- Email: [email protected]
On Hopf type maximum principles for convex domains
Nonlinear
Analysis,
Theory. Methods & Applications.
ON HOPF
TYPE
Vol. 1 No. (. pp 517-559.
MAXIMUM
Pergamoo
Press, 1977.
PRINCIPLES L. E.
Cornell
University,
Cornell
University
FOR
CONVEX
DOMAINS
PAYNE*
Ithaca,
R~ti
Printed in Great Britain.
NY 14853, U.S.A.
SPERB~ and University
of Delaware
and IVAR STAKGOLD$ University
of Delaware
(Received 25 October 1976) Key words
Hopf, maximum
principle,
convex domains
1. INTRODUCTION
IN PREVIOUSpapers Payne and Stakgold [ 1,2] obtained bounds for the gradient Vu of a solution
u(x) of the equation Au + f(u) = 0, where A is the Laplace operator and the solution vanishes on the boundary aD of a bounded domain D in R”. Their method consisted of showing, by use of the Hopf strong maximum principle, that a certain functional of u and Vu must takes its maximum at a critical point of u (that is, at a point where Vu = 0). Because of the simple nature of the dependence of the functional on Vu it was then immediately possible to deduce a pointwise bound for Vu in terms of u and uM, the maximum of u. Various applications were presented, in particular to the membrane equation Au + lu = 0. We now extend and sharpen some of these results for domains having nonnegative average curvature at every point of the boundary. Since this class of domains is only slightly more general than the class of convex domains we shall confine ourselves to convex domains for the sake of simplicity. We shall use the same method as in [l] and [2] with a different functional leading to improved bounds. In fact, for convex domains, the results in [2] are the simplest but least sharp special cases of those of the present paper. In Section 2, we derive the new maximum principle and the corresponding gradient bounds, whose consequences are discussed in Sections 3 and 4. In Section 3 we deal with direct applications of the gradient bounds, whereas in Section 4 we combine these bounds with a level surface analysis. We also extend the Payne-Rayner inequalities [3] which do not require any gradient bounds. In Section 5 we obtain concrete results for special choices of f(u). In particular, we sharpen some classical inequalities pertaining to the torsion problem Au + 2 = 0 and to the membrane problem. Throughout this paper we are concerned with solutions of the problem Au +f(u)= *Research TResearch IResearch
supported supported supported
0,x~D;u = O,x~i3D,
by the National Science Foundation under grant NSF GPp33031X. by Schweizerischer Nationalfonds and AR0 grant 29-76-6-0102 by AR0 grant 29-7664102. 547
(1.1)
548
L. E. PAYNE, R. SPERBAND
1. STAKGOL~
where D is a bounded convex domain of R” whose boundary i?D is a C2+& surface. We shall deal with positiw solutions u(x) for which ,f(u(x)) 3 0 in D; with slight and fairly obvious modifications some of our results remain valid for any solution u for which uf(u) > 0 in D. Sufficient conditions on f to guarantee the existence of a unique positive solution can be found in [2] and elsewhere, but our arguments apply even to cases such as f = ve” when (1.1) may have multiple positive solutions. In any event we are not concerned here with questions of existence and uniqueness. Let us now introduce some notation that will be used throughout the article. Since D is convex, the average curvature K is nonnegative at each point of aD. We shall denote by K, the nonnegative lower bound of K; often we shall assume K, > 0 and, in fact, our most powerful results are for that case. In addition we shall make frequent use of the symbol z defined as t = Max lVu[
(1.2)
XESD
and of the function F(U) =
2. MAXIMUM
We first generalize
” f(v) dv. s 0
PRINCIPLE
as follows the maximum
AND
principle
GRADIENT
(1.3)
BOUNDS
derived in [2] :
THEOREM 1. Let u(x) be any positive solution of (1.1) and let the average curvature of the boundary have a positive lower bound K,. For 0 < a < 2 and c 3 [a2/2(m - l)K,] the function
I4 f(u) PI2 @ = (u + c)” + 2 s 0 (u + c)” d” assumes
its maximum
(2.1)
value at a critical point of u.
In order to prove this theorem we first show that CDsatisfies an elliptic second order differential inequality except possibly at the critical points of u. We then make use of Hopfs second principle [4] to show that @ cannot assume its maximum value on CID. Proof: Let x = (x,, x2,. . ., x,,J be a point in R” and denote the derivative by a subscript i. Using the summation convention we get successively
with respect
@‘,i= (U + C)-"[2U,iU,i,j - X(U + C)-'IVUl*Uyi + 2f(U)Uyi]
to xi
(2.2)
and A@ = 2(u + c)-%,~~u,~~ - 4cr(u + c)-‘-~u,~u,~~ + cr(a + 1) (u + c)-=-~IVUI~ - a(u + c)-O-l f(u)lVu12 - 2(u + c)-“f From (2.2) and Schwarz’s
inequality
2(u).
(2.3)
we obtain
2
2(u + c)-=u,~~Q
> ~?!!5_ + ;(u 2pzp
+ c)-“-~IVUI~
- CL(U+ c)-“-‘[Vul’f
+ 2(u + c)-*f ‘(u)
(2.4)
On Hopf type maximum principles for convex domains
549
where ~ = (U +
C)“~,i
+
[2ar(U+ C)-‘lVUl’ - 4f(U)]U,~
A substitution of the expression 4a(u + ~)-(~+i) u,~u,~u,~~ from (2.2) together with (2.4) yields 1 + 2a(u+ c)-~u,~@,~ A@ - 21vu12 2 JVu14a 1 - t (u + c)-(“+2) + 4(Vul’af(u)(u + c)-6+1). (2.5) ( ) It follows then that for 0 < a < 2 the right hand side is nonnegative and we conclude that CD must take its maximum value either on c?Dor at a point where Vu = 0, that is, at a critical point of u. To complete the proof we must show that the maximum of CDoccurs at a critical point of u. The result is trivially true if 0 is identically constant so suppose this is not the case. Then if CD took its maximum at a point on the boundary, Hopfs second maximum principle would tell us that X)/an is positive at that point, but the explicit calculation we are about to perform shows instead that &D/&r < 0 on the boundary. From (2.1), we find
(2.6) The smoothness of the boundary permits us to write the differential equation on the boundary as
$ +(m -
l)K g = -f(O),
(2.7)
where the tangential terms in A vanish since u is constant on the boundary. Substituting in (2.6) the expression for a2u/an2 from (2.7) we find c” g
= (Vu/’ [F /Vu/ - 2(m - 1)K].
