Abstract
We consider shape functionals of the form \(F_q(\Omega )=P(\Omega )T^q(\Omega )\) on the class of open sets of prescribed Lebesgue measure. Here \(q>0\) is fixed, \(P(\Omega )\) denotes the perimeter of \(\Omega \) and \(T(\Omega )\) is the torsional rigidity of \(\Omega \). The minimization and maximization of \(F_q(\Omega )\) is considered on various classes of admissible domains \(\Omega \): in the class \(\mathcal {A}_{all}\) of all domains, in the class \(\mathcal {A}_{convex}\) of convex domains, and in the class \(\mathcal {A}_{thin}\) of thin domains.
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1 Introduction
In this paper, given an open set \(\Omega \subset \mathbb {R}^d\) with finite Lebesgue measure, we consider the quantities
The perimeter \(P(\Omega )\) is defined according to the De Giorgi formula
The scaling property of the perimeter is
and the relation between \(P(\Omega )\) and the Lebesgue measure \(|\Omega |\) is the well-known isoperimetric inequality:
where B is any ball in \(\mathbb {R}^d\). In addition, the inequality above becomes an equality if and only if \(\Omega \) is a ball (up to sets of Lebesgue measure zero).
The torsional rigidity \(T(\Omega )\) is defined as
where u is the unique solution of the PDE
Equivalently, \(T(\Omega )\) can be characterized through the maximization problem
Moreover T is increasing with respect to the set inclusion, that is
and T is additive on disjoint families of open sets. The scaling property of the torsional rigidity is
and the relation between \(T(\Omega )\) and the Lebesgue measure \(|\Omega |\) is the well-known Saint-Venant inequality (see for instance [16, 17]):
Again, the inequality above becomes an equality if and only if \(\Omega \) is a ball (up to sets of capacity zero). If we denote by \(B_1\) the unitary ball of \(\mathbb {R}^d\) and by \(\omega _d\) its Lebesgue measure, then the solution of (1.2), with \(\Omega =B_1\), is
which provides
We are interested in the problem of minimizing or maximizing quantities of the form
on some given class of open sets \(\Omega \subset \mathbb {R}^d\) having a prescribed Lebesgue measure \(|\Omega |\), where \(\alpha ,\beta \) are two given exponents. Similar problems have been considered for shape functionals involving:
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the torsional rigidity and the first eigenvalue of the Laplacian in [2, 3, 6, 8, 11, 19, 20, 22];
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the torsional rigidity and the Newtonian capacity in [1];
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the perimeter and the first eigenvalue of the Laplacian in [14];
The case \(\beta =0\) reduces to the isoperimetric inequality, and we have, denoting by \(\Omega ^*_m\) a ball of measure m,
Similarly, in the case \(\alpha =0\), the Saint Venant inequality yields
while
Indeed if we choose \(\Omega _n=\cup _{k=1}^n B_{n,k}\) where \(B_{n,k}\) are disjoint balls of measure m/n each, we get for every \(n\in \mathbb {N}\)
The case when \(\alpha \) and \(\beta \) have a different sign is also immediate; for instance, if \(\alpha >0\) and \(\beta <0\) we have from (1.1) and (1.3)
and similarly, if \(\alpha <0\) and \(\beta >0\) we have
The cases we will investigate are the remaining ones; with no loss of generality we may assume \(\alpha =1\), so that the optimization problems we consider are for the quantities
In order to remove the Lebesgue measure constraint \(|\Omega |=m\) we consider the scaling free functionals
In the following sections we study the minimization and the maximization problems for the shape functionals \(F_q\) on various classes of domains. More precisely we consider the cases below.
The class of all domains \(\Omega \) (nonempty)
will be considered in Sect. 2; we show that for every \(q>0\) both the maximization and the minimization problems for \(F_q\) on \(\mathcal {A}_{all}\) are ill posed.
The class of convex domains \(\Omega \)
will be considered in Sect. 3; we show that for \(0<q<1/2\) the maximization problem for \(F_q\) on \(\mathcal {A}_{convex}\) is ill posed, whereas the minimization problem is well posed. On the contrary, when \(q>1/2\) the minimization problem for \(F_q\) on \(\mathcal {A}_{convex}\) is ill posed, whereas the maximization problem is well posed. In the threshold case \(q=1/2\) the precise value of the infimum of \(F_{1/2}\) is provided; concerning the precise value of the supremum of \(F_{1/2}\) an interesting conjecture is stated. At present, the conjecture has been shown to be true in the case \(d=2\), while the question is open in higher dimensions.
The class of thin domains \(\mathcal {A}_{thin}\), suitably defined, will be considered in Sect. 4. If h(s) represents the asymptotical local thickness of the thin domain as s varies in a \(d-1\) dimensional domain A, the maximization of the functional \(F_{1/2}\) on \(\mathcal {A}_{thin}\) reduces to the maximization of a functional defined on nonnegative functions h defined on A; this allows us to prove the conjecture for any dimension d on the class of thin convex domains.
