Some estimates of the minimizing properties of web functions

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Some estimates of the minimizing properties of web functions Graziano CRASTA Dipartimento di Matematica Pura e Applicata - via Campi 213/B - 41100 Modena (Italy)

Filippo GAZZOLA∗ Dipartimento di Scienze T.A. - via Cavour 84 - 15100 Alessandria (Italy)

Abstract We study a general class of functionals over W 1,1 and we approximate their infimum by means of the minimum in the space of web functions. We provide several tools for estimating this approximation and we study in detail some meaningful models.

1

Introduction

Let Ω be an open bounded convex domain of IRn (n ≥ 2), let f : IR+ → IR be a lower semicontinuous (l.s.c.) function and consider the functional J defined by Z J(u) = [f (|∇u|) − u] dx . Ω

This kind of functionals (with possibly nonconvex functions f ) arises from various fields of mathematical physics and optimal design, see [3, 5, 23, 24]. We study the following problem of existence of minima, min J(u) . u∈W01,1 (Ω)

It is well-known that if f is not convex and superlinear then the minimum may not exist. In such case, one usually introduces the relaxed functional and considers its minimum, which coincides with the minimum of J if the latter exists. In particular, we mention a problem from elasticity [5, 22, 23]: we wish to place two different linearly elastic materials (of different shear moduli) in the plane domain Ω so as to maximize the torsional rigidity of the resulting rod; moreover, the proportions of these materials are prescribed. Such a problem may not have a solution, therefore one may construct new composite materials by mixing them together on a microscopic scale. Mathematically, this corresponds to the introduction of the relaxed problem which does have a minimum, hence there exists an optimal design if one is allowed to incorporate composites. However, the resulting design may not be so easy to manufacture and therefore one may try to find an optimal design in a simpler class of possible designs. ∗

Supported by MURST project “Metodi variazionali ed Equazioni Differenziali Non Lineari”.

1

Motivated by this application, we will be concerned with the following minimization problem min J(u) , u∈K

(1)

where K is the subset of W01,1 (Ω) of functions depending only on the distance from the boundary ∂Ω. As in [19], we call web functions the functions in K and we recall that when Ω is a ball, web functions are none other than radially symmetric functions. Web functions were introduced in order to approximate the infimum of the functional J over W01,1 (Ω). As proved in [14, 19], under very mild assumptions on f , the minimum of J over K always exists even for nonconvex functions f . More precisely, in [19] it is proved that if f is superlinear at infinity and Ω is a regular polygon in IR2 then J admits a unique minimum over K whose explicit form is given in terms of f ; this result was extended in [14] to general convex domains Ω ⊂ IRn (n ≥ 2) and to functions f with at least linear growth at infinity. The aim of this paper is to give estimates of the “error” one makes by means of such approximation. More precisely, thanks to a “normalization” (see Proposition 1 below) we may always reduce to functions f satisfying f (0) = 0 so that J(0) = 0; then, by Proposition 2 below we exclude the trivial cases where the minimum of J over W01,1 (Ω) exists and is u ≡ 0 (which is a web function!), so that the following ratio is well-defined minu∈K J(u) . (2) E= inf u∈W 1,1 (Ω) J(u) 0

W01,1 (Ω)

Since K ⊂ one has E ∈ [0, 1] and E represents the relative error of the above mentioned approximation: the closer E is to 1, the better the approximation is. The level 0 for the functional is chosen as a reference level because u ≡ 0 is the rest function; the value of E yields the relative error with respect to the rest function. The outline of this paper is as follows. In next section we establish some general estimates of E: we first obtain a result by a symmetrization method, namely we estimate the infimum over W01,1 (Ω) by means of the (explicit) minimum of the corresponding problem in the symmetrized ball; since we also obtain the explicit value of (1), this yields the estimate. We apply this result to some particular domains Ω and we show that for “thin” domains this estimate loses interest: then, we obtain a different estimate by “truncating and shifting” the function f and by applying previous results by Cellina [10]; we show that this estimate improves the first one for thin domains. Finally, we study in more detail the case where f (s) = sp /p (p > 1) and we obtain an estimate of E which involves optimal Sobolev embedding constants: in Section 4 we show that in some cases this estimate is sharper than the first two. In Section 3 we consider some meaningful models and we apply our results: we consider the three cases where the function f is convex superlinear, convex asymptotically linear, nonconvex. The results obtained show that the approximation by means of web functions is in general satisfactory (E is close to 1).

2

Notation and general results

Let Ω ⊂ IRn (n ≥ 2) be a bounded open convex set and let WΩ denote its width, namely the supremum of the radii of the open balls contained in Ω. The Lebesgue measure and the m-dimensional Hausdorff

2

measure of a set A ⊂ IRn will be denoted respectively by L(A) and Hm (A). We denote by ωn =

2π n/2 nΓ(n/2)

the Lebesgue measure of the unit ball B1 in IRn and we recall that Hn−1 (∂B1 ) = nωn . Assume that f 6≡ +∞

is a l.s.c. function s.t. ∃M >

L(Ω)

Hn−1 (∂Ω)

, ∃b ∈ IR , f (s) ≥ M s − b

∀s ≥ 0 :

(3)

the second condition is certainly satisfied if f is superlinear at infinity or if M > WΩ , see [14]. We denote by f ∗ the polar function of f and by f ∗∗ the bipolar function of f , see [17]; it is well-known that f ∗∗ is the greatest convex l.s.c. function which is pointwise less or equal than f . Let σ = max{s ≥ 0; f ∗∗ (s) = min f ∗∗ } ;