(2.8)
By choosing c 2 [az/2(m- l)K,] we have (iT@/iTn) < 0 and hence @ cannot take its maximum on aD and must therefore assume its maximum at a critical point of u. Remark.If K, = 0, (2.8) will yield the desired result only if a = 0,in which case c disappears from (2.1) and we reproduce the principle of Payne and Stakgold [2] for convex D :If u(x) is a positive
solution of (Ll), then Y = IVul’ + 2 ’ f(u)du s0
(2.9)
assumes its maximum at a critical point of u. Of course the principle applies also for K, > 0 since in that case the value a = 0 is admissible in (2.1) but we shall see that one can then do better by choosing a different from 0. We can now easily obtain gradient bounds. Let uM be the maximum value of u. Since f(u) is nonnegative on 0 < u G uy. it is clear that the point where u = uM is the critical point at which
550
L. E. PAYNE. R. SPERB AND I. STAKGOLD
CDand Y are largest. We conclude (I) If K, > 0,
0 < N < 2,
c 2
/VU12 < where we have introduced
that : C(f 2(m - l)K,’
then
2(u+ c)":"&du s
= 2(u+ c)"(ff~ - H),
(2.10)
the notation (2.11)
(II) If K, 2 0,
3
IVu12 < 2 UMf(v)dv U
= 2(F,
(2.12)
- F)
where F is defined by (1.3). We may regard (2.12) as a special case of (I) with t( = 0 but with validity extended to include K, = 0. An easy calculation shows that the right side of (2.10) takes on its smallest allowable value when c( = 2, [c = r/(m - l)K,]. With these optimal values (2.10) becomes IVu12 < 2(u + c)~ :“&dL., s
c = (m _rl)K,.
(2.13)
An apparent difficulty with (2.10) and (2.13) is that the bounds are not explicit, the quantity c being related to the unknown maximum of Vu on the boundary. To remedy this, we apply (2.10) on the boundary to obtain
or (m -
1)‘K;
~
2
J
UM 0 [v +
f(v) dv zK,'(m- 1))‘I2
which, in specific examples (see Section 5) can be used to find an explicit upper bound The corresponding form of (2.13) is Ivul2 < 2(u + E),’
J
uhf f(v)
----=-Zdv, U (v + c)
-
hf)
c = (m _ l)K,’
(2.14) r < ?(u~).
(2.15)
we shall which relates Vu anywhere in D to the local value of u and to Up. In concrete applications only use the bounds (2.12) and (2.15), but in the general development it is convenient to deal with (2.10) which includes (2.12) and (2.15) as special cases. 3. DIRECT
APPLICATIONS
OF
GRADIENT
BOUNDS
The gradient bounds of the previous section enable us to obtain a number of useful global inequalities for the solution of (1.1). Multiply (1.1) by p(x), an arbitrary, sufficiently smooth, positive function on the closure of D; integrate over D,apply Green’s theorem and use (2.10) on
On Hopf type maximum
principles
for convex domains
551
aD to find
j-D pf(Wx Different
+
s,
uApdx
choices of p will lead to inequalities
for Us For instance
(3.1)
p s 1 gives
2
1 H&f-
6,pds.
< (2c’H,)ii2
2c”S2 (S
f(u)dx
(3.2)
,
>
D
where S denotes the “surface area” of aD. Instead of multiplying (1.1) by p(x), let us now multiply by a nondecreasing Cl-function $(u) satisfying Ii/(O) = 0 and repeat the previous steps. We then obtain (3.3) which, for various choices of $, may lead to upper and lower bounds for Up as well as to other types of inequalities. This will be illustrated in Section 5. More precise and additional information can be found by integrating (2.10) along rays. In the one-dimensional case with c1 = 0, equality holds in (2.10) or equivalently in (2.12). That result follows by multiplying the equation u” + f(u) = 0 by u’ and noting that the left side is the derivative of uf2/2 + F(u). One might therefore expect that for a “thin” region the inequality (2.10) would be sharp for points on the ray joining x~, at which u assumes its maximum, to the nearest point on dD. Thus any inequality resulting from integration along such a ray should be sharper than those obtained by integrating (2.10) either over D or 2D. Let $(u) be an arbitrary C’-function defined on 0 < u < Us and satisfying G(u) 2 0,
1(/(Q) = 0,
II/‘(u) < 0.
(3.4)
Let r measure the distance along a ray from X~ (at which u = UJ and an arbitrary integration along the ray gives
J
/~--xMl
4%4x))=
$74
0
au ar
J
Ix-xhfl
dr <
point
x;
lx--xMl
- $‘(u)lVu/dr < ,/2
0
J0
Q(uW,
(3.5)
where we have set
Q(u) = [- @(u)] (u + c,““[H,
- H(u)]“~.
(3.6)
If we choose *(u) =
““(t. + c)+‘[H~ s”
- H(o)]-1’2du,
(3.7)
then Icl(u(x)) < J2)x Let us now take x on the boundary and set d(x,, point on the boundary. We then conclude that ti(O) G J2d(x,,
- $41.
aD) equal to the distance aD) < J2R
(3.8) from X~ to the nearest
(3.9)
L. E. PAYNE, R. SPERBAND
552
I.
STAKGOLD
where R is the radius of the largest sphere that can be inscribed in D. We shall apply (3.9) in Section 5, see (5.6). Another interesting choice for II/is I&) =
;
Au) (u + c)-“H- 1’2(~~) [HM - H(u)] - “‘du,
(3.10)
s which leads to $(O) = x; then (3.5) becomes n<
J2
Ix--x&XI f(u) (U + c)-a’2H-1’2(u)dr.
