Inequalities between torsional rigidity and principal eigenvalue of the p-Laplacian

We consider the torsional rigidity and the principal eigenvalue related to the p-Laplace operator. The goal is to find upper and lower bounds to products of suitable powers of the quantities above in various classes of domains. The limit cases p=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=1$$\end{document} and p=∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=\infty $$\end{document} are also analyzed, which amount to consider the Cheeger constant of a domain and functionals involving the distance function from the boundary.


Introduction
In this paper we consider the problem of minimizing or maximizing the quantity λ α p ( )T β p ( ) on the class of open sets ⊂ R d having a prescribed Lebesgue measure, where α, β are two real parameters, and λ p ( ), T p ( ) are respectively the principal eigenvalue and the torsional rigidity, which are defined below, relative to the p-Laplace operator In all the paper, we use the following notation: • p is the conjugate exponent of p given by p := p/( p − 1); • P( ) is the distributional perimeter of in the De Giorgi sense, defined by When = B 1 , the solution w p to the boundary problem (1.2) is explicit and given by w p (x) = 1 − |x| p p d 1/( p−1) (1.5) which leads to The p-principal eigenvalue λ p ( ) is defined through the Rayleigh quotient Equivalently, λ p ( ) denotes the least value λ such that the nonlinear PDE − p u = λ|u| p−2 u in , u ∈ W 1, p 0 ( ), has a nonzero solution; we recall that in dimension 1 we have (see for instance [22]) while in higher dimension the following estimate holds true, see [20,Theorem 3.1]: It is easy to see that the two quantities above scale as λ p (t ) = t − p λ p ( ), T p (t ) = t p+d( p−1) T p ( ). (1.9) By using a symmetrization argument and the so-called Pólya-Szegö principle (see [18]) it is possible to prove that balls maximize T p (respectively minimize λ p ) among all sets of prescribed Lebesgue measure, which can be written in a scaling free form as (1.10) where B is any ball in R d . The inequalities (1.10) are known respectively as Faber-Krahn inequality and Saint-Venant inequality. Moreover, we have: To prove (1.11) and (1.12) it is enough to take into account of the scaling properties (1.9) and use the fact that if is the disjoint union of a family of open sets i with i ∈ I , then Then, choosing n as the disjoint union of n balls with measure 1/n each and taking the limit as n → ∞, gives Thus, a characterization of inf / sup of the quantity λ α p ( )T β p ( ), among the domains ⊂ R d with unitary measure, when α = 0 or β = 0 or αβ < 0, follows by (1.10), (1.11) and (1.12).
It remains to consider the case α > 0 and β > 0. Setting q = β/α > 0 we can limit ourselves to deal with the quantity Using the scaling properties (1.9) we can remove the constraint of prescribed Lebesgue measure on by normalizing the quantity λ p ( )T q p ( ), multiplying it by a suitable power of | |. We then end up with the scaling invariant shape functional that we want to minimize or maximize over the class of open sets ⊂ R d with 0 < | | < ∞. The limit cases, when p = 1 and p = +∞, are also meaningful. When p → 1 the quantities λ p ( ) and T p ( ) are related to the notion of Cheeger constant h( ), see definition (5.1). In particular we obtain as a natural "limit" functional F 1,q ( ) = h( )| | 1/d 1−q whose optimization problems are well studied in the literature. Concerning the case p = +∞, we show that the family F 1/ p p,q pointwise converges, as p → ∞, to the shape functional and we study the related optimization problems in the class of all domains and in that of convex domains. The study of the functionals F p,q has been already considered in the literature. The case when p = 2 has been extensively discussed in [30][31][32][33] (see also [9]) and our results can be seen as natural extensions. An interesting variant, where the shape functionals involve the L ∞ norm of the function w p solution of (1.2) has been considered in [19] in the case p = 2.
The paper is organized as follows. In the first three sections we study the optimization problems for F p,q , when 1 < p < ∞ and in different classes of domains. More precisely: in Sect. 2 we consider the class of all open sets of R d with finite Lebesgue measure, in Sect. 3 we consider the class of bounded convex open sets and in Sect. 4 that of thin domains which will be suitable defined. The analysis of the optimization problems in the extremal cases (respectively when p = 1 and p = +∞) are contained in Sects. 5 and 6. Finally Sect. 7 contains a list of several open problems which we believe may be interest for future researches. For the sake of completeness we add an appendix section devoted to clarify the assumptions we use for the limit case of Sect. 6.