(4)

we also define the normalized non-decreasing bipolar function f∗∗ of f by   0 if 0 ≤ s ≤ σ f∗∗ (s) =  f ∗∗ (s) − f ∗∗ (σ) if s ≥ σ :

obviously, if f ∗∗ is non-decreasing and f ∗∗ (0) = 0, then f ∗∗ = f∗∗ . Finally, we denote by f∗ the polar function of f∗∗ which coincides with f ∗ if f (0) = 0 and f is non-decreasing and convex. Our first result states that any function f satisfying (3) may be normalized without altering the minimization problem: Proposition 1. Assume that f satisfies (3), let σ be as in (4) and let f∗∗ be the normalized nondecreasing bipolar function of f ; then Z Z [f∗∗ (|∇u|) − u] + L(Ω)f ∗∗ (σ) . (5) [f (|∇u|) − u] = min inf u∈W01,1 (Ω) Ω

u∈W01,1 (Ω) Ω

Define the set of web functions relative to Ω K = {u ∈ W01,1 (Ω); u(x) = u(d(x, ∂Ω)) ∀x ∈ Ω} , where d(·, ∂Ω) denotes the distance function from the boundary. We also consider the one-parameter family of subsets of Ω defined by Ωt = {x ∈ Ω; d(x, ∂Ω) > t}

∀t ∈ [0, WΩ ]

and their boundaries ∂Ωt ; we clearly have Ω0 = Ω and ΩWΩ = ∅. In the sequel a major role is played by the functions L(Ωt ) ν(t) = n−1 , α(t) = Hn−1 (∂Ωt ) t ∈ [0, WΩ ] . H (∂Ωt ) Consider the functionals

J(u) =

Z

Ω

[f (|∇u|) − u]

J∗∗ (u) = 3

Z

Ω

[f∗∗ (|∇u|) − u]

and the corresponding values IK = min J∗∗ (u)

I∗∗ =

u∈K

min

u∈W01,1 (Ω)

J∗∗ (u) .

Note that by the results in [14, 19] and by Proposition 1 we have IK = min J(u) + L(Ω)f ∗∗ (σ) . u∈K

We are concerned with the estimate of the relative error E defined in (2): with the above notations we have E = IK /I∗∗ . If I∗∗ = 0, then the problem “degenerates” (it loses interest) because the web function u ≡ 0 minimizes J∗∗ ; in other words, we also have IK = 0 and there is no need to approximate the minimum! In order to avoid this trivial case, we have to take into account a necessary condition: Proposition 2. Assume that f satisfies (3) and that I∗∗ < 0; then f∗ (WΩ ) > 0. In order to have the relative error E well-defined, Proposition 2 tells us that we have to assume that f∗ (WΩ ) > 0; as we will see, in some cases more stringent assumptions are needed. Note that the condition f∗ (s) = 0 for some s > 0 implies that the right derivative of f∗∗ at 0 is greater or equal than s. Hence, if this derivative vanishes then we have f∗ (s) > 0 for every s > 0. The basic tool which will be used in this paper in order to estimate E is

1/n ; assume that f Theorem 1. Let Ω ⊂ IRn be an open bounded convex set and let R = ( L(Ω) ωn ) satisfies (3) and f∗ ( R n ) > 0, then R WΩ α(t)f∗ (ν(t))dt = E1 . (6) E ≥ 0 RR nωn 0 tn−1 f∗ ( nt )dt

This result is obtained by means of a symmetrization method, see Section 5 below. In order to use Theorem 1, the explicit form of the functions α(t) and ν(t) is needed: in the next two corollaries we apply Theorem 1 to some particular cases. We first deal with cubes in IRn : Corollary 1. Let ℓ > 0, Ω = (0, ℓ)n and assume that f satisfies (3) and f∗ ( E≥

R ℓωn−1/n 0

1/n

tn−1 f∗ ( ωn2

R ℓωn−1/n 0

t n )dt

ℓ 1/n ) nωn

> 0. Then,

.

tn−1 f∗ ( nt )dt

Next, we deal with domains being planar regular polygons: as stated in [19, Theorem 2], when the number of sides tends to infinity, the corresponding sequence of minimizing web functions converges (in a suitable sense) to the unique radial minimum in the circumscribed ball. Here, we rephrase this result in a more precise fashion: Corollary 2. Let ρ > 0, m ∈ IN (m ≥ 3) and let Ωm be a regular polygon of m sides inscribed in the ball Bρ ⊂ IR2 ; assume that f satisfies (3) and f∗ (0) > 0. Let Em be the corresponding relative error, then π R ρ cos m r tf∗ ( 2t )dt m π 0 , σm = Em ≥ R ρ cos π tan ; σm t π m m )dt tf ( ∗ 0 2

in particular,

lim Em = 1 .

m→∞

4

In the previous result, the assumption f∗ (0) > 0 can be relaxed to f∗ ( ρ4 wants the asymptotic behavior of Em , to f∗ ( ρ2 ) > 0.

q

√ 3 3 π )

or, if one merely

We now study the asymptotic behavior of E1 in (6) in the case of “vanishing domains”. To this end, we make a further assumption on f : ∃γ, δ > 0

s.t. f∗ (s) ∼ γsδ

as s → 0 ;

(7)

by this we mean that f∗ (s) = γsδ + o(sδ ) as s → 0. This assumption implies that f∗ (s) > 0 for all s > 0 and no further positiveness requirement on f∗ is needed. An interesting fact is that if we take Ω to be a cube, then E1 in (6) tends to a constant (depending only on n, δ but not on the particular function f considered) as the measure of the cube tends to 0: Theorem 2. Let ℓ > 0, let Ω = (0, ℓ)n and assume that f satisfies (3) and (7); let E1 be as in (6), then  ω δ/n n lim E1 = . (8) ℓ→0 2n

Note that the term inside parenthesis in the limit in (8) is just the ratio between the measure of the unit ball and the measure of the circumscribed cube: of course, this is a consequence of the symmetrization method. The next result shows that Theorem 1 loses interest for “thin” domains (i.e. domains Ω with a small width WΩ ):

Theorem 3. Let Ω ⊂ IRn be an open bounded convex set and assume that f satisfies (3) and f∗ (WΩ ) > 1/n and assume that 0. Let M be as in (3), let R = ( L(Ω) ωn ) 



.

−1

.