s0
(3.11)
Let us now restrict our attention to the case m = 2 and try to obtain bounds for integrals of u over the whole domain D. We shall use polar coordinates with origin at X~ and let r = z(6) be the equation of the boundary. Then with x on c?D,(3.5) becomes
s =(W
s s =(W
WV s 42 o Q(u)dr = -J2
au
=(e)~+~‘IQdr < 2 _._q ' s0
rQtardr
0
(3.12)
where we have assumed that $‘(O) = 0. Integrating (3.12) with respect to the polar angle we are led to
Ii/(O)s
;
D
IQ’(u)lQ(u) -1c/‘b4
dA
’
(3.13)
where dA is an element of area. We shall apply (3.13) in Section 5, see (5.18). 4. LEVEL
SURFACE
COORDINATES
In [2] and [5] introduction of the level surfaces of u led to interesting inequalities for solutions of (1.1). Many of these inequalities can how be made sharper by use of the improved gradient bound (2.10). At the same time we take the opportunity of generalizing some of the results of [2]. Let D(t) be the domain on which u > t, r(t) its boundary, and V(t) its volume. By definition, we have V(t) =
dx, s u(t)
and V(t) is clearly a decreasirig function satisfying dV --=
dt
(4.1)
We shall also make use of quantity
for which we find (4.2)
On Hopf type maximum
Integration
principles
553
for convex domains
of (1.1) over D(t) yields (4.3)
x(r) = By combining
(4.1) and (4.3) we obtain
the basic inequalities
-XV’
s‘Wdss$
=
uo
and
(4.4)
r(*)
(4.5) Our first set of results will not require (4.4) we find
any gradient --XV
where S(t) is the area of r(t). By appealing obtain
Schwarz’s
inequality
> P(t)
in (4.6)
isoperimetric
2 k,f(t)V2(“‘-I)‘“‘,
inequality,
we
(4.7)
related to the surface area of the unit sphere. For m = 2, we have the simpler
-XX’ 2 where V(t) is the area of the plane domain
4nf(t)W,
D(t). Integration
2 47c s
(4.8) of (4.8) from EL= 0 to u = uM yields
“M
IIM $2(O)
Using
to (4.2) and the classical
--xx’ 2 f(t)S2(t) where k, is a constant relation
bounds.
= -47~
V(t)f(t)dt
V’(t)F(t)dt s
0
= 47~
Wdx, sD
where F is defined from (1.3) or, equivalently [b Inequality
(4.9) is a generalization
f(u)d.y,2
2 8ij-
F(u)dx.
(4.9)
D
of the Payne-Rayner
inequality
for the linearized
membrane (4.10)
It is sometimes isoperimetric
convenient
to relate
jDudx
and
ID f (u)dx. For m = 2, we use the classical
inequality
in (4.9 and integrate from t = 0 to uM to find UM Uhf V(t)dt = -4x tV’dt = 4n X(u)dx 2 471 udx. sD sD s0 s 0
It is easily seen that
s
X(u)dx < A
D
f(u)dx sD
(4.11)
:
L. E. PAYNE, R. SPERB AND I. STAKGOLD
554
where A is the area of D. We thus obtain
the nonisoperimetric
s
D f(u)dx
> ;
result in R*
udx
(4.12)
sD
which gives the crude bound L > 4n/A for the membrane problem. A completely different set of inequalities can be deduced from (4.5) by using bound (2.10). We find first -(V/x)
> ;(L + c)-“[H,
which, on using (4.2) and multiplying
An integration
- H(t)]_’
by 2f(HM - H)x-’ $(H,
the gradient
- H)X-2]
becomes
2 0.
(4.13)
from 0 to t then gives
h
- ff(t)] [x(t)] - 2 3
[S 1 D f(u)dx
H,,.,
or
s
H(WH,]1’2
x(t) d [l -
- 2.
f(uWx.
(4.14)
D
For a > 0 let us multiply manipulations,
(4.14) by H’H”-’
H”f(u)dx sD
< uH,“~
and integrate
from t = 0 to uM to obtain.
jji'H'-'
bj(z4)d.x
after some
jrr,- H)“‘dt (4.15)
Special cases of (4.15) were given in [2] and [5]. We could of course obtain a more general inequality by multiplying (4.14) by an arbitrary nonnegative Cl-function $(t) and integrating. For instance if $(O) = 0 it is easily shown that
M (1 - H/H,)“2$‘(t)dt,
(4.16)
of which a special case is
s D
where
G(u)f(u)dx <
G(u) =
uM
s Df
s”,[l - FI1”dy.
(Wx
(4.17)
(4.18)
On Hopf type maximum 5.
principles
555
for convex domains
EXAMPLES
We now consider in some detail a number of examples. First we look at the problem of torsion of an elastic beam of convex cross section. The corresponding boundary value problem is Au + 2 = 0,
xinD;
u = OonaD,
(5.1)
where D is a convex domain in R2. In terms of the general formulation of Section f(u) = 2, m = 2. The optimal H is given by (2.11) with U.= 2, c = r/K,, that is 2u H(u) = ___ c(u + C)’ Hhf - H = 2(u, We obtain
an upper bound
“+“$“+
2, we have
c)’
for T from (2.14):
T+uM+(~)y1’2 _E!pf; which is an improvement
of an earlier result by Payne
(5.2)
[6] :
T d 2Ju, which can be derived either by applying (2.12) or by taking inequality (3.2) for tl = 2, c = s/K,, now states that
(5.3) the limit as K, + 0 in (5.2). The
A2 (5.4)
uM ’ L2 - [A2K,/~]'
where A is the area of D and L is the length of dD. Since (5.4) implies uM 2 A2/L2, (5.3) then gives T d 2A/L, which when substituted in (5.4) yields 2A2 (5.5)
uM ’ 2L2 - K,AL again an improvement over the corresponding Still in connection with the torsion problem.
inequality in [6]. we apply (3.9) which gives after a little calculation
R 2 Jn,[yarcsin(i)],
(5.6)
where y is given by Y=[;+(:+&y”]“‘.
It is worth noting that for the torsion the rigidity (Polya [7]): P < A2/271.
problem,
Next we turn to the fixed membrane
problem
Au+lu=O
(5.7) (4.9) gives a classical
xinD;u=OonSD,
isoperimetric
inequality
for
(5.8)
where again D is a convex domain in R2.We confine our attention to the first (positive) eigenfunction of (5.8) where, without loss of generality, we have taken uM = 1. We indicate first how we can compute an upper bound for t that is sharper than the one given in [ 11. Taking the optimal
L. E. PAYNE, R. SPERBAND I. STAKGOLD
556
values u = 2, c = r/K,,
we find from (2.14) with M = 2, f(u) = Au,
s 1
2a2 <
V
dv = ln(1 + x) - &,
0 [u + rK,‘-J2
(5.9)
where we have set a = K0/2JA
We want to use (5.9) to determine
and x = K,/r.
