0, is (« = T/(|Vw|3) and hence),

x u2n(zf )uss(zf )'

Using the hypothesis of the second paragraph that k > 0, we have that usa(zf) < 0 from which unn(zf) > — 1 from which the result follows from the first part of the theorem.

Remarks, (i) Theorem 6.4 extends so that, for any z € dfl at which

dP0 dP0a 7 (z) = °- " ^ ( 2 ) > 0 ’

the same conclusions hold with 2 replacing Z f .

THE ELASTIC TORSION PROBLEM: SOLUTIONS IN CONVEX DOMAINS 53

(ii) For nonconvex domains there are examples with Zf £ fii: for example the lune with the centre of one disk on the circumference of the other, the torsion function for which is given in [MK1].

A widely held belief is that, for convex Cl, Zf should occur at some point where the boundary curvature is relatively small. The item below is related to this and is one of several sent to us by Kosmodem’yanskii. It accords with special results found before, in connection with segments, special cases of lenses with one straight boundary: see [KM1].

Theorem 6.5. Let u be any solution of the torsion problem (P ) in a convex domain Cl with C 3 boundary. Suppose that ze is a local extremum of |Vu| on the boundary of and suppose, further, that the curvature of the boundary there, K,(ze) is zero. Then the extremum at ze is a maximum.

Proof. In [ML] it is shown that the function Q = T + 2uH is superharmonic. (See Section 10 for the proof.) On the boundary of Cl, Q = —ku^ > 0. Hence the point ze is a point of absolute minimum of Q. From this, the outward normal derivative Qn satisfies Qn (ze) < 0. Choose axes as in [Kosl] with y in the inward normal direction. Then using ux(ze) = 0 = uxy(ze),

Q y {ze) = 'U’yU xxy

From this we have, with, as usual, s denoting arclength along dCl,

d2unQn(Ze) ~ Qy{,ze) == '^n(ze) ~ ^rc(^e)

Hence un(ze) > 0 and un has a minimum at ze, so |Vu| has a maximum.

7. Domain Comparison

Theorem 7.1. Suppose that Cl C Cl. Let u and u denote their respective torsion functions. Then u < u in Cl.

Furthermore|Vw| < |Vw| on dCindCl.

The proof is a routine application of Maximum Principle arguments. It follows that the functionals um and S depend monotonically on the domain, that is

Cl C Cl = > um < um and S < S.

In general, the functional pm and the concavity set Cl\ do not depend mono­tonically on the domain. The easiest counterexamples are non-convex domains. Theorem 7.1 does, however, provide some information on pm. Let Rf be the radius of the smallest disk containing Cl and tangent to dCl at Z f . Then

54 GRANT KEADY and ALEX McNABB

For lens domains this inequality is the same as that in Lemma 6.1. Let p / be the radius of the largest disk contained in Cl and tangent to dCl at Z f . Then

Furthermore, if one considers families of lenses in which the arc of lesser curvature is fixed, and the radius of the other arc varies, then some monotonicity of pm can be infered.

8. Some Isoperimetric Inequalities and ApproximationsIn this section p = 1. We have compared, for lens domains in [KM1], the

calculated values of and similar functionals against several inequalities and ap­proximations which we now describe. Some of these are ‘isoperimetric inequalities’ , that is inequalities which become exact when SI is a disk. The solution in a disk is given in Section 2 and various functionals can be obtained from the ellipse versions in Section 3.' The functionals um and S both increase under symmetrisation: see [PS]. In

general, the functional pm and the concavity set f2i have neither this property, nor that of depending monotonically on the domain, with the easiest counterexamples being non-convex domains.

The domain monotonicity result establishes that um > p2/ 4. (inequality (5.1), urn < p2/ 2, holds for convex domains.)

One improvement on the preceding is as follows: amongst all domains with a given area A, the disk maximises the quantity um,

See Polya and Szego [PS], Payne [Pal], Bandle [Ba], Hersch [He]. These references give numerous other isoperimetric inequalities on other functionals, some of which we now describe.

The St Venant inequality

1— P f < p m .