R WΩ < min M, 2(n + 1) Let E1 be as in (6), then E1 ≤



1 R −1 n + 1 WΩ

(9)

For fixed L(Ω), as WΩ → 0 the ratio R/WΩ tends to infinity and (9) states that E1 approaches zero: therefore, for thin domains the estimate (6) is not satisfactory. In particular, if 0 < λ < ℓ and Ω = (0, ℓ)n−1 × (0, λ) and we let λ → 0, then E1 → 0. Of course, this does not mean that the same is true for E: on the contrary, we conjecture that the minimum over K well approximates the infimum over W01,1 (Ω) for thin domains, see Propositions 4 and 5 below. When dealing with thin domains, the next result seems more useful: Theorem 4. Let Ω ⊂ IRn be an open bounded convex set such that WΩ < M , assume that f satisfies (3) and f∗ (WΩ ) > 0. Then RW WΩ 0 Ω α(t)f∗ (ν(t))dt E≥ = E2 . (10) RW f∗ (WΩ ) 0 Ω L(Ωt ) dt To see how Theorem 4 improves Theorem 1 for thin domains, consider the following example

5

Example 1. Let f (s) = s2 /2 and let Ω = (0, 1) × (0, 2WΩ ), 0 < WΩ ≤ 1/2. Let us denote by E1 (WΩ ) and E2 (WΩ ) respectively the r.h.s. of (6) and (10). We have that E2 is monotone decreasing on [0, 1/2], limWΩ →0 E2 (WΩ ) = 2/3 and E2 (1/2) = 3/8. On the other hand, E1 is monotone increasing on [0, 1/2], approaches 0 as WΩ tends√to 0 and E1 (1/2) = π/4. The explicit computation of E1 and E2 gives that 2 −12π ≈ 0.181. E2 > E1 for 0 < WΩ < 3π− 9π 4π As a consequence of Theorem 4, we obtain

Corollary 3. Under the assumptions of Theorem 4, one has f∗ (ζ)/ζ E≥ , f∗ (WΩ )/WΩ

1 ∀0 1, in order to estimate E, a method involving optimal embedding constants may also be used: Theorem 5. Let p > 1, Ω ⊂ IRn be an open bounded convex set and assume that f (s) = R |∇u|p
p . RΩ S = S(Ω) = inf u∈W01,p (Ω) Ω |u|

sp p .

Let

u6=0

Then

1

E ≥ S p−1

Z

WΩ

0

p

α(t)ν p−1 (t) dt = E3 .

(13)

In spite of its simple and elegant form, the previous result may not be so easily applied: besides the already mentioned problem of determining explicitly the functions α and ν, here a major problem is also how to determine the (possibly sharp) constant S. Nevertheless, in Section 4, we quote some examples of planar rectangles for which Theorem 5 improves Theorems 1 and 4.

3

Estimates and applications

In this section we discuss the general results given in the previous section and we apply them to some particularly interesting models.

3.1

Some convex superlinear problems s2 2

so that the functional J becomes  Z  |∇u|2 J(u) = − u dx . 2 Ω

Consider first the case where f (s) =

To the minimization problem of J is associated the Euler equation   −∆u = 1 in Ω  u = 0 on ∂Ω ; 6

(14)

the unique solution of (14) is precisely the minimum of the functional J over the space W01,1 (Ω) (in fact the solution is smooth). This equation describes a viscous incompressible fluid moving in straight parallel streamlines through a straight pipe of cross section Ω, see [28]. In the two-dimensional context, (14) also has different interpretations, see e.g. [4]: consider the torsion problem of a long cylindrical beam in IR3 whose axis is the x3 axis and whose uniform cross-section Ω is a simply connected region of the plane x1 , x2 ; the state of stress in the interior of the beam is determined by a warping function u which satisfies (14). We restrict our attention to some particular domains Ω which allow us to obtain the “exact” value of E and, as a consequence, to evaluate how fine the estimate of Theorem 1 is. We first consider the case of a square: Proposition 3. If f (s) =

s2 2

and Ω = (0, 1)2 , then E ≈ 0.889.

Therefore, in this case the approximation by means of web functions is satisfactory since E is close to 1. In order to compare the value of E obtained in Proposition 3 with the lower bound for E obtained in Corollary 1, we refer to Proposition 6 below: in the case n = p = 2 it yields the estimate E > 0.785 which is approximately the 88.3% of the value 0.889. If we shrink the length of two parallel sides of the square to 0, then the approximation tends to become optimal: Proposition 4. Let f (s) =

s2 2,

let ℓ ∈ (0, 1) and let Ω = (0, ℓ) × (0, 1). Then E → 1 as ℓ → 0.

Proposition 4 does not come unexpected; indeed, when ℓ → 0 we can say in some sense that the problem becomes 1-dimensional. Apart circles and regular polygons, the simplest planar domains seem to be ellipses; however, as will be shown in the proof of the next result, the explicit forms of the functions α and ν are complicated and one has to proceed numerically: with the aid of Mathematica we obtain Proposition 5. Let f (s) =

s2 2,

let 0 < b < 1 and let   y2 Ω = (x, y) ∈ IR2 ; x2 + 2 < 1 . b

Then, we have the following approximate values of E; b

0.01

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

E≈

0.895

0.897

0.903

0.914

0.928

0.945

0.962

0.977

0.989

0.997

Moreover, lim E = 1. b→1

2

We point out that even in the simple case f (s) = s2 the behavior of E for thin rectangles (small ℓ in Proposition 4) and thin ellipses (small b in Proposition 5) is not the same: we believe that the non optimal behavior of thin ellipses is due to their curvature. Next, we deal with a slightly more general class of functions f : we consider the case where f (s) = sp /p for some p > 1 so that the functional to minimize is  Z  |∇u|p − u dx ; J(u) = p Ω 7

here, the corresponding Euler equation is the degenerate elliptic problem   −∆p u = 1 in Ω  u = 0 on ∂Ω ,

(15)

where ∆p u = div(|∇u|p−2 ∇u). This operator may be used for the description of some phenomena in glaciology [26, 27] and for the study of non-Newtonian fluids in rheology [2]; we also refer to the introduction in [16] for further applications. The unique solution of (15) is the minimum of the functional J. An application of Corollary 1 yields Proposition 6. Let p > 1; if f (s) =

sp p

and Ω = (0, 1)n , then

E≥

1/n ωn

2

!

p p−1

.