an upper bound
ln(1 + x) - -!? 1+x
(5.10)
for r. In view of the relation
1
we find 1
2a2 < -__
x2
21 +x
or x > 2a2 + [4a4 + 4a2]‘12 = 2a2 + 2a[l
+ a2]‘12
which leads to 7 =
2
d JA [(1 + a2)l/2 -
a sharper bound than r < JA which appears K,/2 JA is fairly small. Indeed, elementary (see [S]) show that K
a] = JJ. [(l+g2-$]>
in [ 11. It should, however, be noted that the quantity considerations involving the Minkowski constant
~g’l lim
<
0'
[ v
where V,,,(l) is the volume of the m-dimensional states that among all m-dimensional domains mental eigenvalue) then gives K,
where 1 is the fundamental then (x)“’ - 2.4 and
eigenvalue
< (A/X)1’2,
JA[(l
+
a2)l12 - a]<
(5.12)
of the m-dimensional
Ko
T <
’
1
unit ball. The Faber-Krahn inequality (which of equal volume the ball has the lowest funda-
a = 2Jjl
which means that (5.10) can be rewritten
(5.11)
unit ball. In the special case m = 2,
< 0.21,
(5.13)
as Jl.p--a(l--t)]<
From (5.14) we easily derive an upper bound E(u)
Ji[l_cO.89)$].
for the mean-to-peak
s
‘A
udx.
D
(5.14)
ratio (5.15)
On Hopf type maximum
Indeed
principles
551
for convex domains
from (5.8) or (3.2), we find E(u) = &
aD lVu[ds < $
< &
(5.16)
[1 - (0.89) $1.
s We could add here the following * = @:
inequality
E@2P+
1)
which is a direct consequence
<
dk
‘(’
+
‘)
of (3.3) with a = 0 and
(5.17)
,l7(u),
r(~ + 3/2)
2
which can be used in conjunction with (5.16) to obtain an explicit upper bound for E(uZp+ ‘). A lower bound for E(u) can be found from (3.13) and (3.6) by setting CI = 0 and $(u) = u + arccos u - 1. For the unit disk, this calculation gives
udx sD or 71-2 E(u) 2 ~ 2 which is admittedly rather crude. We now turn to some nonlinear
problems
(5.18)
(in two dimensions)
of the form (l.l), repeated
here :
in D; u = 0 on aD.
Au + f(u) = 0 The mean-to-peak
- 0.20,
ratios defined by udx
u x D E, = J--‘( ---,E,=L)d
s UMA
u,A
both have applications in reactor analysis. We shall confine ourselves to fairly rough estimates, but sharper results follow from using the area within a level line as the independent variable [ 1, 2, 51. From (4.12) we immediately deduce the relation
sf
(u)dx
E&E,= which, together
D 47TUM ’
with (3.2) gives (u)
E
2
<
.
PffhP2~ 4ln4,
A particularly interesting example occurs in combustion theory real positive parameter. All solutions of (1.1) are then necessarily to be a convex domain in E,, and we shall let u = 0 in our gradient H(u) = F(u) = v(e” -
l), H,
(5.19)
.
where f(u) = ve”, with v a positive. Again we take D bounds. We then have
= v(eUM- 1)
558
L. E. PAYNE, R. SPERB AND I. STAKGOLD
and (3.2) gives
[J 1 J 2
eyM > JY
2L2 whereas (4.15) yields (with a = l),
J
e2”dx < 5(2e”-
D
An application
D
eUdx
+ I
+ I)
e”dx.
(5.20)
(5.21)
D
of the Schwarz inequality
J
to the left side of (5.21) leads to
eUdx < : (2e”” + 1).
(5.22)
D
Finally
we note that another
a result already
observed
bound
by Bandle
stems from (4.9):
[8]. 6. CONCLUSIONS
The main purpose of this article was to give a range of applications of the new gradient bound (2.10) obtained from the maximum principle based on the functional (2.1) which is more powerful than that used in [2]. Further extensions are being made in a number ofdirections. In [9] Schaefer and Sperb consider a still more general class of functionals that, under favourable circumstances, lead to improved gradient bounds for (1.1); in [lo] Payne and Phillipin treat a more difficult nonlinear equation than (1.1); in [ll] Sperb and Stakgold deal with the membrane of varying density whose fundamental mode satisfies Au + lp(x)u = 0, in [12] Payne carries further the consequence of another maximum principle and also considers equations of the fourth order. REFERENCES 1, PAYNE L. E. & STAKGOLD I., On the mean value of the fundamental mode in the fixed membrane problem. J. appl. Anal. 3 295-306 (1973). 2. STACKGOLD I. & PAYNEL. E., Nonlinear problems in nuclear reactor analysis, Proc. Confi?rencr on Nonlinear Problems in Physical Sciences and Biology, pp. 298 -307. Springer Lecture Notes in Mathematics g322 (1972). 3. PAYNE L. E. & RAYNEK M. E.. An isoperimetric inequality for the first eigenfunctlon in the fixed membrane problem, ZAMP 23 13-15 (1972). 4. HOPF E., A remark on linear elliptic differential equations of the second order, Proc. Am. math. Sot. 3 791-793 (1952). 5. STAKGOLD I., Global estimates for nonlinear reaction and diffusion, Proc. Dundee Conf. on Ordinary and Partial Dij@rential Equarions pp. 252-266. Springer Lecture Notes in Mathematics #415 (1974). 6. PAYNE L. E., Bounds for the maximum stress in the Saint Venant torsion problem, Ind. J. Mech. Mafh., Special issue, 51-59 (1968). 7. P~LYA G., Torsional rigidity. principal frequency, electrostatic capacity and symmetrization, Q. appl. Math. 6 267-277, (1948 ). 8. BANDLE C., Existence theorems, some qualitative results and a priori bounds for a class of nonlinear Dirichlet problems, Arch. Rat. Mech. Anal. 58, 219-238 (1975). 9. SCIIAE~ERP. & S~EKB R., Maximum principles for some functionals associated with the solution of elliptic boundary value problems, Arch. Rat. mesh. Anal.. to appear. 10. PAYNE L. E. & PHILIPPIN G. A., Some applications of the maximum principle in the problem of torsional creep, to appear. 11. SPERB R. & STAKGOLD I., Estimates for membranes of varying density, to appear.
On Hopf type maximum
12. 13.
principles
559
for convex domains
PAYNE
L. E., Some remarks on maximum principles, J. Anal. to appear. G. & SZEG~ G., Isoperitnetric Inequalities in Mathematical Physics. Annals Princeton University Press (1951).