(8.1)

(8.2)

was conjectured in 1856 but proved (by Polya) only in 1948. We also have

27T J 47TL Y < A (8.3)

The next three paragraphs discuss some approximations, which appear to be useful for convex f2, but which are not bounds. The style of approximation is suggested by Polya and Szego [PS], p. 113.

THE ELASTIC TORSION PROBLEM: SOLUTIONS IN CONVEX DOMAINS 55

The functional um might be estimated as follows. As noted in Section 3, for problem (P) in an ellipse, um is equal to u r (R, p) where u r (R, p) is defined by equation (3.1). .

We have, for various domains, compared the values of um with those of ellipses with the same inradius p and circumradius R, as in formula (3.1), or similar compar­isons associated with formula (3.2). For the equilateral triangle, for any symmetric lens and for the semicircle um < ur. However for a convex domain formed as the union of a long thin rectangle with two small circular-arc caps at its ends, um > u r . Thus ur is neither an upper bound nor a lower bound for um for arbitrary convex domains. See [KM1] for numerical results in lens domains, where it is shown that

~ ur can have either sign. For an ellipse, um is also equal to u r^(R , p) defined in equation (3.2).

The expression at the right of equation (3.2) is perhaps best considered as a function of the area A and the ratio p/R. The tables in [KM1], associated with the lens domains indicate that ur:a is likely to be a satisfactory approximation to Um for most applications.

Other “ellipse approximation” formulae, analogous to (3.1) and (3.2), can be found for pm and for other like quantities. One other item, related to the centre- on-circumference lenses, which we include in the tables of [KM1], is the “ellipse approximation” formula estimating uxx(zm)/uyy(zm), namely (p/R)2. This quan­tity is related to the eccentricity of the near-elliptical level curves near zm.

(The accuracy of the “ellipse approximations” is a favourable portent for similar approximations for the more general problem treated in [KM2].)

We now consider inequalities involving pm-

Sperb [Sp, p. 85] gives numerous inequalities for pm. We have, from inequality (6.1) um < Pm and inequality (6.2) (2S/A) < p^. For these, equality holds for a disk.

Payne and Wheeler [PaWh] established that

and again equality holds for a disk. Inequality (6.2) follows from (8.4) and the St Venant inequality.

For convex Q we have

(8.4)

4 K (8.5)

where Kf is the curvature at the principal fail point. See Lemma 6.1. Equality holds for a disk.

Fu and Wheeler [FW] established that, for convex f2,

56 GRANT KEADY and ALEX McNABB

which is exact both when Cl is a strip and when it is a disk. Various improvements to (4.3) are available if one allows further functionals of Cl in them. The inequality

where T is defined in Section 2, is proved in [PP]: equality is attained for any ellipse. [We] gives

in which equality is attained for a disk.For certain lens shaped Cl, the numerical values of several of these expressions,

including, in particular, the left-hand expression in inequalities (8.3) and the right- hand expression in inequality (8.6), are given in the tables in [KM1].

Further inequalities are given in [Pa3], [We] and the other references cited in this subsection.

9. The hessian H and the concavity set Cl\

(We remark that many of our results in this section and in the next do not depend on the boundary conditions applied at the boundary of Cl, but apply to any positive function defined on Cl and satisfying the torsion equation there.)

Theorem 9.1. For any solution u of the torsion equation (-D(^z)), H <2

At any point where H is not equal to

The second sentence follows from the following.(i) If v is harmonic, log(— det D2v) is harmonic.

(ii) With v = u — u b , (a) v is harmonic, and (b) det D2v = H — /i2/4.

Theorem 9.2. Let u solve the torsion equation (Z?(/i)) in a bounded domain D.

K'minPm Pm n'U'mPm “I” 2Um ^ 0,

/ Plog(—— H ) is harmonic.

Proof. The first sentence is proved by the identity

4H -- (UjXX 'Uyy') (^M Wy y ) 4ux,

If H = p2/A on any open subset E of D, then H — //2/4 throughout D. Hence either (i) {z | H(z) = p2/A} has no interior, or (ii) u — \{c — x2 — y2) for some constant c.