Note that the lower bound for E in Proposition 6 tends to 0 as p → 1 (i.e. as the functional “loses coercivity”) and tends to √ π (n−1)/n 1/n 2 n Γ1/n (n/2) as p → ∞: in particular, when n = 2 this limit is about 0.886.

3.2

A convex asymptotically linear problem

Consider the function f (s) =

√

1 + s2 − 1 so that the functional to minimize is the following Z p  1 + |∇u|2 − 1 − u dx ; J(u) = Ω

associated to the functional J we have the Euler equation    ∇u   −div p =1 1 + |∇u|2   u=0 on ∂Ω

in Ω (16)

whose solutions are the celebrated Delaunay surfaces of constant mean curvature, see [15]. We take √ √ f (s) = 1 + s2 − 1 instead of the usual f (s) = 1 + s2 so that f (0) = 0 and E is well-defined. In this case we have M = 1 and if we wish to fulfill (3) the set Ω must be “sufficiently small”, namely νΩ (0) =

L(Ω) β > 0, γ > 0) and f (t) = min{h1 (t), h2 (t)} , and consider the functional J(u) =

Z

Ω

t≥0,

(19)

[f (|∇u|) − u] dx .

The problem of minimizing J over the space W01,1 (Ω) arises from elasticity [5, 22, 23]: we wish to 1 1 place two different linearly elastic materials (of shear moduli 2α and 2β ) in the plane domain Ω so as to maximize the torsional rigidity of the resulting rod; moreover, the proportions of these materials are prescribed. By applying Corollary 1 we obtain Proposition 8. Let ℓ > 0, Ω = (0, ℓ)2 and assume that f is as in (19), then E≥ E≥

π 4

√ if ℓ ≤ 2 πa

πβ(α − β)ℓ4 4α(α − β)ℓ4 − 128παβγ(α − β)ℓ2 + 1024π 2 α2 β 2 γ 2 E≥

π(α − β)ℓ4 − 128π(α − β)βγℓ2 + 4096παβ 2 γ 2 4(α − β)ℓ4 − 128π(α − β)βγℓ2 + 1024π 2 αβ 2 γ 2

√ if 2 πa < ℓ < 4a if ℓ ≥ 4a .

If ℓ is small, then the minimum of u of J∗∗ has small gradients |∇u| in the main part of the square Ω; therefore we have f ≈ h1 and the problem reduces to that of Proposition 6 (with p = 2); this is why we find π4 as a lower bound for E. Similarly, if ℓ → ∞ then we have f ≈ h2 and the problem tends again to that of Proposition 6.

9

Finally, we consider the case where    0 f (t) = 1   ∞

if t = 1 if t = 2 elsewhere ;

(20)

this function was studied in [8] in an attempt to simplify the function f in (19) by retaining its essential feature of lacking convexity. For simplicity, we only deal with the case of squares Ω = (0, ℓ)2 for ℓ > 0. By [10, Theorem 1] we know that if ℓ ≤ 2 then E = 1; on the other hand, the functional J is known to have no minimum in W01,1 (Ω) if ℓ ∈ (2, 2 + 2ε) for sufficiently small ε, see [8] (we write 2 + 2ε instead of 2 + ε for later convenience). Therefore, we wish to prove that E → 1 as ℓ → 2+ ; the next result also gives the rate of such convergence: Proposition 9. Assume that f is as in (20) and let ε ∈ (0, 1), Ω = (0, 2 + 2ε)2 ; then E≥

1 . 1+ε

We point out that this result is not proved by means of the general statements of Section 2.

4

Some applications of Theorem 5

Throughout this section E1 , E2 and E3 are respectively the constants defined in (6), (10) and (13): we show that in some cases we have E3 > E1 or E3 > E2 so that Theorem 5 gives a finer estimate of E than Theorems 1 or 4. In order to apply Theorem 5 we need to determine the constant S; we give a lower bound for such constant in planar rectangles: Lemma 1. Let 0 < ℓ < 1 and let Ω = (0, 1) × (0, ℓ); then S≥

2p/2 (1 + ℓ)p . ℓ2p−1

Proof. By a density argument it suffices to prove that 2p/2 (1 + ℓ)p k∇vkpp p ≥ ℓ2p−1 kvk1

∀v ∈ Cc∞ (Ω) s.t. v ≥ 0 .

(21)

So, take v ∈ Cc∞ (Ω) such that v(x) ≥ 0 in Ω and denote by ∂i = ∂/∂xi (i = 1, 2); then, since v = 0 on ∂Ω, if we denote s+ = max[s, 0] and s− = min[s, 0], we have Z

1

+

[∂1 v(ξ1 , x2 )] dξ1 +

0

Z

1

−

[∂1 v(ξ1 , x2 )] dξ1 =

0

Z

1

0

∂1 v(ξ1 , x2 )dξ1 = v(1, x2 ) − v(0, x2 ) = 0 ∀x2 ;

furthermore, Z

0

1

+

[∂1 v(ξ1 , x2 )] dξ1 −

Z

1

−

[∂1 v(ξ1 , x2 )] dξ1 =

Z

0

0

10

1

|∂1 v(ξ1 , x2 )|dξ1

∀x2 .

These two equations show that Z 1 Z 1 1 + [∂1 v(ξ1 , x2 )] dξ1 = |∂1 v(ξ1 , x2 )|dξ1 2 0 0 if we proceed similarly with the other variable we obtain Z ℓ Z 1 ℓ [∂2 v(x1 , ξ2 )]+ dξ2 = |∂2 v(x1 , ξ2 )|dξ2 2 0 0 Since v(x1 , x2 ) =

Z

x1

∂1 v(ξ1 , x2 )dξ1 =

0

Z

∀x2 ;

(22)

∀x1 .