Optimization for general domains
The crucial inequality to provide a lower bound to F p,q is the Kohler-Jobin inequality, first proved for p = 2 in [25,26] and then for a general p in [7], which asserts that balls minimize principal frequency among all sets of prescribed torsional rigidity. More precisely we have (2.1) where B is any ball in R d .

Proof
We denote for the sake of brevityq = p /( p + d). Notice that By Kohler-Jobin inequality (2.1) and Saint-Venant inequality (1.10) we get the thesis for 0 < q ≤q. Now, let be the disjoint union of B 1 and N disjoint balls of radius ε ∈ (0, 1]. Taking into account (1.13) we have Taking now N ε d+ p/( p−1) = 1 gives which vanishes as ε → 0 as soon as q >q.
In dealing with the supremum of F p,q a natural threshold arises from the Polya inequality whose brief proof we recall.

Proposition 2.2 For every
Proof Let w be the solution to (1.2). By the definition of λ p ( ) and by Hölder inequality we have The conclusion follows by (1.3).
Proof Let N be the disjoint union of N balls of unitary radius. By (1.13) we have Taking the limit as N → ∞ we have F p,q ( N ) → +∞ whenever 0 < q < 1. Moreover, when q ≥ 1, using Proposition 2.2 and the Saint-Venant inequality (1.10), we have which concludes the proof.
When p = 2 and q = 1 the upper bound given in the Proposition 2.2 is sharp as first proved in [32]. Using the theory of capacitary measures, a shorther proof was given in [30]. The latter extends, naturally, to the case when p ≤ d and q = 1 as we show in the proposition below.
Proof By repeating the construction made in [13] (see also Remark 4.3.11 and Example 4.3.12 of [10], and references therein) we have that for every p-capacitary measure μ (that is a nonnegative Borel measure, possibly taking the value +∞, with cap p (E) = 0 ⇒ μ(E) = 0) there exists a sequence ( n ) of (smooth) domains such that Taking the ball B 1 and μ c = c dx B 1 for every c > 0, we have Clearly λ p (μ c ) = c + λ p (B 1 ). Now, consider for δ > 0 the function We have By letting c → +∞ and then δ ↓ 0 we obtain the thesis.

Optimization in convex domains
We now deal with the optimization problems in the class of convex domains. Notice that adding in (1.11) and in (1.12) a convexity constraint on the admissible domains does not change the values of inf and sup. To see this one can take a unit measure normalization of the following convex domains (slab shape) being A a convex d −1 dimensional open set with finite d −1 dimensional measure and use the following Lemma, which will be proved in a slightly more general version in Proposition 4.1 of Sect. 4.