P roof. This follows from the fact that H is real-analytic in D. Our next concern is with the set Cli which, for solutions of the torsion problem, is closed in Cl and has nonempty interior.

THE ELASTIC TORSION PROBLEM: SOLUTIONS IN CONVEX DOMAINS 57

Theorem 9.3. For solutions of the torsion equation (.D(/i)) in a simply connected plane domain ft, any component (maximal open connected subset) of the set fti is simply connected.

Proof. Let<b{z) = log (l -

At all points </> is either harmonic or negative infinite. Thus if </> is nonpositive on the boundary of a domain it is nonpositive throughout the domain.

Consider a component C of the set ft i. On its boundary 0 is nonpositive Assume that there exists a curve T not homotopic to a point in C. T doesn’t intersect the boundary of ft. Let (<9C)outer be the part of the boundary of C in the annular region outside T but inside or on the boundary of ft. Let Cbigger be the simply connected region inside (dC)outer- Since (f) is nonpositive on (<9C%uter it is positive everywhere in Cbigger> including all of T. T is homotopic to a point in Cbigger > but, by the definition of component Cbigger Q C. Thus T is empty: there isn’t a curve T with the property asserted, and the corollary is proved.

Remark. A generalization of this result to iV-dimensions is given in [Kea3]: the proof techniques are different.

Another application of the Maximum Principle establishes that either H = /i2/4 at a point in the interior of ft, or H attains its maximum on the boundary. For the solutions of problem (P) in the equilateral triangle H = /i2/4 at the centroid. Calculations in [KM1] indicate that in the lens domains it appears that H attains its maximum on the axis of symmetry on the arc of the greater curvature.

The same proof techniques as used in Theorem 9.3 establish the following:

Theorem 9.4. Define

Cll = {z | (H{u))(z) < 0 } .

Let D be an open connected subset of ft in which H < /i2/4. If dD C ftf, then D C ftf.

Here is a remark showing that some care is needed in Theorem 9.4. Consider the torsion function for the equilateral triangle. Then consider D to be the domain inside a curve through the midpoints of the sides and just outside the incircle. Observe that the hypothesis excluding H = [i2 /4 is necessary in the Theorem.

Theorem 9.5. Consider the torsion problem on a bounded domain ft. Any com­ponent D of fti over the closure of which H < n2/4 touches dft.

Proof. Assume not. Then dD is contained in the open set ft and H = 0 on dD. Since log(/i2 — 4H) is harmonic in D, H = 0 throughout D. This cannot happen (unless ft is an infinite strip which is not allowed by our assumption that ft is bounded). Thus the initial assumption yields a contradiction and Theorem 9.5 is proved.

58 GRANT KEADY and ALEX McNABB

Theorem 9.6. Consider the torsion problem on a bounded domain fl.

(i) If dfl contains a segment T of a straight line, then H < 0 on T.

(ii) In any polygonal fl, H < 0 everywhere on dfl away from comers.

(iii) If H < 0 everywhere on dfl, then H = n2/ 4 at some point in fl.

Proof, (i) On straight line boundaries, H — — w2s, where n and s are defined in Section 2.

(ii) Follows from (i).

(iii) Assume H < /x2/4 everywhere in fl.

Then log(p2—4if) is harmonic throughout fl, and with the hypothesis that H < 0 on dfl, we have H < 0 in fl. (The case H — 0 everywhere in fl is excluded by the hypothesis that fl is bounded.) However, at zm, H > 0, and this contradiction ensures that the assumption at the beginning of the proof of (iii) is false. Hence(iii) is proved.

Another use for the calculation of H is that the value of H at zm determines the eccentricity of the nearly elliptical level curves of u near zm. The eigenvalues A+ , A_ of the second-derivative matrix D2u satisfy

A+ + A_ = —[l and A+A_ — H <4 ’

so that

. - ^ ± y V - 4 H

The closer that H is to /x2/4 the more nearly circular are the contour lines of u near zm. The ratio A+/A_ determines the ratio of the lengths of the major and minor axes of the ellipses. In [KM1] for certain families of lens shapes, this ratio is calculated and tabulated.