P~LYA
of Mathematics
Studies No. 27.
Analysis,
Theory. Methods & Applications.
ON HOPF
TYPE
Vol. 1 No. (. pp 517-559.
MAXIMUM
Pergamoo
Press, 1977.
PRINCIPLES L. E.
Cornell
University,
Cornell
University
FOR
CONVEX
DOMAINS
PAYNE*
Ithaca,
R~ti
Printed in Great Britain.
NY 14853, U.S.A.
SPERB~ and University
of Delaware
and IVAR STAKGOLD$ University
of Delaware
(Received 25 October 1976) Key words
Hopf, maximum
principle,
convex domains
1. INTRODUCTION
IN PREVIOUSpapers Payne and Stakgold [ 1,2] obtained bounds for the gradient Vu of a solution
u(x) of the equation Au + f(u) = 0, where A is the Laplace operator and the solution vanishes on the boundary aD of a bounded domain D in R”. Their method consisted of showing, by use of the Hopf strong maximum principle, that a certain functional of u and Vu must takes its maximum at a critical point of u (that is, at a point where Vu = 0). Because of the simple nature of the dependence of the functional on Vu it was then immediately possible to deduce a pointwise bound for Vu in terms of u and uM, the maximum of u. Various applications were presented, in particular to the membrane equation Au + lu = 0. We now extend and sharpen some of these results for domains having nonnegative average curvature at every point of the boundary. Since this class of domains is only slightly more general than the class of convex domains we shall confine ourselves to convex domains for the sake of simplicity. We shall use the same method as in [l] and [2] with a different functional leading to improved bounds. In fact, for convex domains, the results in [2] are the simplest but least sharp special cases of those of the present paper. In Section 2, we derive the new maximum principle and the corresponding gradient bounds, whose consequences are discussed in Sections 3 and 4. In Section 3 we deal with direct applications of the gradient bounds, whereas in Section 4 we combine these bounds with a level surface analysis. We also extend the Payne-Rayner inequalities [3] which do not require any gradient bounds. In Section 5 we obtain concrete results for special choices of f(u). In particular, we sharpen some classical inequalities pertaining to the torsion problem Au + 2 = 0 and to the membrane problem. Throughout this paper we are concerned with solutions of the problem Au +f(u)= *Research TResearch IResearch
supported supported supported
0,x~D;u = O,x~i3D,
by the National Science Foundation under grant NSF GPp33031X. by Schweizerischer Nationalfonds and AR0 grant 29-76-6-0102 by AR0 grant 29-7664102. 547
(1.1)
548
L. E. PAYNE, R. SPERBAND
1. STAKGOL~
where D is a bounded convex domain of R” whose boundary i?D is a C2+& surface. We shall deal with positiw solutions u(x) for which ,f(u(x)) 3 0 in D; with slight and fairly obvious modifications some of our results remain valid for any solution u for which uf(u) > 0 in D. Sufficient conditions on f to guarantee the existence of a unique positive solution can be found in [2] and elsewhere, but our arguments apply even to cases such as f = ve” when (1.1) may have multiple positive solutions. In any event we are not concerned here with questions of existence and uniqueness. Let us now introduce some notation that will be used throughout the article. Since D is convex, the average curvature K is nonnegative at each point of aD. We shall denote by K, the nonnegative lower bound of K; often we shall assume K, > 0 and, in fact, our most powerful results are for that case. In addition we shall make frequent use of the symbol z defined as t = Max lVu[
(1.2)
XESD
and of the function F(U) =
2. MAXIMUM
We first generalize
” f(v) dv. s 0
PRINCIPLE
as follows the maximum
AND
principle
GRADIENT
(1.3)
BOUNDS
derived in [2] :
THEOREM 1. Let u(x) be any positive solution of (1.1) and let the average curvature of the boundary have a positive lower bound K,. For 0 < a < 2 and c 3 [a2/2(m - l)K,] the function
I4 f(u) PI2 @ = (u + c)” + 2 s 0 (u + c)” d” assumes
its maximum
(2.1)
value at a critical point of u.
In order to prove this theorem we first show that CDsatisfies an elliptic second order differential inequality except possibly at the critical points of u. We then make use of Hopfs second principle [4] to show that @ cannot assume its maximum value on CID. Proof: Let x = (x,, x2,. . ., x,,J be a point in R” and denote the derivative by a subscript i. Using the summation convention we get successively
with respect
@‘,i= (U + C)-"[2U,iU,i,j - X(U + C)-'IVUl*Uyi + 2f(U)Uyi]
to xi
(2.2)
and A@ = 2(u + c)-%,~~u,~~ - 4cr(u + c)-‘-~u,~u,~~ + cr(a + 1) (u + c)-=-~IVUI~ - a(u + c)-O-l f(u)lVu12 - 2(u + c)-“f From (2.2) and Schwarz’s
inequality
2(u).
(2.3)
we obtain
2
2(u + c)-=u,~~Q
> ~?!!5_ + ;(u 2pzp
+ c)-“-~IVUI~
- CL(U+ c)-“-‘[Vul’f
+ 2(u + c)-*f ‘(u)
(2.4)
On Hopf type maximum principles for convex domains
549
where ~ = (U +
C)“~,i
+
[2ar(U+ C)-‘lVUl’ - 4f(U)]U,~
A substitution of the expression 4a(u + ~)-(~+i) u,~u,~u,~~ from (2.2) together with (2.4) yields 1 + 2a(u+ c)-~u,~@,~ A@ - 21vu12 2 JVu14a 1 - t (u + c)-(“+2) + 4(Vul’af(u)(u + c)-6+1). (2.5) ( ) It follows then that for 0 < a < 2 the right hand side is nonnegative and we conclude that CD must take its maximum value either on c?Dor at a point where Vu = 0, that is, at a critical point of u. To complete the proof we must show that the maximum of CDoccurs at a critical point of u. The result is trivially true if 0 is identically constant so suppose this is not the case. Then if CD took its maximum at a point on the boundary, Hopfs second maximum principle would tell us that X)/an is positive at that point, but the explicit calculation we are about to perform shows instead that &D/&r < 0 on the boundary. From (2.1), we find
(2.6) The smoothness of the boundary permits us to write the differential equation on the boundary as
$ +(m -
l)K g = -f(O),
(2.7)
where the tangential terms in A vanish since u is constant on the boundary. Substituting in (2.6) the expression for a2u/an2 from (2.7) we find c” g
= (Vu/’ [F /Vu/ - 2(m - 1)K].