2 Optimization in the Class of All Domains
In this section we show that the minimization and the maximization problems for the shape functionals \(F_q\) are both ill posed, for every \(q>0\).
Theorem 2.1
There exist two sequences \((\Omega _{1,n})\) and \((\Omega _{2,n})\) of smooth domains such that for every \(q>0\) we have
In particular, we have
Proof
In order to show the \(\sup \) equality it is enough to take as \(\Omega _{2,n}\) a perturbation of the unit ball \(B_1\) such that
Then we have
where we used the monotonicity of the torsional rigidity. Then
In order to prove the \(\inf \) equality we take as \(\Omega _{c,{\varepsilon }}\) the unit ball \(B_1\) from which we remove a periodic array of holes; the centers of two adjacent holes are at distance \({\varepsilon }\) and the radii of the holes are
where c is a positive constant. It is easy to see that, as \({\varepsilon }\rightarrow 0\), we have
Concerning the torsion \(T(\Omega _{c,{\varepsilon }})\), we have (see [10])
where \(u_c\) is the nonnegative function which solves
being \(K_c\) the constant
Since for every \(c>0\) we have that
we get that
Therefore, a diagonal argument allows us to construct a sequence \((\Omega _{1,n})\) such that
which concludes the proof. \(\square \)
3 Optimization in the Class of Convex Domains
In this section we consider only domains \(\Omega \) which are convex. A first remark is in the proposition below and shows that in some cases the optimization problems for the shape functional \(F_q\) is still ill posed.
Proposition 3.1
We have
Proof
Let A be a smooth convex \(d-1\) dimensional set and for every \({\varepsilon }>0\) consider the domain \(\Omega _{\varepsilon }\in \mathcal {A}_{convex}\) given by
We have (for the torsion asymptotics see for instance [2])
so that
Letting \({\varepsilon }\rightarrow 0\) achieves the proof. \(\square \)
We show now that in some other cases the optimization problems for the shape functional \(F_q\) is well posed. Let us begin to consider the case \(q=1/2\).
Proposition 3.2
We have
and the infimum is asymptotically reached by domains of the form
as \({\varepsilon }\rightarrow 0\), where A is any \(d-1\) dimensional convex set.
Proof
Thanks to a classical result by Polya ( [21], see also Theorem 5.1 of [11]) it holds
Then
for any bounded open convex set. Taking into account (3.1), we get (3.2). \(\square \)
Concerning the supremum of \(F_{1/2}(\Omega )\) in the class \(\mathcal {A}_{convex}\) we can only show that it is finite.
Proposition 3.3
For every \(\Omega \in \mathcal {A}_{convex}\) we have
Proof
By the John’s ellipsoid Theorem [18], there exists an ellipsoid that, without loss of generality, we may assume centered at the origin,
such that \(E_a\subset \Omega \subset dE_a\). Then we have
Since the solution of (1.2) for \(E_a\) is given by
we obtain
while
To estimate \(P(E_a)\) we notice that \(E_a\) is contained in the cuboid \(Q_a=\prod _1^d]-a_i,a_i[\), so that
Combining these formulas we have from (3.4)
and finally, by Jensen inequality,
as required. \(\square \)
On the precise value of \(\sup \big \{F_{1/2}(\Omega ):\Omega \in \mathcal {A}_{convex}\big \}\) we make the following conjecture.
Conjecture 3.4
We have
and it is asymptotically reached by taking for instance
as \({\varepsilon }\rightarrow 0\), where A is the unit ball in \(\mathbb {R}^{d-1}\).
Remark 3.5
We recall that Conjecture 3.4 has been shown to be true in the case \(d=2\) (see [21, 23], and the more recent paper [12]). In Sect. 4 we prove the conjecture above for every \(d\ge 2\) in the class of convex thin domains.
We show now that for \(F_q\) in the class \(\mathcal {A}_{convex}\) the minimization problem is well posed when \(q<1/2\) and the maximization problem is well posed when \(q>1/2\). From the bounds obtained in Propositions 3.2 and 3.3 we can prove the following results.
Proposition 3.6
We have
Proof
We have
Hence it is enough to apply the bounds (3.2) and (3.3), together with the Saint-Venant inequality (1.3) to get that for every \(\Omega \in \mathcal {A}_{convex}\)
By the expression (1.4) for T(B) we conclude the proof. \(\square \)
We now prove the existence of a convex minimizer when \(q<1/2\) and of a convex maximizer when \(q>1/2\).