10. Other Com binations o f Second Derivatives

The formulae in this section apply to any solution of (D(/j,)). We shall be interested only in certain special combinations of the second derivatives. Recall the definitions of H , T and Q(k) from Section 2. There are various formulae for Q(k) = T + kuH including:

UXx VixyUXy llyyMX Uy

ux D 2u uxuy — ILyku My ku

THE ELASTIC TORSION PROBLEM: SOLUTIONS IN CONVEX DOMAINS 59

The quantities H, T and T + 2uH — Q(2) — Q are related to P2 by

r - i v u - v f 2

h + ] a f4

I^T - H\Vu\2 -^ \ V P 2\

(:T + 2uH) + it i2d iv (tr ‘ VP2

p(T + 2 u H ) + 1- P 2A(\ogP2

Makar-Limanov’s [ML] proof of Theorem 1.1, which we will give in Section 11, uses the following identities, or variants thereof:

Theorem 10.1. (Makar-Limanov). For solutions of the torsion equation (D(/i)) the following identities hold:

0 , (10.1)

0, (10.2)

0, (10.3)

0, (10.4)

0. (10.5)

4|Vtf|~ A H = ii2 - a h = Uxxy + Uxyy^ 1̂0'6^

div(VT + 2HVu) + 2Vu -V H = 0,- A (T + 2uH) + 2uAH = 0. (10.7)

-A {T + 2uH ) - S^ T+q 2uH^ = 0 ,

wherePi0 < © = - 8 ( T + 2uH).u

The left-hand equality in equation (10.6) is merely a restatement of the result of the preceding section that log (/la 2 — AH) is harmonic. Equations (10.6) and (10.7) state that H and, when u > 0, (T + 2uH) respectively are superharmonic.

11. Further Results on Second Derivatives in Convex Q,Many of the results of Sections 9 and 10 apply to the torsion problem in general,

not necessarily convex, domains. We now return to the case of the domain being convex.

The set for the equilateral triangle is the domain inside the incircle. For a regular polygon with n sides, we expect to be connected but to have n components. We do not yet know, for general convex f2, if f2i must be connected, though we expect that this will be the case.

(We expect that Qi need not be connected for nonconvex domains, domains such as the union of two equal disks overlapping on a small set.)

In [KM1] we give numerical evidence, presented graphically, that, for the torsion function of the semicircle, the incircle of the semicircle is (as in the case of the equilateral triangle) again contained in f2i.

60 GRANT KEADY and ALEX McNABB

In general, for convex Cl, Cl\ need not contain the incircle, the largest inscribed circle of Cl. There is a connection with the question concerning when Zf is the minimum of pi discussed in Section 6.

For any domain with a flat boundary at which uns / 0, n the normal, s the arc length, at the point of intersection of the inscribed circle and dCl has H < 0 at that point. Since zm € Cl\ it is natural to ask if other sets to which zm is known to belong are related to fii. One such set is the set M(Cl), defined in Section 12 for which zm € M(Cl).

For some domains, such as ellipses, u is concave over the whole domain, that is, Cl\ — Cl. For further work, characterising domains for which the torsion function is concave over the whole domain, see [Kosl]. He proves the following:

Theorem 11.1. (Kosmodem’yanskii 1). The torsion function u for a convex do­main fi is concave if every parabola osculating with the boundary of Cl contains the domain Cl in its interior.

A key ingredient of the proof is that the minimum of H is attained at Cl. As­suming that the minimum is bounded, e.g. nonnegative, leads to restrictions on the domain Cl. For many domains, though, we have H tending to minus infinity: this occurs, for example, at the corners in the lens shapes.

Considerations of fii do not exhaust the possibilities for studying the concavity properties of the solutions.

Proof of Theorem 1.1. A calculation shows that T + 2uH, has the same sign as, det (D2y/u), and that when T + 2uH > 0 the function y/u is concave.