(23)

x2

∂2 v(x1 , ξ2 )dξ2

0

∀(x1 , x2 ) ∈ Ω ,

by Fubini’s Theorem and (22)-(23) we obtain  Z x2 Z  Z x1 Z 1 ℓ ∂1 v(ξ1 , x2 )dξ1 + ∂2 v(x1 , ξ2 )dξ2 dx1 dx2 kvk1 = v(x1 , x2 )dx1 dx2 = 1+ℓ Ω 0 0 Ω   Z Z x1 Z x2 1 + + ≤ ℓ [∂1 v(ξ1 , x2 )] dξ1 + [∂2 v(x1 , ξ2 )] dξ2 dx1 dx2 1+ℓ Ω 0 0 Z
ℓ ≤ [∂1 v(x1 , x2 )]+ + [∂2 v(x1 , x2 )]+ dx1 dx2 1+ℓ Ω Z ℓ (|∂1 v(x1 , x2 )| + |∂2 v(x1 , x2 )|) dx1 dx2 ; = 2(1 + ℓ) Ω hence, if we use H¨ older’s inequality and the inequality (a + b)p ≤ 2p/2 (a2 + b2 )p/2 (which holds for all a, b ≥ 0) we get ℓ ℓ(2p−1)/p kvk1 ≤ √ [L(Ω)](p−1)/p k∇vkp = √ k∇vkp , 2(1 + ℓ) 2(1 + ℓ) which proves (21).



Now we prove that if Ω = (0, 1)2 and p is “close” to 1, then E3 > E2 . Indeed, Lemma 1 and (13) yield E3 ≥

p − 1 (p−2)/(2p−2) 2 ; 3p − 2

on the other hand, if E2 is as in (10), we get

E2 =

p−1 3 1/(p−1) 3p − 2 2

and we have E3 > E2 for p sufficiently close to 1. Finally, for all p > 1 we have lim E3 =

ℓ→0

p − 1 (2−3p)/(2p−2) 2 >0 2p − 1

which, together with Theorem 3, shows that E3 > E1 for ℓ small enough. 11

5

Proof of Theorem 1

We first recall the basic definitions of the Schwarz symmetrization, see e.g. [4]. Given the set Ω, we denote by Ωs the ball centered at the origin such that L(Ωs ) = L(Ω). Let u be a real-valued function defined in Ω, then we define its symmetrized function us : Ωs → IR by us (x) = sup{µ; x ∈ Dµs }, where Dµ = {y ∈ Ω; u(y) ≥ µ}. In the sequel, we denote Z SJ (u) = [f (|∇u|) − u] , Is = min SJ (u) . u∈W01,1 (Ωs )

Ωs

Explicit form of the minimizing web function. Consider the functions ν and α defined in Section 2 then, the unique web function w which minimizes J over K is given by Z d(x,∂Ω) w(x) = φ(d(x, ∂Ω)) = (f∗ )′ (ν(s))ds ; (24) 0

this follows from formula (6) in [14] and by taking into account that (f∗ )′ (t) = (f∗ )′− (t) for a.e. t. Indeed, we recall that by Theorem 4.1 in [14] we know that the function ν is strictly decreasing and therefore (f∗ )′ (ν(s)) = (f∗ )′− (ν(s)) for a.e. s ∈ [0, WΩ ]. Moreover, by (7) and (8) in [14], the corresponding (minimum) value of the functional is given by Z WΩ IK = J(w) = α(t)[f (φ′ (t)) − φ(t)]dt ; 0

indeed, from (24), we see that φ′ (t) ≥ 0 for a.e. t and we have |φ′ (t)| = φ′ (t) for a.e. t ∈ [0, WΩ ]. Using again the strict monotonicity of ν we have f ((f∗ )′ (ν(t))) = ν(t) (f∗ )′ (ν(t)) − f∗ (ν(t))

for a.e. t ∈ [0, WΩ ] ;

then, integrating by parts the term in φ (see Lemma 5.6 in [14] where A(t) = α(t)ν(t)) we obtain Z WΩ IK = − α(t)f∗ (ν(t))dt . (25) 0

In the case of regular polygons in IR2 an equivalent explicit form of w and IK may be given, see [19]. The symmetrization method. Thanks to well-known results in [1, 7] we obtain Lemma 2. Assume that f satisfies (3) and that f is convex and non-decreasing. Then Z Z [f (|∇u|) − u] . [f (|∇u|) − u] ≥ min inf u∈W01,1 (Ωs ) Ωs

u∈W01,1 (Ω) Ω

Proof. We show that, for every u ∈ W01,1 (Ω) we have J(u) ≥ SJ (us ). Since J(|u|) ≤ J(u) for all u ∈ W01,1 (Ω), we may assume that u ≥ 0; then, by the properties of the symmetrized function (see [4]) we have Z Z u= us . (26) Ωs

Ω

12

Next, by Proposition 2.1 in [7] and by arguing as in the proof of Lemma 3.1 in [7], one has Z Z f (|∇u|) dx ≥ f (|∇us |) dx ; Ωs

Ω

this, together with (26) concludes the proof.



1/n and Conclusion. For any bounded convex set Ω ⊂ IRn we have Ωs = BR (0) with R = ( L(Ω) ωn )

αB (t) = nωn (R − t)n−1 ,

νB (t) =

R−t ; n

therefore, according to (25) we have Is = −nωn

R

Z

0

(R − t)n−1 f∗



R−t n



dt = −nωn

Z

0

R

tn−1 f∗

  t dt . n

(27)

Since f∗ ( R n ) > 0 we have Is < 0 so that, by Lemma 2, we also have I∗∗ < 0. Moreover, Lemma 2 yields IK IK ≥ . (28) E= I∗∗ Is Theorem 1 follows now directly from (25), (28) and (27).



Remark 1. By Theorem 3.6 in [13] (see also [11]) the minimization problem Z [f (|∇u|) − u] min u∈W01,1 (Ωs ) Ωs

admits a unique solution us which is radially symmetric (and decreasing); moreover, by (11) in [13] we know that its derivative u′s (r) (with respect to r = |x|) satisfies the Euler-Lagrange inclusion −

r ∈ ∂f∗∗ (u′s (r)) : n

when f ∈ C 1 (IR+ ) and f is strictly increasing and strictly convex the above inclusion simply becomes r u′s (r) = −(f ′ )−1 , n

see Remark 1 in [20] for related results concerning the corresponding Euler equation.

6



Proofs of Theorems 2-5

Proof of Theorem 2. The constants WΩ , R and the functions α, ν relative to Ω are given in (34). Then, by (7) and (27) we get 1/n

|Is | ∼ nωn

Z

0

ℓ/ωn

tn−1 γ

tδ γ ℓn+δ dt = δ δ/n δ−1 n (n + δ)ωn n

13

as ℓ → 0 .

