On the relations between

principal eigenvalue and torsional

rigidity

Giuseppe ButtazzoDipartimento di Matematica

Università di Pisabuttazzo@dm.unipi.it

http://cvgmt.sns.it

Chair of Applied Analysis at FAU Erlangen-NürnbergErlangen (Zoom webinar), July 1, 2020

Research jointly made with

• Michiel van den BergUniversity of Bristol, UK

• Aldo PratelliUniversità di Pisa, Italy

• (numerical experiments by Ginevra Biondi,Università di Pisa, Master thesis).

1

Our goal is to present some relations be-tween two important quantities that arisein the study of elliptic equations. We al-ways consider the Laplace operator �� withDirichlet boundary conditions; other ellipticoperator can be considered, while consider-ing other boundary conditions (Neumann orRobin) adds to the problem severe extra dif-ficulties, essentially due to the fact that inthe Dirichlet case functions in H10(⌦) canbe easily extended to Rd while this is not ingeneral true in the other cases.

2

To better understand the two quantities wedeal with, let us make the following two mea-surements.

• Take in ⌦ an uniform heat source (f = 1),fix an initial temperature u0(x), wait a longtime, and measure the average temperaturein ⌦.

• Consider in ⌦ no heat source (f = 0), fixan initial temperature u0(x), and measurethe decay rate to zero of the temperature in⌦.

3

The first quantity is usually called torsionalrigidity and is defined as

T (⌦) =Z

⌦u dx

where u is the solution of

��u = 1 in ⌦, u 2 H10(⌦).

In the thermal di↵usion model T (⌦)/|⌦| isthe average temperature of a conducting medium⌦ with uniformly distributed heat sources(f = 1).

4

The second quantity is the first eigenvalueof the Dirichlet Laplacian

�(⌦) = min

(

R

⌦ |ru|2 dxR

⌦ u2 dx

: u 2 H10(⌦) \ {0})

In the thermal di↵usion model, by the Fourieranalysis,

u(t, x) =X

k�1e��kthu0, ukiuk(x),

so �(⌦) represents the decay rate in time ofthe temperature when an initial temperatureis given and no heat sources are present.

5

If we want, under the measure constraint|⌦| = m, the highest average temperature,or the slowest decay rate, the optimal ⌦ isthe same and is the ball of measure m. Also,it seems consistent to expect a slow (resp.fast) heat decay related to a high (resp. low)temperature. We then want to study if

�(⌦) ⇠ T�1(⌦),or more generally

�(⌦) ⇠ T�q(⌦),where by A(⌦) ⇠ B(⌦) we mean

0 < c1 A(⌦)/B(⌦) c2 < +1 for all ⌦.

6

We further aim to study the so-called Blaschke-Santaló diagram for the quantities �(⌦) andT (⌦). This consists in identifying the setE ⇢ R2

E =n

(x, y) : x = T (⌦), y = �(⌦)o

where ⌦ runs among the admissible sets. Inthis way, minimizing a quantity like

F⇣

T (⌦), �(⌦)⌘

is reduced to the optimization problem in R2

minn

F (x, y) : (x, y) 2 Eo

.

7

The di�culty consists in the fact that char-acterizing the set E is hard. Here we onlygive some bounds by studying the inf andsup of

�↵(⌦)T�(⌦)

when |⌦| = m.

Since the two quantities scale as:

T (t⌦) = td+2T (⌦), �(t⌦) = t�2�(⌦)

it is not restrictive to reduce ourselves tothe case |⌦| = 1, which simplifies a lot thepresentation.

8

For the relations between T (⌦) and �(⌦):

• Kohler-Jobin ZAMP 1978

• van den Berg, B., Velichkov in Birkhäuser2015

• van den Berg, Ferone, Nitsch, TrombettiIntegral Equations Operator Theory 2016

• Lucardesi, Zucco preprint

9

The Blaschke-Santaló diagram has been stud-ied for other pairs of quantities:

• for �1(⌦) and �2(⌦) by D. Bucur, G.B., I.Figueiredo (SIAM J. Math. Anal. 1999);

• for �1(⌦) and Per(⌦) by M. Dambrine, I.Ftouhi, A. Henrot, J. Lamboley (on HAL);

• for T (⌦) and cap(⌦) by M. van den Berg,G.B. (on arxiv and cvgmt);

• for T (⌦) and Per(⌦) by L. Briani, G.B., F.Prinari (on arxiv and cvgmt).