Theorem 3.7
There exists a solution for the following optimization problems:
Proof
Suppose \(q<1/2\) and consider \((\Omega _n)\) a minimizing sequence for \(F_q(\Omega )\). By the John’s ellipsoid Theorem we can assume that there exists a sequence of ellipsoids \(E_{a_n}\) such that
By rotations, translations and scaling invariance of \(F_q\) we can assume without loss of generality that
Observe that this implies that the diameter of \(\Omega _n\) is uniformly bounded in n. We claim that
where c is a positive constant. Then the proof is achieved by extracting a subsequence \((\Omega _{n_k})\) which converges both in the sense of characteristic functions and in the co-Hausdorff metric to some open, non empty, convex, bounded set \(\Omega ^-\) and by using the continuity properties of torsional rigidity, perimeter and volume (see for instance, [7, 17]).
To prove the claim we use a strategy similar to the one already used in the proof of Proposition 3.3. Let \(Q_{a_n}\) be the cuboid \(\prod _{i=1}^d]-a_{in},a_{in}[\). Since
we have, for n large enough,
An explicit computation shows
Observe that, by Cauchy–Schwarz inequality,
while for the last term it holds
Therefore, putting together (3.5)–(3.7) and using the fact that \(q<1/2\) we obtain that, if n is large enough, the sequence \(a_{1n}\) must be greater than some positive constant c, which proves the claim.
The case \(q>1/2\) can be proved in a similar way. If \((\Omega _n)\) is a maximizing sequence for \(F_q(\Omega )\) and \(E_{a_n}\) are ellipsoids such that \(E_{a_n}\subset \Omega _n\subset dE_{a_n}\), we have
If \(Q_{a_n}\) is the cuboid \(\prod _{i=1}^d]-a_{in},a_{in}[\) we have \(E_{a_n}\subset Q_{a_n}\), so that
Hence (3.8) implies, for a suitable constant \(C_{q,d}\) depending only on q and on d,
where in the last inequality we used the Cauchy–Schwarz inequality (3.6). Finally, since \(a_{in}\le a_{dn}=1\), we obtain
and, since \(q>1/2\), the conclusion follows as in the previous case. \(\square \)
4 Optimization in the Class of Thin Domains
In this section we consider the class of thin domains, that we define below through the families of domains
where \({\varepsilon }\) is a small positive parameter, A is a (smooth) domain of \(\mathbb {R}^{d-1}\), and \(h_-,h_+\) are two given (smooth) functions. We denote by h(s) the local thickness
and we assume that \(h(s)\ge 0\). More precisely, we call thin domain a family \((\Omega _{\varepsilon })_{{\varepsilon }>0}\) as above; in other words a thin domain is characterized by the \(d-1\) dimensional domain A and by the local thickness function h.
The following asymptotics hold for the quantities we are interested to (for the torsional rigidity we refer to [5]):
which together give the asymptotic formula when \(q=1/2\)
where we use the notation
We then define the functional \(F_{1/2}\) on the thin domain \((\Omega _{\varepsilon })_{{\varepsilon }>0}\) associated with the \(d-1\) dimensional domain A and the local thickness function h by
By Hölder inequality we have
and the value \(3^{-1/2}\) is actually reached by taking the local thickness function h constant, which corresponds to \(\Omega _{\varepsilon }\) a thin slab.
A sharp inequality from above is also possible for \(F_{1/2}(A,h)\), if we restrict the analysis to convex domains, that is to local thickness functions h which are concave. The following result will be used, for which we refer to [4, 15].
Theorem 4.1
Let \(1\le p\le q\). Then for every convex set A of \(\mathbb {R}^N\) \((N\ge 1)\) and every nonnegative concave function f on A we have
where the constant \(C_{p,q}\) is given by
In addition, the inequality above becomes an equality when A is a ball of radius 1 and \(f(x)=1-|x|\).
We are now in a position to prove the Conjecture 3.4 for convex thin domains.
Theorem 4.2
If \((\Omega _{\varepsilon })_{{\varepsilon }>0}\) is a thin convex domains given by (4.1), we have
In addition, the inequality above becomes an equality taking for instance as A the unit ball of \(\mathbb {R}^{d-1}\) and as the local thickness h(s) the function \(1-|s|\).
Proof
Since the local thickness function h is concave, by Theorem 4.1 with \(N=d-1\), \(q=3\), \(p=1\), we obtain
so that
as required. Finally, an easy computation shows that in (4.3) the inequality becomes an equality if A is the unit ball of \(\mathbb {R}^{d-1}\) and \(h(s)=1-|s|\). \(\square \)
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Acknowledgements
The work of GB is part of the project 2017TEXA3H “Gradient flows, Optimal Transport and Metric Measure Structures” funded by the Italian Ministry of Research and University. The authors are member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
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Briani, L., Buttazzo, G. & Prinari, F. Some Inequalities Involving Perimeter and Torsional Rigidity. Appl Math Optim 84, 2727–2741 (2021). https://doi.org/10.1007/s00245-020-09727-7
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DOI: https://doi.org/10.1007/s00245-020-09727-7