Furthermore, by (7) and (35) we obtain 1/n

|IK | ∼ nωn γ

Z

ℓ/ωn

δ/n

tn−1

0

ω n tδ γ dt = ℓn+δ δ δ 2 n (n + δ)2δ nδ−1

as ℓ → 0 .

Taking the ratio and applying (28) shows that E1 (defined in (6)) tends to the value in (8) as ℓ → 0. Proof of Theorem 3. Since 0 ≤ ν(t) ≤ WΩ for every t ∈ [0, WΩ ], we have that 0 ≤ f∗ (ν(t)) ≤ f∗ (WΩ )

∀t ∈ [0, WΩ ] ,

hence by (25) we get |IK | =

Z

WΩ

α(t)f∗ (ν(t)) dt ≤ f∗ (WΩ )

0

Z

WΩ

0

α(t) dt = f∗ (WΩ )L(Ω) = f∗ (WΩ )ωn Rn .

(29)

By (3) and the assumptions WΩ < M and f∗ (WΩ ) > 0, we may find β > 0 such that f∗∗β(β) = WΩ . Since f∗ is the polar function of f∗∗ , we have f∗ (z) ≥ βz − f∗∗ (β) for every z ≥ 0; then, by (27) we get |Is | = nωn

Z

0

R

t

n−1

      n+1 Z R t t R n−1 n f∗ β − f∗∗ (β) dt = ωn β t . dt ≥ nωn − WΩ R n n n+1 0

¿From the definition of β we have that β ≥ f∗ (WΩ )/WΩ , hence   f∗ (WΩ ) Rn+1 |Is | ≥ ωn − WΩ R n . WΩ n+1

(30)

The conclusion now follows from (29) and (30).



Proof of Theorem 4. By the very definition of polar function we have that f∗∗ (s) ≥ sWΩ − f∗ (WΩ ) for every s ≥ 0, hence f∗∗ (s) ≥ fˆ(s) = max{0, sWΩ − f∗ (WΩ )} ∀s ≥ 0 . Let us define the functional ˆ J(u) =

Z

Ω

u ∈ W01,1 (Ω).

[fˆ(|∇u|) − u] dx,

By [9, 10, 29] we have that the function u ˆ(x) =

f∗ (WΩ ) d(x, ∂Ω), WΩ

x ∈ Ω,

is a minimizer of Jˆ on W01,1 (Ω). Since f∗∗ ≥ fˆ we deduce that ˆ u) = − f∗ (WΩ ) inf J ≥ J(ˆ 1,1 WΩ W0 (Ω)

Z

0

WΩ

L(Ωt ) dt,

where we have used (25) applied to Jˆ and the fact that fˆ∗ (s) = (10) then follows. 14

f∗ (WΩ ) WΩ s

for s ∈ [0, WΩ ]; the estimate 

Proof of Theorem 5. First note that f ∗ (s) = f∗ (s) =

p p − 1 p−1 s . p

(31)

Next, we remark that if u ∈ W01,1 (Ω) minimizes J, then u solves (15), u ∈ W01,p (Ω) and u > 0 on Ω; by multiplying the equation in (15) by u and by integrating by parts we obtain Z Z u, (32) |∇u|p = Ω

Ω

so that

sp f (s) = , u minimizes J p

1−p J(u) = p

=⇒

Since u > 0 in Ω, by (32) and by definition of S, we have Z 1 u ≤ S 1−p ;

Z

u.

(33)

Ω

Ω

by (33) this gives I∗∗ = J(u) ≥

1 1 − p 1−p . S p

Therefore, by definition of E and by (25) (with f∗ given by (31)) we get E=

7

1 IK ≥ S p−1 I∗∗

Z

WΩ

p

α(t)ν p−1 (t) dt .

0

Proofs of the Corollaries

Proof of Corollary 1. Take ℓ > 0 and Ω = (0, ℓ)n , then Ωs = BR and WΩ = n

L(Ωt ) = (ℓ − 2t)

α(t) = H

n−1

ℓ 2

R=

ℓ 1/n ωn n−1

ℓ − 2t ν(t) = 2n

(∂Ωt ) = 2n(ℓ − 2t)

Then, by (25) and by the change of variables s =

ℓ−2t 1/n , ωn

IK = −nωn

0

ℓ/ωn

(34)

we infer

1/n

Z

  ℓ t ∈ 0, . 2

1/n

tn−1 f∗

ωn 2

This, together with (28) and (27), yields the estimate of E.

t n

!

dt .

(35) 

Remark 2. A more elegant form of the estimate of E may be obtained by using the convexity of f∗ ; indeed, by (27) we obtain ! Z 1/n   1/n ℓ/ωn t ωn n ′ t (f∗ ) ωn dt . Is − IK ≤ 1 − 2 n 0 15

This, together with (28) and (27), yields 1/n 1/n R
ℓ/ω (1 − ωn2 ) 0 n tn (f∗ )′ nt dt IK . E≥ ≥1−
R ℓ/ω1/n Is n 0 n tn−1 f∗ nt dt

However, the estimate in the statement of Corollary 1 is sharper and therefore we will not make use of the latter.  Proof of Corollary 2. The constants and functions relative to the polygon Ωm are given by s m sin 2π π m Rm = ρ WΩm = ρ cos m 2π    π π 1 π αm (t) = 2m ρ cos − t tan νm (t) = ρ cos − t . m m 2 m π Therefore, (25) and the change of variables s = ρ cos m − t yield π IK = −2m tan m

Z

ρ cos

0

π m

sf∗

s 2

ds ;

moreover, (27) and the change of variables s=

r

π π cot t, m m

yield π Is = −2m tan m

ρ cos

Z

0

π m

 s sf∗ σm ds . 2

Since f∗ (0) > 0, we have Is < 0 (for all m) and the estimate of Em follows from (28).