(2.8)
By choosing c 2 [az/2(m- l)K,] we have (iT@/iTn) < 0 and hence @ cannot take its maximum on aD and must therefore assume its maximum at a critical point of u. Remark.If K, = 0, (2.8) will yield the desired result only if a = 0,in which case c disappears from (2.1) and we reproduce the principle of Payne and Stakgold [2] for convex D :If u(x) is a positive
solution of (Ll), then Y = IVul’ + 2 ’ f(u)du s0
(2.9)
assumes its maximum at a critical point of u. Of course the principle applies also for K, > 0 since in that case the value a = 0 is admissible in (2.1) but we shall see that one can then do better by choosing a different from 0. We can now easily obtain gradient bounds. Let uM be the maximum value of u. Since f(u) is nonnegative on 0 < u G uy. it is clear that the point where u = uM is the critical point at which
550
L. E. PAYNE. R. SPERB AND I. STAKGOLD
CDand Y are largest. We conclude (I) If K, > 0,
0 < N < 2,
c 2
/VU12 < where we have introduced
that : C(f 2(m - l)K,’
then
2(u+ c)":"&du s
= 2(u+ c)"(ff~ - H),
(2.10)
the notation (2.11)
(II) If K, 2 0,
3
IVu12 < 2 UMf(v)dv U
= 2(F,
(2.12)
- F)
where F is defined by (1.3). We may regard (2.12) as a special case of (I) with t( = 0 but with validity extended to include K, = 0. An easy calculation shows that the right side of (2.10) takes on its smallest allowable value when c( = 2, [c = r/(m - l)K,]. With these optimal values (2.10) becomes IVu12 < 2(u + c)~ :“&dL., s
c = (m _rl)K,.
(2.13)
An apparent difficulty with (2.10) and (2.13) is that the bounds are not explicit, the quantity c being related to the unknown maximum of Vu on the boundary. To remedy this, we apply (2.10) on the boundary to obtain
or (m -
1)‘K;
~
2
J
UM 0 [v +
f(v) dv zK,'(m- 1))‘I2
which, in specific examples (see Section 5) can be used to find an explicit upper bound The corresponding form of (2.13) is Ivul2 < 2(u + E),’
J
uhf f(v)
----=-Zdv, U (v + c)
-
hf)
c = (m _ l)K,’
(2.14) r < ?(u~).
(2.15)
we shall which relates Vu anywhere in D to the local value of u and to Up. In concrete applications only use the bounds (2.12) and (2.15), but in the general development it is convenient to deal with (2.10) which includes (2.12) and (2.15) as special cases. 3. DIRECT
APPLICATIONS
OF
GRADIENT
BOUNDS
The gradient bounds of the previous section enable us to obtain a number of useful global inequalities for the solution of (1.1). Multiply (1.1) by p(x), an arbitrary, sufficiently smooth, positive function on the closure of D; integrate over D,apply Green’s theorem and use (2.10) on
On Hopf type maximum
principles
for convex domains
551
aD to find
j-D pf(Wx Different
+
s,
uApdx
choices of p will lead to inequalities
for Us For instance
(3.1)
p s 1 gives
2
1 H&f-
6,pds.
< (2c’H,)ii2
2c”S2 (S
f(u)dx
(3.2)
,
>
D
where S denotes the “surface area” of aD. Instead of multiplying (1.1) by p(x), let us now multiply by a nondecreasing Cl-function $(u) satisfying Ii/(O) = 0 and repeat the previous steps. We then obtain (3.3) which, for various choices of $, may lead to upper and lower bounds for Up as well as to other types of inequalities. This will be illustrated in Section 5. More precise and additional information can be found by integrating (2.10) along rays. In the one-dimensional case with c1 = 0, equality holds in (2.10) or equivalently in (2.12). That result follows by multiplying the equation u” + f(u) = 0 by u’ and noting that the left side is the derivative of uf2/2 + F(u). One might therefore expect that for a “thin” region the inequality (2.10) would be sharp for points on the ray joining x~, at which u assumes its maximum, to the nearest point on dD. Thus any inequality resulting from integration along such a ray should be sharper than those obtained by integrating (2.10) either over D or 2D. Let $(u) be an arbitrary C’-function defined on 0 < u < Us and satisfying G(u) 2 0,
1(/(Q) = 0,
II/‘(u) < 0.
(3.4)
Let r measure the distance along a ray from X~ (at which u = UJ and an arbitrary integration along the ray gives
J
/~--xMl
4%4x))=
$74
0
au ar
J
Ix-xhfl
dr <
point
x;
lx--xMl
- $‘(u)lVu/dr < ,/2
0
J0
Q(uW,
(3.5)
where we have set
Q(u) = [- @(u)] (u + c,““[H,
- H(u)]“~.
(3.6)
If we choose *(u) =
““(t. + c)+‘[H~ s”
- H(o)]-1’2du,
(3.7)
then Icl(u(x)) < J2)x Let us now take x on the boundary and set d(x,, point on the boundary. We then conclude that ti(O) G J2d(x,,
- $41.
aD) equal to the distance aD) < J2R
(3.8) from X~ to the nearest
(3.9)
L. E. PAYNE, R. SPERBAND
552
I.
STAKGOLD
where R is the radius of the largest sphere that can be inscribed in D. We shall apply (3.9) in Section 5, see (5.6). Another interesting choice for II/is I&) =
;
Au) (u + c)-“H- 1’2(~~) [HM - H(u)] - “‘du,
(3.10)
s which leads to $(O) = x; then (3.5) becomes n<
J2
Ix--x&XI f(u) (U + c)-a’2H-1’2(u)dr.