10

For the inf/sup of

�↵(⌦)T�(⌦)

the case � = 0 is well-known and reduces tothe Faber-Krahn result (B ball with |B| = 1)

minn

�(⌦) : |⌦| = 1o

= �(B),

while

supn

�(⌦) : |⌦| = 1o

= +1

(take many small balls or a long thin rectan-gle).

11

Similarly, the case ↵ = 0 is also well-knownthrough a symmetrization argument (Saint-Venant inequality):

maxn

T (⌦) : |⌦| = 1o

= T (B),

while

infn

T (⌦) : |⌦| = 1o

= 0

(take many small balls or a long thin rectan-gle).

The case when ↵ and � have a di↵erent signis also easy, since T (⌦) is increasing for theset inclusion, while �(⌦) is decreasing.

12

So we can reduce the study to the case

�(⌦)Tq(⌦)

with q > 0. If we want to remove the con-straint |⌦| = 1 the corresponding scaling freeshape functional is

Fq(⌦) =�(⌦)Tq(⌦)

|⌦|(dq+2q�2)/d

that we consider on various classes of admis-sible domains.

13

We start by considering the class of all do-mains (with |⌦| = 1). The known cases are:

• q = 2/(d + 2) in which the minimum of�(⌦)Tq(⌦) is reached when ⌦ is a ball(Kohler-Jobin ZAMP 1978);

• q = 1 in which (Pólya inequality)

0 < �(⌦)T (⌦) < 1.

14

When 0 < q 2/(d +2):8

<

:

min�(⌦)Tq(⌦) = �(B)Tq(B)

sup�(⌦)Tq(⌦) = +1.

For the minimum

�(⌦)Tq(⌦) = �(⌦)T (⌦)2/(d+2)T (⌦)q�2/(d+2)

� �(B)T (B)2/(d+2)T (B)q�2/(d+2)

= �(B)Tq(B),

by Kohler-Jobin and Saint-Venant inequali-ties.For the sup take ⌦ = N disjoint small balls.

15

When 2/(d +2) < q < 1:8

<

:

inf �(⌦)Tq(⌦) = 0

sup�(⌦)Tq(⌦) = +1.

For the sup take again ⌦ = N disjoint balls.

For the inf take as ⌦ the union of a fixedball BR and of N disjoint balls of radius ".We have

�(⌦)Tq(⌦) = R�2�(B1)Tq(B1)

✓

Rd+2+N"d+2◆q

and choosing first " ! 0 and then R ! 0 wehave that �(⌦)Tq(⌦) vanishes.

16

When q = 1:

inf �(⌦)T (⌦) = 0, sup�(⌦)T (⌦) = 1.

For the inf take as ⌦ the union of a fixedball BR and of N disjoint balls of radius ", asabove.

The sup equality, taking ⌦ a finely perfo-rated domain, was shown by van den Berg,Ferone, Nitsch, Trombetti [Integral Equa-tions Opera- tor Theory 2016]. A shorterproof can be given using the theory of ca-pacitary measures.

17

The finely perforated domains:" = distance between holes r" =radius of a hole

r" ⇠ "d/(d�2) if d > 2, r" ⇠ e�1/"2if d = 2.

18

When q > 1:

inf �(⌦)Tq(⌦) = 0, sup�(⌦)Tq(⌦) < +1.For the inf take as ⌦ the union of a fixedball BR and of N disjoint balls of radius ", asabove.

For the sup (using Pólya and Saint-Venant):

�(⌦)Tq(⌦) = �(⌦)T (⌦)Tq�1(⌦)

Tq�1(⌦) Tq�1(B)It would be interesting to compute explicitlysupFq(⌦) for q > 1 (is it attained?).

Summarizing: for general domais we have19

8 G. BUTTAZZO

General domains �

0 < q � 2/(d + 2) min Fq(�) = Fq(B) sup Fq(�) = +�

2/(d + 2) < q < 1 inf Fq(�) = 0 sup Fq(�) = +�

q = 1 inf Fq(�) = 0 sup Fq(�) = 1

q > 1 inf Fq(�) = 0 sup Fq(�) < +�

Table 1. Bounds for Fq(�) when � varies among all domains.tableall

4. Convex domainssconv

In the case of convex domains, some of the bounds seen in Section 3 remain:taking as � a slab A�] � �/2, �/2[ we obtain

infn

Fq(�) : � convexo

= 0 if q > 1,

supn

F1(�) : � convexo

= +� if q < 1.