Proof of Corollary 3. For every fixed β > 0 we have that f∗ (z) ≥ βz − f∗∗ (β), for every z ≥ 0. Then Z

0

WΩ

α(t)f∗ (ν(t)) dt ≥

Z

WΩ

α(t)[βν(t) − f∗∗ (β)] dt

0

=β

Z

(36)

WΩ

α(t)ν(t) dt − f∗∗ (β)L(Ω) .

0

¿From (10) and (36) we have that # " f∗∗ (β)L(Ω) WΩ . β − RW E≥ Ω f∗ (WΩ ) α(t)ν(t) dt

(37)

0

The right hand side in (37) is maximized choosing β ∈ ∂f∗ (ζ0 ), with ζ0 = obtaining f∗ (ζ0 )/ζ0 E≥ . f∗ (WΩ )/WΩ

1 L(Ω)

Since the map z 7→ f∗ (z)/z is monotone increasing on (0, +∞), then (11) follows. 16

R WΩ 0

α(t)ν(t) dt,

Concerning (12), by (11) it is enough to prove that 1 L(Ω)

Z

WΩ

α(t)ν(t) dt ≥

0

We recall that A(t) = α(t)ν(t) =

Z

WΩ L(Ω) . n+1

(38)

WΩ

α(s) ds = L(Ωt )

t

t ∈ [0, WΩ ].

From the Brunn-Minkowski theorem (see [6]) we have that the function γ(t) = A(t)1/n is concave on [0, WΩ ]. Since γ(0) = L(Ω)1/n and γ(WΩ ) = 0 we deduce that γ(t) ≥ hence Z

WΩ

A(t) dt =

0

WΩ

0

and (38) is proved.

8

Z

L(Ω) (WΩ − t), WΩ

L(Ω) γ (t) dt ≥ WΩ n n

t ∈ [0, WΩ ], Z

WΩ

0

(WΩ − t)n dt =

L(Ω)WΩ , n+1 

Proofs of the Propositions

Proof of Proposition 1. Any minimizing sequence {uRm } of both theR l.h.s. and the r.h.s. of (5) may be chosen so that |∇um (x)| ≥ σ for a.e. x ∈ Ω; hence, f (|∇um |) = fσ (|∇um |) and the result follows.  Proof of Proposition 2. For contradiction, if f∗ (WΩ ) = 0 then by the very definition of polar function we have that f∗∗ (s) ≥ sWΩ for every s ≥ 0. Since f∗∗ (0) = 0, from [10, Theorem 1] (see also [9, 29]) we conclude that the function u ≡ 0 is a minimizer of J∗∗ over W01,1 (Ω), contradiction.  Proof of Proposition 3. By separating the variables one finds that the unique solution u of (14) is given by ∞  sin[(2k + 1)πx] 4 X x − x2 − 3 exp[(2k + 1)πy] + exp[(2k + 1)π(1 − y)] . u(x, y) = 2 π (2k + 1)3 (e(2k+1)π + 1) k=0

Therefore, by (33) we have I∗∗ = J(u) = −

1 2

Z

Ω

u=−

∞ 8 X e(2k+1)π − 1 1 1 + 5 ≈ −0.0175721 . (2k+1)π + 1 (2k + 1)5 24 π e k=0

On the other hand, we have WΩ =

1 2

α(t) = 4(1 − 2t)

ν(t) =

1 − 2t 4

therefore, (25) yields 1 IK = − 8

Z

0

1/2

(1 − 2t)3 dt = − 17

1 . 64

f∗ (s) =

s2 : 2

(39)

This, together with (39), gives E ≈ 0.889.



Proof of Proposition 4. By arguing as in the proof of Proposition 3 we find that the unique solution u of (14) is given by u(x, y) =

∞  ℓx − x2 4ℓ2 X sin[(2k + 1)πx] − 3 exp[(2k + 1)πy/ℓ] + exp[(2k + 1)π(1 − y)/ℓ] . (2k+1)π/ℓ 3 2 π (2k + 1) (e + 1) k=0

Therefore, by (33) we have 1 J(u) = − 2

Z

Ω

u=−

∞ 1 ℓ3 8ℓ4 X e(2k+1)π/ℓ − 1 + 5 ; (2k+1)π/ℓ 24 π e + 1 (2k + 1)5 k=0

since the sum of the series is bounded away from +∞ and 0 as ℓ → 0, we infer that J(u) = −

ℓ3 + o(ℓ3 ) 24

as ℓ → 0 .

(40)

In order to evaluate the minimum of J over K, note that in this case we have WΩ = α(t) = 2(1 + ℓ − 4t) ,

ν(t) =

ℓ 2

and

(ℓ − 2t)(1 − 2t) . 2(1 + ℓ − 4t)

Then, from (25) we get Z 1 1−ℓ ℓ 1 ℓ/2 (ℓ − 2t)2 (1 − 2t)2 dt = (1 − ℓ)4 log + (1 + ℓ2 − 4ℓ) IK = IK (ℓ) = − 4 0 1 + ℓ − 4t 256 1 + ℓ 128 ℓ3 = − + o(ℓ3 ) as ℓ → 0 ; 24 this, together with (40) proves that E → 1 as ℓ → 0.



Proof of Proposition 5. The unique solution u of (14) is given by u(x, y) =

b2 − b2 x2 − y 2 ; 2(1 + b2 )

then, if B1 denotes the unit ball in IR2 , by (33) we infer Z Z 1 b3 πb3 2 2 u=− I∗∗ = J(u) = − . (1 − x − y )dxdy = − 2 Ω 4(1 + b2 ) B1 8(1 + b2 ) The boundary of the ellipse Ω may be represented parametrically as x = cos θ

y = b sin θ

θ ∈ [0, 2π] .

The outward normal vector has components b cos θ p

sin2 θ + b2 cos2 θ

,p

sin θ sin2 θ + b2 cos2 θ

18

!