Lemma 3.1 Let A ⊂ R d−1 be a bounded open set and let
Then we have where π p is given in (1.7). In addition, as ε → 0, we have By using the previous lemma we have also Hence the only interesting optimization problems in the class of convex domains are the following ones We denote respectively by m p,q and M p,q the two quantities above.
With the convexity constraint, the so called Hersch-Protter inequality holds (for a proof see for instance [8,22]): Moreover, the p-torsional rigidity of a bounded convex open set satisfies the following generalization of Makai inequality (see [28,Theorem 4.3]): Both inequalities are sharp and the equality is asymptotically attained by taking, for instance, the sequence C A,ε of Lemma 3.1. Taking advantage of (3.2) and (3.3) we can show the following bounds.
Proof Let ⊂ R d be any bounded convex set. Without loss of generality, we can suppose 0 ∈ . We denote by j (x) the Minkowski functional (also known as gauge function) of , that is The main properties of j are summarized in Lemma 2.3 of [9]. In particular we recall that j is a convex, Lipschitz, 1-positively homogeneous function, H d−1 -a.e. differentiable in ∂ , and satisfies being ν (x) the outer normal unit versor at the point x ∈ ∂ . We consider By using coarea formula (3.6) and the divergence theorem it is easy to prove that where the last inequality follows by the fact that see Lemma 2.1 in [9]. Hence by testing (1.1) with the function u we have Taking into account (3.2), we obtain To prove the second inequality we use (3.3) and the inequality to obtain which, together with Proposition 2.2, gives (3.5). Remark 3. 3 We stress here that inequality (3.7) has been already proved in [12,28]. However, their results are given in the more general anisotropic setting where the proofs become more involved.

Remark 3.4
Combining inequalities (3.5) and (1.8), we obtain Thereby, as soon as p is large enough, we have M p,1 < 1.
When q = 1 the values m p,q and M p,q are achieved by some optimal domains, as shown in the next theorem.
Moreover, there exist convex domains m p,q and M p,q such that Proof The first part follows at once using Saint-Venant inequality (1.10) together with the equality Concerning the existence of optimal convex domains, we can repeat the argument used in [30]. First we notice that Moreover, any convex open set contains a two-sided cone with base area equal to a d − 1 dimensional disk of radius ρ( ) and total height equal to diam( ), hence Thus, suppose 0 < q < 1 and let ( n ) be a minimizing sequence for F p,q made up of convex domains. By scaling invariance we can suppose ρ( n ) = 1. For n large enough we have Combining the last estimate with (3.9) we have sup n diam( n ) < +∞.
Hence, up to translations, the whole sequence ( n ) is contained in a compact set and we can extract a subsequence ( n k ) which converges in both Hausdorff and co-Hausdorff distance to some m p,q (see [17], for details about these convergences). Using the well-known continuity properties for λ p , T p and Lebesgue measure with respect to Hausdorff metrics on the class of bounded convex sets, we conclude that If q > 1 we can follow the similar strategy and consider a maximizing sequence ( n ) with unitary inradius. By (3.8) and (3.2) we have, for n large enough, , which, thanks to (3.9), implies again sup n diam( n ) < +∞.

Optimization for thin domains
In this section we study the optimization problems for the functionals F p,1 in the class of the so-called thin domains, which has been already considered in [30] for p = 2. By a thin domain we mean a family of open sets ( ε ) ε>0 , of the form Moreover we say that the thin domain ( ε ) ε>0 is convex if the corresponding domain A is convex and the local thickness function h is concave. The volume of ε is clearly given by while we can compute the behaviour (as ε → 0) of T p ( ε ) and λ p ( ε ) by means of the following proposition (in the case p = 2 a more refined asymptotics can be found in [5,6]). From now on, we write the norms · p , omitting the dependence on the domain.