The case q = 1 was studied in [2], where the following bounds have been obtained:8

<

:

infn

F1(�) : � convexo

= C� > 0,

supn

F1(�) : � convexo

= C+ < 1.(4.1) bounds

The explicit values for C� and C+ are not yet known and only conjectured, as below.

conje Conjecture 4.1. The values C� and C+ are given by

• C� = �2/24 asymptotically reached by a thin “triangle”, obtained by A ad � 1 rectangle and h(s1, . . . , sd�1) = s1;

• C+ = �2/12 asymptotically reached by a thin “slab”, obtained by any Aand h(s1, . . . , sd�1) = 1.

The other cases follow easily from the bounds above.

Proposition 4.2. We have:

infn

Fq(�) : � convexo

� C��

d(d + 2)�2/dd�1�q

if q < 1,

supn

Fq(�) : � convexo

� C+�

d(d + 2)�2/dd�1�q

if q > 1.

Proof. Since

Fq(�) = F1(�)⇣ T (�)

|�|1+2/d⌘q�1

20

The Blaschke-Santaló diagram with d = 2, for x =

�(B)/�(⌦) and y = T (⌦)/T (B) is contained in the

colored region.

21

If we limit ourselves to consider only domains⌦ that are union of disjoint disks of radii rkwe find

x =maxk r

2k

P

k r2k

, y =

P

k r4k

⇣

P

k r2k

⌘2 .

It is not di�cult to show that in this case weobtain the region

x2 y x2[1/x] +⇣

1 � x[1/x]⌘2

where [s] is the integer part of s.In this way in the Blaschke-Santaló diagramwe can reach the colored region in the pic-ture below.

22

In the Blaschke-Santaló diagram with d = 2, the col-

ored region can be reached by domains ⌦ made by

union of disjoint disks.

23

The case d = 1

In the one-dimensional case every domain ⌦is the union of disjoint intervals of half-lengthrk, so that we have

x =maxk r

2k

⇣

P

k rk⌘2 , y =

P

k r3k

⇣

P

k rk⌘3

and we deduce that the full Blaschke-Santalóset is given by the region

x3/2 y x3/2[x�1/2] +⇣

1 � x1/2[x�1/2]⌘3

where [s] is the integer part of s.

24

The full Blaschke-Santaló diagram in the case d = 1,

where x = ⇡2/�(⌦) and y = 12T (⌦).

25

The case ⌦ convex

If we consider only convex domains ⌦, theBlaschke-Santaló diagram is clearly smaller.For the dimension d = 2 the conjecture is

⇡2

24

�(⌦)T (⌦)

|⌦|

⇡2

12for all ⌦

where the left side corresponds to ⌦ a thintriangle and the right side to ⌦ a thin rect-angle.

26

If the Conjecture for convex domains is true, the

Blaschke-Santaló diagram is contained in the colored

region.

27

At present the only available inequalities arethe ones of [BFNT2016]: for every ⌦ ⇢ R2

convex

0.2056 ⇡⇡2

48

�(⌦)T (⌦)

|⌦| 0.9999

instead of the bounds provided by the con-jecture, which are

8

<

:

⇡2/24 ⇡ 0.4112 from below⇡2/12 ⇡ 0.8225 from above.

28

In dimensions d � 3 the conjecture is

⇡2

2(d +1)(d +2)

�(⌦)T (⌦)

|⌦|

⇡2

12

• the right side asymptotically reached by athin slab

⌦" =n

(x0, t) : 0 < t < "o

with x0 2 A", being A" a d � 1 dimensionalball of measure 1/"

• the left side asymptotically reached by a thincone based on A" above and with height d".

29

The convexity assumption on the admissibledomains provides a strong extra compact-ness that allows to prove the existence ofoptimal domains in the cases:

8

<

:

maxn

�(⌦)Tq(⌦) : ⌦ convex, |⌦| = 1o

if q > 1

minn

�(⌦)Tq(⌦) : ⌦ convex, |⌦| = 1o

if q < 1.