,

(41)

and therefore a parametric representation of ∂Ωt (0 ≤ t < b = WΩ ) for x, y ≥ 0 is given by  cos θ     x = a cos θ − tb p 2 2 h πi a sin θ + b2 cos2 θ , θ ∈ θt , sin θ  2  y = b sin θ − ta p   a2 sin2 θ + b2 cos2 θ

where

θt = arcsin

s

(42)

 t2 − b 4 max 2 ,0 ; b (1 − b2 ) 

this value of θt comes from the fact that if t > b2 then two different inward normal segments to ∂Ω of length t may intersect: in particular, for these values of t, ∂Ωt is not a regular curve and this tells us that any web function which is not constant on Ωb2 is not in C 1 (Ω). ¿From (42) we infer that for x, y ≥ 0 we have  sin θ    x˙ = − sin θ + tb h πi 2 (sin θ + b2 cos2 θ)3/2 θ ∈ θt , cos θ  2   y˙ = b cos θ − tb2 2 (sin θ + b2 cos2 θ)3/2

and therefore

α(t) = 4

π/2 p

Z

θt

Moreover,

2 |x(θ)| ˙

+

2 dθ |y(θ)| ˙

=4

=2

π/2  b−

θt

2

sin θ +

b2 cos2 θ

θt

1 L(Ωt ) = 2 Z

π/2 p

Z

Z

∂Ωt

 tb dθ . − sin2 θ + b2 cos2 θ

(x dy − y dx)

 p t2 b tb2 2 2 2 dθ . − t sin θ + b cos θ + (sin2 θ + b2 cos2 θ)3/2 sin2 θ + b2 cos2 θ

On the other hand, by (25) we have IK = −

1 2

Z

b

0

L2 (Ωt ) dt α(t)

and a numerical computation with Mathematica allows to determine the approximate values of E given in the statement of Proposition 5. ¿Finally, if b → 1, by the explicit value of α(t) and L(Ωt ) found above and by Lebesgue Theorem, we have limb→1 IK = −π/16 which, together with (41), gives limb→1 E = 1.  Proof of Proposition 6. By (31) we see that f satisfies (7) with δ = p/(p − 1). By a rescaling argument one sees that E1 in (6) is independent of ℓ; then, the result follows from Theorem 2.  Proof of Proposition 7. Consider the modified functional Z p J+ (u) = 1 + |∇u|2 − u , Ω

19

so that J+ (u) = J(u) + L(Ω) for all u ∈ W01,1 (Ω) and we can argue instead on the functional J+ (see also Proposition 1) and obtain minu∈K J+ (u) − ℓn . (43) E= inf u∈W 1,1 (Ω) J+ (u) − ℓn 0 √ √ Therefore, we consider the function g(s) = 1 + s2 : then, g∗ (s) = − 1 − s2 which is defined if s ∈ [0, 1] (we only need to consider s ≥ 0). If u minimizes J+ and us minimizes the “symmetrized functional” SJ , by Lemma 2 and (27) we get J+ (u) ≥

Z

Z p 2 1 + |∇u| − u = nωn

R

rn−1

dr 1 + |u′s (r)|2 Z R Z ℓ/nωn1/n p p n−1 n+1 2 2 r = ωn n − r dr = ωn n rn−1 1 − r2 dr . min

u∈W01,1 (Ωs ) Ωs

p

0

0

0

On the other hand, (25) yields Z Z ℓ/2 p n n+1 n−1 2 2 4n − (ℓ − 2t) dt = 2 n (ℓ − 2t) IK =

ℓ/2n

0

0

The estimate of E now follows by applying (28) and (43).

rn−1

p 1 − r2 dr .

Proof of Proposition 8. We have

where t1 =

s

    h1 (t) if 0 ≤ t ≤ t1     f∗∗ (t) = at + b if t1 ≤ t ≤ t2        h2 (t) if t2 ≤ t βγ α(α − β)

t2 =

r

s

αγ β(α − β)

a=2

αβγ α−β

b=

βγ . β−α

Moreover, the polar function is given by

f∗ (t) = and by Corollary 1 we get

      

E≥ Hence,

t2 4α t2 4β

if 0 ≤ t ≤ a − γ if t ≥ a

R ℓ/√π

√

π t)dt 4 R ℓ/√π t tf∗ ( 2 )dt 0

0

tf∗ (

.

√ π if ℓ ≤ 2 πa 4 πβ(α − β)ℓ4 E≥ 4α(α − β)ℓ4 − 128παβγ(α − β)ℓ2 + 1024π 2 α2 β 2 γ 2 E≥

20

√ if 2 πa < ℓ < 4a



E≥

π(α − β)ℓ4 − 128π(α − β)βγℓ2 + 4096παβ 2 γ 2 4(α − β)ℓ4 − 128π(α − β)βγℓ2 + 1024π 2 αβ 2 γ 2

Proof of Proposition 9. Let f be as in (20), then     0     f∗∗ (s) = s−1        ∞

if ℓ ≥ 4a . 

if s ∈ [0, 1] if s ∈ [1, 2] elsewhere ;

for all ε ∈ (0, 1) consider the function

fε (s) =

    0    

if s ∈ [0, 1 + ε]

s−1−ε

if s ∈ [1 + ε, 2]

1−ε        ∞

elsewhere .

Denote by J∗∗ e Jε the functionals associated to f∗∗ and fε Z Z J∗∗ (u) = [f∗∗ (|∇u|) − u] , Jε (u) = [fε (|∇u|) − u] . Ω

Ω

1 1 (s − 1 − ε)} and since 1−ε ≥ 1 + ε, by Theorem 1 in [10] we have that the Since fε (s) ≥ max{0, 1−ε 1,1 (unique) minimum uε of Jε over W0 (Ω) is given by uε (x) = (1 + ε)d(x, ∂Ω) and hence

4 I∗∗ ≥ min Jε = Jε (uε ) = − (1 + ε)4 , 1,1 3 W0 (Ω) where we have used the fact that fε ≤ f . Next, note that in this case we have WΩ = 1 + ε, α(t) = 8(1 + ε − t), ν(t) = (1 + ε − t)/2 and     s if s ∈ [0, 1] f∗ (s) =    2s − 1 if s ∈ [1, ∞) ; hence, by (25) we infer

IK = −

Z

1+ε

0

α(t)f∗ (ν(t)) dt = −4

We can now conclude that E=

Z

0

1+ε

4 (1 + ε − t)2 dt = − (1 + ε)3 . 3

1 IK ≥ . I∗∗ 1+ε 21



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