2)
where π p is given in (1.7). In addition, as ε → 0, we have Proof First we deal with inequalities (4.2). Let φ ∈ C ∞ c ( ε ); since the function φ(x, ·) is admissible to compute T p (εh − (x), εh + (x)), by (1.1) we obtain Taking into account (1.5) we have and thus, integrating on A in (4.4), we deduce Hölder inequality now gives Since φ is arbitrary and p + 1 = (2 p − 1)/( p − 1), we conclude that To get the second inequality in (4.2) we notice that, by (1.6), for every φ ∈ C ∞ c ( ε ) we have |∇φ(x, ·)| p dy. Since integrating on A and minimizing on φ, we obtain We now prove (4.3) for T p ( ε ). To this end we consider the function where w denotes the solution to (1.2) when = (0, 1) and d = 1 (for the sake of brevity we omit the dependence on p). Notice that w ε (x, ·) solves (1.2) in the interval (εh − (x), εh + (x)). In particular, by using (1.3) and (4.5), we have A simple computation shows that In particular By exploiting the change of variable z = y−εh − (x) εh(x) in the latter identity, we conclude that, as ε → 0, Moreover, by using basically the same argument as above, we have also that ε |∇ (w ε (x, y)φ(x))| p dxdy ≈ ε ∇ y w ε (x, y) p |φ(x)| p dxdy, as ε → 0. (4.7) By combining (4.6) and (4.7) we obtain Finally, by taking φ which approximates 1 A in L p (A) in the right hand side of the inequality above, we conclude that and the thesis is achieved taking into account (4.2). The asymptotics in (4.3) for λ p can be treated with similar arguments.
Actually, by means of a density argument, we can drop the regularity assumptions on h + and h − and extend the formulas (4.2) and (4.3) to any family ( ε ) ε>0 defined as in (4.1), with h + and h − bounded and measurable functions. We thus have: We then define the functional F p,1 on the thin domain ( ε ) ε>0 associated with the d − 1 dimensional domain A and the local thickness function h by (4.8) Our next goal is to give a complete solution to the optimization problems for the functional F p,1 in the class of convex thin domains. To this aim we recall the following result (see Theorem 6.2 in [4]). and such that h L ∞ (E) = 1, it holds

In addition, equality occurs if E is a ball of radius 1 and h(x) = 1 − |x|.
As an application we obtain the following lemma, which generalizes Proposition 5.2 in [30].

Lemma 4.3 Let E ⊂ R N be a bounded open convex set and let 1 < r < ∞. Then for every concave function h
(4.10)

In addition, the inequality above becomes an equality when E is a ball of radius 1 and h(x) = 1 − |x|.
Proof First we assume that E ⊂ R N is a ball centered in the origin and h is a radially symmetric, decreasing, concave function h : E → [0, 1] with h(0) = 1. Then h satisfies (4.9) and we can apply Theorem 4.2 with s = 1, to get In order to get the inequality (4.10) in the general case, let h * : B → [0, 1] be the radially symmetric decreasing rearrangement of h, defined on the ball B centered at the origin and with the same volume as E. The standard properties of the rearrangement imply that Moreover, it is well-known that h * is concave. Since h * satisfies all the assumptions of the previous case, we get that h * (hence h) satisfies (4.10). Finally, it is easy to show that the inequality in (4.10) holds as an equality for every cone function h(x) = 1 − |x|.
We are now in a position to show the main theorem of this section.

In addition, the first equality is attained taking h(x) to be any constant function while the second equality is attained taking as A the unit ball and as the local thickness function h(x) the function 1 − |x|.
Proof Using definition (4.8) it is straightforward to prove that and to verify that, if h is constant, then Finally, by applying Lemma 4.3 with N = d − 1, E = A and r = p + 1 we obtain the second part of the theorem.

The case p = 1
Given an open set ⊂ R d with finite measure we define its Cheeger constant h( ) as where E means thatĒ ⊂ . Notice that in definition (5.1), thanks to a well-known approximation argument, we can evaluate the quotient P(E)/|E| among smooth sets which are compactly contained in . Following [24] we have lim p→1 λ p ( ) = h( ), (5.2) for every open set with finite measure.

Remark 5.1
A caveat is necessary at this point: the usual definition of Cheeger constant as is not appropriate to provide the limit equality (5.2), which would hold only assuming a mild regularity on (for instance, it is enough to consider which coincides with its essential interior, see [27]). To prove that in general h( ) = c( ), one can consider = By the same argument used in [24] to prove (5.2) we can show that T p ( ) → h −1 ( ) as p → 1. For the sake of completeness we give the short proof below (see also Theorem 2 in [11]).