s0
(3.11)
Let us now restrict our attention to the case m = 2 and try to obtain bounds for integrals of u over the whole domain D. We shall use polar coordinates with origin at X~ and let r = z(6) be the equation of the boundary. Then with x on c?D,(3.5) becomes
s =(W
s s =(W
WV s 42 o Q(u)dr = -J2
au
=(e)~+~‘IQdr < 2 _._q ' s0
rQtardr
0
(3.12)
where we have assumed that $‘(O) = 0. Integrating (3.12) with respect to the polar angle we are led to
Ii/(O)s
;
D
IQ’(u)lQ(u) -1c/‘b4
dA
’
(3.13)
where dA is an element of area. We shall apply (3.13) in Section 5, see (5.18). 4. LEVEL
SURFACE
COORDINATES
In [2] and [5] introduction of the level surfaces of u led to interesting inequalities for solutions of (1.1). Many of these inequalities can how be made sharper by use of the improved gradient bound (2.10). At the same time we take the opportunity of generalizing some of the results of [2]. Let D(t) be the domain on which u > t, r(t) its boundary, and V(t) its volume. By definition, we have V(t) =
dx, s u(t)
and V(t) is clearly a decreasirig function satisfying dV --=
dt
(4.1)
We shall also make use of quantity
for which we find (4.2)
On Hopf type maximum
Integration
principles
553
for convex domains
of (1.1) over D(t) yields (4.3)
x(r) = By combining
(4.1) and (4.3) we obtain
the basic inequalities
-XV’
s‘Wdss$
=
uo
and
(4.4)
r(*)
(4.5) Our first set of results will not require (4.4) we find
any gradient --XV
where S(t) is the area of r(t). By appealing obtain
Schwarz’s
inequality
> P(t)
in (4.6)
isoperimetric
2 k,f(t)V2(“‘-I)‘“‘,
inequality,
we
(4.7)
related to the surface area of the unit sphere. For m = 2, we have the simpler
-XX’ 2 where V(t) is the area of the plane domain
4nf(t)W,
D(t). Integration
2 47c s
(4.8) of (4.8) from EL= 0 to u = uM yields
“M
IIM $2(O)
Using
to (4.2) and the classical
--xx’ 2 f(t)S2(t) where k, is a constant relation
bounds.
= -47~
V(t)f(t)dt
V’(t)F(t)dt s
0
= 47~
Wdx, sD
where F is defined from (1.3) or, equivalently [b Inequality
(4.9) is a generalization
f(u)d.y,2
2 8ij-
F(u)dx.
(4.9)
D
of the Payne-Rayner
inequality
for the linearized
membrane (4.10)
It is sometimes isoperimetric
convenient
to relate
jDudx
and
ID f (u)dx. For m = 2, we use the classical
inequality
in (4.9 and integrate from t = 0 to uM to find UM Uhf V(t)dt = -4x tV’dt = 4n X(u)dx 2 471 udx. sD sD s0 s 0
It is easily seen that
s
X(u)dx < A
D
f(u)dx sD
(4.11)
:
L. E. PAYNE, R. SPERB AND I. STAKGOLD
554
where A is the area of D. We thus obtain
the nonisoperimetric
s
D f(u)dx
> ;
result in R*
udx
(4.12)
sD
which gives the crude bound L > 4n/A for the membrane problem. A completely different set of inequalities can be deduced from (4.5) by using bound (2.10). We find first -(V/x)
> ;(L + c)-“[H,
which, on using (4.2) and multiplying
An integration
- H(t)]_’
by 2f(HM - H)x-’ $(H,
the gradient
- H)X-2]
becomes
2 0.
(4.13)
from 0 to t then gives
h
- ff(t)] [x(t)] - 2 3
[S 1 D f(u)dx
H,,.,
or
s
H(WH,]1’2
x(t) d [l -
- 2.
f(uWx.
(4.14)
D
For a > 0 let us multiply manipulations,
(4.14) by H’H”-’
H”f(u)dx sD
< uH,“~
and integrate
from t = 0 to uM to obtain.
jji'H'-'
bj(z4)d.x
after some
jrr,- H)“‘dt (4.15)
Special cases of (4.15) were given in [2] and [5]. We could of course obtain a more general inequality by multiplying (4.14) by an arbitrary nonnegative Cl-function $(t) and integrating. For instance if $(O) = 0 it is easily shown that
M (1 - H/H,)“2$‘(t)dt,
(4.16)
of which a special case is
s D
where
G(u)f(u)dx <
G(u) =
uM
s Df
s”,[l - FI1”dy.
(Wx
(4.17)
(4.18)
On Hopf type maximum 5.
principles
555
for convex domains
EXAMPLES
We now consider in some detail a number of examples. First we look at the problem of torsion of an elastic beam of convex cross section. The corresponding boundary value problem is Au + 2 = 0,
xinD;
u = OonaD,
(5.1)
where D is a convex domain in R2. In terms of the general formulation of Section f(u) = 2, m = 2. The optimal H is given by (2.11) with U.= 2, c = r/K,, that is 2u H(u) = ___ c(u + C)’ Hhf - H = 2(u, We obtain
an upper bound
“+“$“+
2, we have
c)’
for T from (2.14):
T+uM+(~)y1’2 _E!pf; which is an improvement
of an earlier result by Payne
(5.2)
[6] :
T d 2Ju, which can be derived either by applying (2.12) or by taking inequality (3.2) for tl = 2, c = s/K,, now states that
(5.3) the limit as K, + 0 in (5.2). The
A2 (5.4)
uM ’ L2 - [A2K,/~]'
where A is the area of D and L is the length of dD. Since (5.4) implies uM 2 A2/L2, (5.3) then gives T d 2A/L, which when substituted in (5.4) yields 2A2 (5.5)
uM ’ 2L2 - K,AL again an improvement over the corresponding Still in connection with the torsion problem.
inequality in [6]. we apply (3.9) which gives after a little calculation
R 2 Jn,[yarcsin(i)],
(5.6)
where y is given by Y=[;+(:+&y”]“‘.
It is worth noting that for the torsion the rigidity (Polya [7]): P < A2/271.
problem,
Next we turn to the fixed membrane
problem
Au+lu=O
(5.7) (4.9) gives a classical
xinD;u=OonSD,
isoperimetric
inequality
for
(5.8)
where again D is a convex domain in R2.We confine our attention to the first (positive) eigenfunction of (5.8) where, without loss of generality, we have taken uM = 1. We indicate first how we can compute an upper bound for t that is sharper than the one given in [ 11. Taking the optimal
L. E. PAYNE, R. SPERBAND I. STAKGOLD
556
values u = 2, c = r/K,,
we find from (2.14) with M = 2, f(u) = Au,
s 1
2a2 <
V
dv = ln(1 + x) - &,
0 [u + rK,‘-J2
(5.9)
where we have set a = K0/2JA
We want to use (5.9) to determine
and x = K,/r.