Theorem 10.1 shows that T + 2 uH is super harmonic. The proof for the torsion problem that (T + 2uH) > 0 follows using the two further ingredients (1) that, with zero Dirichlet boundary conditions, T > 0 on the boundary of the convex Cl, and (2) the maximum principle for superharmonic functions.

Interestingly, Makar Limanov’s Theorem 1.1 concerning convex fI, is best pos­sible in the sense that, for an equilateral triangle, y/u is concave while ua is not concave for any exponent a > 1/2. For positive, quasiconcave, superharmonic functions, the local concavity exponent, a(u,z), is defined in Kennington [Ken] as

a(u, z) = max{a € II | (D2ua)(z ) is positive semidefinite}.

For the torsion function u of a convex Cl, a(u, z) has the following properties:(i) a(u, z) is always greater than or equal to a half;

(ii) a(u, z) < 1 for points z on any straight line segment in the boundary;(iiia) a(u,z) < 1 in a sufficiently small neighbourhood of a corner point zc on

the boundary; and(iiib) a(u, z) — 1/2 if the included angle at the corner is a right-angle or less.

There are many open questions concerning the behaviour of a(u, z) as z varies over Cl. For an equilateral triangle, the level curves of a(u, •) are particularly simple to calculate. A discussion of a(u, •) for lenses is given in [KM1].

THE ELASTIC TORSION PROBLEM: SOLUTIONS IN CONVEX DOMAINS 61

12. Results from reflection argumentsThe approach in this section was initiated by Gidas, Ni and Nirenberg [GNN].

There is a forthcoming introductory book: see [Fra]. There are considerable sim­plifications possible when Cl is convex: see [BaS, KK]. Indeed we have already had occasion to mention the methods in Section 6 of this survey.

Variational approaches, involving symmetrisation, show that, for a solution of problem (P) in a symmetrised domain Cl, zm lies on the axis of symmetry. The reflection methods give an alternative proof of this which also yield results on zm when Cl is not necessarily symmetric.

The results on zm have a different character than those, due to Payne [Pa2] and Sperb [Sp], which we described in Sections 4 to the first part of 6.

Theorem 12.1. Let H be a half-plane. Let f2r(II) denote the reflection of Cl in on. if nr( n ) n n c n then 2m6 0nn.

By considering various II it is possible to define a set called M(Cl) in [KK] (and called C(Cl) in [BaS]) which will contain any critical points of the solutions of the Dirichlet problem.

Theorem 12.2. Let u solve Problem (P ). There is a closed convex set M{Cl) C Cl with the following properties.

(i) Let II = {z | v ■ z > d} be any half-plane with M(Cl) in its complement. Then for z 6 Cl fl II, v • Vu(z) < 0.

(ii) zm G M(Cl).

In Keady and Kloeden [KK] it is shown that, for convex 1), the centroid of Cl lies in M(Cl).

We expect that it would also be true that, for any convex domain Cl with a unique in-centre, its in-centre also belongs to M(Cl). In our numerical computations we have found so far that zm is closer to the in-centre than it is to the centroid. For the lens solutions we expect that M(Cl) C Cl\.

Theorem 12.3. Let u solve Problem (P ). Suppose that Cl is convex and sym­metric about x = 0. Then ux < 0 in x > 0, and M(Cl) is a subset of the axis of symmetry.

We are confident that Maximum Principles, possibly in connection with other techniques, will continue to yield results for the torsion problem for many years to come.

Acknowledgements. This paper was commenced during a visit by Grant Keady to Massey University in July-August 1988: see [MK1]. Except for some matters of exposition, it was completed while Grant Keady was working at the University of Waikato. Grant Keady acknowledges the support of Massey University’s Visiting Research Associates Fund for the former and the New Zealand U.G.C. for the latter period.

Added in Proof. Further results are given inA.A. Kosmodem’yanskii, Doklady Akad. Nauk 282(1993), 139-140.

62 GRANT KEADY and ALEX McNABB

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Grant Keady Alex McNabbUniversity of Western Australia Massey UniversityNedlands, 6009 Private BagAUSTRALIA Palmerston North

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