Proposition 5.2 Let ⊂ R d be an open set with finite measure. Then, as p → 1,
Proof First we notice that for any u ∈ C ∞ c ( ), it holds: Indeed, by assuming without loss of generality that u ≥ 0, by coarea formula and Cavalieri's principle, we have that Since the sets {u > t} are smooth for a.e. t ∈ u( ), (5.4) follows straightforwardly from (5.1). By combining (1.1) with (5.4) and Hölder inequality we then have (5.5) for any 1 < p < ∞. Now, let E k be a sequence of smooth sets of such that P(E k )/|E k | → h( ). For a fixed k and any ε > 0 small enough, we can find a Lipschitz function v compactly supported in , such that, where E k,ε = E k + B ε . Hence, by (1.1), we have By first passing to the limit as p → 1, and then as ε → 0 we get which implies, as k → ∞, Finally we conclude, taking into account (5.5).
The limits (5.2) and (5.3) justify the following definition: Notice that F p,q ( ) → F 1,q ( ) as p → 1. In the next proposition we solve the optimization problems for F 1,q in the class of general domains and in that of convex domains.

Proposition 5.3 For
For q > 1, we have Proof The minimality (respectively maximality) of B, for 0 < q < 1 (respectively for q > 1), is an immediate consequence of the well known inequality which holds for any ⊂ R d with finite measure. To prove the other cases we use the inequality which holds for any ⊂ R d open, bounded, convex set (see [8,Corollary 5.2]). Then taking C A,ε as in (3.1) we get from which the thesis easily follows.

The case p = ∞
The limit behaviour of the quantities λ p ( ), T p ( ), as p → ∞, are well known for bounded open sets ⊂ R d : in [14] and in [21] the authors prove that while, following [3] (see also [23]) it holds w p → d uniformly in , which implies Actually, in all these results, the boundedness assumption on is not needed, as it is only used to provide the compactness of the embedding W 1, p 0 ( ) into the space C 0 ( ) defined as the completion of C c ( ) with respect to the uniform convergence. Indeed, this holds under the weaker assumption that | | < +∞ (see Appendix A for more details and for a -convergence point of view of both limits (6.1) and (6.2)). According to (6.1) and to (6.2) we define the shape functional F ∞,q as Moreover, both inequalities are sharp. In particular For its proof, we recall the following result, for which we refer to [4,15].
where the constant C p,q is given by

In addition, the inequality above becomes an equality when E is a ball of radius 1 and
Proof of Proposition 6.1 In order to prove the right-hand side inequality in (6.4), for every t ≥ 0, we denote by (t) the interior parallel set at distance t from ∂ , i.e. Then for a.e. t ∈ (0, ρ( )) there exists the derivative A (t) and it coincides with −L(t). Moreover, being a convex set, L is a monotone decreasing function. Then A is a convex function such that A(ρ( )) = 0 and A(0) = | |. As a consequence we have Integrating by parts, we get The value 1/2 is asymptotically attained in (6.4) by considering a sequence of slab domains as ε → 0. Indeed, we have ρ( ε ) = ε/2 and | ε | = ε. Being Now we prove the left-hand side inequality in (6.4). Since is convex, the distance function d is concave (see [2]); then, applying Theorem (6.2) to d , we obtain Since (6.5) is an identity when = B, C p,1 satisfies As p → ∞ in (6.5), we obtain which is an equality when = B.