This is obtained by showing that maximizing(resp. minimizing) sequences ⌦n are not toothin, in the sense that

inradius(⌦n)

diameter(⌦n)� cd,q ,

where cd,q > 0 depends only on d and q.

30

Summarizing: for convex domais we have

ON THE RELATIONS BETWEEN PRINCIPAL EIGENVALUE AND TORSIONAL RIGIDITY 11

Since r(�) is fixed, (4.15) gives the required uniform upper bound for diam(�). Astraightforward computation shows that

Fq(B) =�

j(d�2)/2�2

(d(d + 2))�q�2(1�q)/dd , (4.16) e14

and (4.6) follows by (4.15) and (4.16). �

Proof of Theorem 4.4. Let 0 < q < 1. We follow the same strategy as in the proof ofTheorem 4.3, and fix the inradius of the elements of a minimising sequence. To obtaina uniform upper bound on the diameter we proceed as follows. For an open, bounded,convex set in Rd we have by Theorem 1.1(i) in [14] in the special case p = q = 2 that

T (�)

|�|M(�) �2

d(d + 2), (4.17) e15

where M(�) is the maximum of the torsion function. On the other hand it is wellknown that M(�) � �(�)�1, see for example [3] and the references therein. It followsthat

Fq(�) �✓

2

d(d + 2)

◆q �(�)1�q

|�|2(q�1)/d . (4.18) e16

By (4.11),

Fq(�) �23q�2�2(1�q)�

d(d + 2)�q

r(�)2(q�1)

|�|2(q�1)/d . (4.19) e17

Furthermore, by the isodiametric inequality (see for instance [15]),

|�| � �d2d

diam(�)d. (4.20) e18

For any element of a minimising sequence of (4.7) we have Fq(B) � Fq(�). This gives,by (4.19)–(4.20),

r(�)

diam(�)� �2(5q�4)/(2(1�q))�1/dd (d(d + 2)

q/(2(q�1))Fq(B)1/(2(q�1)). (4.21) e19

Since r(�) is fixed diam(�) is uniformly bounded from above. This completes the proofof the existence of a minimiser. Finally (4.8) follows from (4.16) and (4.21). �

We may then summarize the results about the case of convex domains in Table 2.

Convex domains �

q < 1 min Fq(�) > 0 sup Fq(�) = +�

q = 1 inf F1(�) = C�d > 0 sup F1(�) = C

+d < 1

q > 1 inf Fq(�) = 0 max Fq(�) < +�

Table 2. Bounds for Fq(�) when � varies among convex domains.tableconvex 31

Thin domains

We say that ⌦" ⇢ R2 is a thin domain if

⌦" =n

(s, t) : s 2]0,1[, "h�(s) < t < "h+(s)o

where " is a small positive parameter andh�, h+ are two given (smooth) functions. Wedenote by h(s) the local thickness

h(s) = h+(s) � h�(s)

and we assume that h(s) � 0.

The following asymptotics hold (as " ! 0):32

�(⌦") ⇡"�2⇡2

khk2L1[Borisov-Freitas 2010]

T (⌦") ⇡"3

12

Z

h3(s) ds

|⌦"| ⇡ "Z

h(s) ds.

Hence, for a thin domain ⌦" we have

�(⌦")T (⌦")

|⌦"|⇡

⇡2

12

R

h3(s) ds

khk2L1R

h ds.

We are able to prove the conjecture abovein the class of thin domains. More precisely:

33

• for every h we haveR

h3(s) ds

khk2L1R

h ds 1 ;

• for every h concave we haveR

h3(s) ds

khk2L1R

h ds�

1

2.

Hence

⇡2

24 lim

"!0�(⌦")T (⌦")

|⌦"|

⇡2

12

where the right inequality holds for all thindomains, while the left inequality holds forconvex thin domains.

34

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1experimental datay=x2Polya s upper bound

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1y=f(x)y=x2y=xadmissible area

Plot of 100 experimental domains (left), union of disks (right).

35

Open questions

• characterize sup|⌦|=1

�(⌦)Tq(⌦) when q > 1;

• prove (or disprove) the conjecture for con-vex sets;

• simply connected domains or star-shapeddomains? The bounds may change;

• full Blaschke-Santaló diagram;

• p-Laplacian instead of Laplacian?

• e�cient experiments (random domains?).36