an upper bound
ln(1 + x) - -!? 1+x
(5.10)
for r. In view of the relation
1
we find 1
2a2 < -__
x2
21 +x
or x > 2a2 + [4a4 + 4a2]‘12 = 2a2 + 2a[l
+ a2]‘12
which leads to 7 =
2
d JA [(1 + a2)l/2 -
a sharper bound than r < JA which appears K,/2 JA is fairly small. Indeed, elementary (see [S]) show that K
a] = JJ. [(l+g2-$]>
in [ 11. It should, however, be noted that the quantity considerations involving the Minkowski constant
~g’l lim
<
0'
[ v
where V,,,(l) is the volume of the m-dimensional states that among all m-dimensional domains mental eigenvalue) then gives K,
where 1 is the fundamental then (x)“’ - 2.4 and
eigenvalue
< (A/X)1’2,
JA[(l
+
a2)l12 - a]<
(5.12)
of the m-dimensional
Ko
T <
’
1
unit ball. The Faber-Krahn inequality (which of equal volume the ball has the lowest funda-
a = 2Jjl
which means that (5.10) can be rewritten
(5.11)
unit ball. In the special case m = 2,
< 0.21,
(5.13)
as Jl.p--a(l--t)]<
From (5.14) we easily derive an upper bound E(u)
Ji[l_cO.89)$].
for the mean-to-peak
s
‘A
udx.
D
(5.14)
ratio (5.15)
On Hopf type maximum
Indeed
principles
551
for convex domains
from (5.8) or (3.2), we find E(u) = &
aD lVu[ds < $
< &
(5.16)
[1 - (0.89) $1.
s We could add here the following * = @:
inequality
E@2P+
1)
which is a direct consequence
<
dk
‘(’
+
‘)
of (3.3) with a = 0 and
(5.17)
,l7(u),
r(~ + 3/2)
2
which can be used in conjunction with (5.16) to obtain an explicit upper bound for E(uZp+ ‘). A lower bound for E(u) can be found from (3.13) and (3.6) by setting CI = 0 and $(u) = u + arccos u - 1. For the unit disk, this calculation gives
udx sD or 71-2 E(u) 2 ~ 2 which is admittedly rather crude. We now turn to some nonlinear
problems
(5.18)
(in two dimensions)
of the form (l.l), repeated
here :
in D; u = 0 on aD.
Au + f(u) = 0 The mean-to-peak
- 0.20,
ratios defined by udx
u x D E, = J--‘( ---,E,=L)d
s UMA
u,A
both have applications in reactor analysis. We shall confine ourselves to fairly rough estimates, but sharper results follow from using the area within a level line as the independent variable [ 1, 2, 51. From (4.12) we immediately deduce the relation
sf
(u)dx
E&E,= which, together
D 47TUM ’
with (3.2) gives (u)
E
2
<
.
PffhP2~ 4ln4,
A particularly interesting example occurs in combustion theory real positive parameter. All solutions of (1.1) are then necessarily to be a convex domain in E,, and we shall let u = 0 in our gradient H(u) = F(u) = v(e” -
l), H,
(5.19)
.
where f(u) = ve”, with v a positive. Again we take D bounds. We then have
= v(eUM- 1)
558
L. E. PAYNE, R. SPERB AND I. STAKGOLD
and (3.2) gives
[J 1 J 2
eyM > JY
2L2 whereas (4.15) yields (with a = l),
J
e2”dx < 5(2e”-
D
An application
D
eUdx
+ I
+ I)
e”dx.
(5.20)
(5.21)
D
of the Schwarz inequality
J
to the left side of (5.21) leads to
eUdx < : (2e”” + 1).
(5.22)
D
Finally
we note that another
a result already
observed
bound
by Bandle
stems from (4.9):
[8]. 6. CONCLUSIONS
The main purpose of this article was to give a range of applications of the new gradient bound (2.10) obtained from the maximum principle based on the functional (2.1) which is more powerful than that used in [2]. Further extensions are being made in a number ofdirections. In [9] Schaefer and Sperb consider a still more general class of functionals that, under favourable circumstances, lead to improved gradient bounds for (1.1); in [lo] Payne and Phillipin treat a more difficult nonlinear equation than (1.1); in [ll] Sperb and Stakgold deal with the membrane of varying density whose fundamental mode satisfies Au + lp(x)u = 0, in [12] Payne carries further the consequence of another maximum principle and also considers equations of the fourth order. REFERENCES 1, PAYNE L. E. & STAKGOLD I., On the mean value of the fundamental mode in the fixed membrane problem. J. appl. Anal. 3 295-306 (1973). 2. STACKGOLD I. & PAYNEL. E., Nonlinear problems in nuclear reactor analysis, Proc. Confi?rencr on Nonlinear Problems in Physical Sciences and Biology, pp. 298 -307. Springer Lecture Notes in Mathematics g322 (1972). 3. PAYNE L. E. & RAYNEK M. E.. An isoperimetric inequality for the first eigenfunctlon in the fixed membrane problem, ZAMP 23 13-15 (1972). 4. HOPF E., A remark on linear elliptic differential equations of the second order, Proc. Am. math. Sot. 3 791-793 (1952). 5. STAKGOLD I., Global estimates for nonlinear reaction and diffusion, Proc. Dundee Conf. on Ordinary and Partial Dij@rential Equarions pp. 252-266. Springer Lecture Notes in Mathematics #415 (1974). 6. PAYNE L. E., Bounds for the maximum stress in the Saint Venant torsion problem, Ind. J. Mech. Mafh., Special issue, 51-59 (1968). 7. P~LYA G., Torsional rigidity. principal frequency, electrostatic capacity and symmetrization, Q. appl. Math. 6 267-277, (1948 ). 8. BANDLE C., Existence theorems, some qualitative results and a priori bounds for a class of nonlinear Dirichlet problems, Arch. Rat. Mech. Anal. 58, 219-238 (1975). 9. SCIIAE~ERP. & S~EKB R., Maximum principles for some functionals associated with the solution of elliptic boundary value problems, Arch. Rat. mesh. Anal.. to appear. 10. PAYNE L. E. & PHILIPPIN G. A., Some applications of the maximum principle in the problem of torsional creep, to appear. 11. SPERB R. & STAKGOLD I., Estimates for membranes of varying density, to appear.
On Hopf type maximum
12. 13.
principles
559
for convex domains
PAYNE
L. E., Some remarks on maximum principles, J. Anal. to appear. G. & SZEG~ G., Isoperitnetric Inequalities in Mathematical Physics. Annals Princeton University Press (1951).
P~LYA
of Mathematics
Studies No. 27.