Remark 6.3
The proof of the right-hand side of (6.4) relies on the convexity properties of the function A(t). In the planar case a general result, due to Sz. Nagy (see [29]), ensures that, if is any bounded k-connected open set, (i.e. c has k bounded connected components), then the function is convex. Therefore, for such an , with the same argument as above it is easy to prove that Hence, it is interesting to notice how, even when k = 0, 1, the upper bound given in (6.4) remains sharp. In other words, in the maximization of F ∞,1 on planar domains, there is no gain in replacing the class of convex domains by the larger one consisting of simply-connected domains or even more in allowing to have a single hole.
In the general case q = 1 the optimization problems for the functional F ∞,q defined in (6.3) are studied below.
If q > 1, then while if q > 1, using again (6.4) we have Finally, let ε be the slab domain as in Proposition 6.1. Then from which the thesis is achieved.
If we remove the convexity assumption on the admissible domains , the picture is similar to those provided by Proposition 2.1 and Proposition 2.3. More precisely, if q > 1/(d +1), the minimization problem for F ∞,q is ill posed. When q > 1, this follows directly by Corollary 6.4, while, in the case 1/(d + 1) < q ≤ 1, by taking n to be the union of n disjoint balls of radius r j = j −1/d with j = 1, . . . , n, one can verify that F ∞,q ( n ) → 0, as n → ∞. On the contrary, when q ≤ 1/(d + 1), the minimum of F ∞,q is attained by any ball. Indeed, since 1/(d + 1) ≤ p /( p + d) for every p > 1, by using Proposition 2.1, we have Hence, passing to the limit as p → +∞, we obtain Concerning the upper bound, Corollary 6.4 implies that the maximization problem is ill posed in the case q < 1, while, when q ≥ 1, using (6.6), we obtain However, working with general domains provides an upper bound larger than in (6.4); for instance, in the two-dimensional case, taking as N the unit disk where we remove N points as in Fig. 1, gives where E is the regular exagon with unitary sides centered at the origin, as an easy calculation shows.

Further remarks and open questions
Several interesting problems and questions about the shape functionals F p,q are still open; in this section we list some of them.  [30]). It seems natural to conjecture that the right sharp inequalities are those given in Theorem 4.4 for F p,1 on the class of thin domain.

Problem 4
In the two-dimensional case with p = ∞ we have seen that the domains N in Fig. 1 give the asymptotic value 1 3 + log 3 4 for the shape functional F ∞,1 . It would be interesting to prove (or disprove) that this number is actually the supremum of F ∞,1 ( ) when varies in the class of all bounded open two-dimensional sets. In addition, in the case of a dimension d > 2, it is not clear how a maximizing sequence ( n ) for F ∞,1 has to be.
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Appendix A
We devote this Appendix to give a proof of the known asymptotics (6.1) and (6.2) by means of -convergence when 0 < | | < +∞. We recall that if p > d and ⊂ R d is any (possibly unbounded) open set with finite measure we have the compact embedding: A quick proof of it can be obtained by combining the Gagliardo-Niremberg inequality in together with the well known facts that the inclusion W Then, as p → ∞, the sequence p -converges to ∞ with respect to the L 1 -convergence.
Proof Let p n → ∞. The -lim sup inequality is trivial since, for every u ∈ W 1,∞ 0 ( ) with u ∞ = 1, the sequence u p n = u −1 p n u converges to u in L 1 and satisfies u n p n = 1, lim sup n→∞ p n (u n ) = ∞ (u).
To prove the -lim inf inequality, without loss of generality, let u ∈ L ∞ ( ), (u p n ) ⊆ W 1, p n 0 ( ) be such that u p n → u in L 1 ( ), u p n p n = 1, and lim inf n→∞ p n (u p n ) = C < ∞. Since for every q ≥ 1 and for n large enough it holds Du p n q ≤ | | 1/q−1/ p n p n (u p n ) we get that u p n → u in L q ( ), u ∈ W 1,q 0 ( ) and The thesis follows by the arbitrariness of the sequence p n .
Next Proposition generalizes Proposition 2.1 in [16]. for every q ≥ 1, which implies w ∞ ≤ 1. Now let J p be the functional defined in (1.4) and J ∞ be the functional given by Since the functional u → u dx is continuous with respect to the L 1 -convergence, thanks to Proposition A.3, we have that lim n→∞ J p n (w p n ) = J ∞ (w ∞ ) = min Moreover, by using (A.5) we have w ∞ (x) ≤ d (x). In addition, since d ∈ W 1,∞ 0 ( ), we have also J ∞ (w ∞ ) ≤ J ∞ (d ), i.e. (w ∞ − d ) dx ≥ 0. Hence d = w ∞ . By the arbitrariness of the sequence p n , we get that w p → d uniformly as p → ∞.