Some Inequalities Involving Perimeter and Torsional Rigidity
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Some Inequalities Involving Perimeter and Torsional Rigidity Luca Briani1 · Giuseppe Buttazzo1
· Francesca Prinari2
Accepted: 10 October 2020 © The Author(s) 2020
Abstract We consider shape functionals of the form Fq () = P()T q () on the class of open sets of prescribed Lebesgue measure. Here q > 0 is fixed, P() denotes the perimeter of and T () is the torsional rigidity of . The minimization and maximization of Fq () is considered on various classes of admissible domains : in the class Aall of all domains, in the class Aconvex of convex domains, and in the class Athin of thin domains. Keywords Torsional rigidity · Shape optimization · Perimeter · Convex domains Mathematics Subject Classification 49Q10 · 49J45 · 49R05 · 35P15 · 35J25
1 Introduction In this paper, given an open set ⊂ Rd with finite Lebesgue measure, we consider the quantities P() = perimeter of ; T () = torsional rigidity of .
B
Giuseppe Buttazzo [email protected] http://www.dm.unipi.it/pages/buttazzo/ Luca Briani [email protected] Francesca Prinari [email protected] http://docente.unife.it/francescaagnese.prinari/
1
Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy
2
Dipartimento di Matematica e Informatica, Università di Ferrara, Via Machiavelli 30, 44121 Ferrara, Italy
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Applied Mathematics & Optimization
The perimeter P() is defined according to the De Giorgi formula P() = sup
div φ d x : φ ∈ Cc1 (Rd ; Rd ), φ L ∞ (Rd ) ≤ 1 .
The scaling property of the perimeter is for every t > 0
P(t) = t d−1 P()
and the relation between P() and the Lebesgue measure || is the well-known isoperimetric inequality: P(B) P() ≥ (d−1)/d || |B|(d−1)/d
(1.1)
where B is any ball in Rd . In addition, the inequality above becomes an equality if and only if is a ball (up to sets of Lebesgue measure zero). The torsional rigidity T () is defined as T () =
u dx
where u is the unique solution of the PDE
−u = 1 in , u ∈ H01 ().
(1.2)
Equivalently, T () can be characterized through the maximization problem T () = max
u dx
2
|∇u|2 d x
−1
: u ∈ H01 () \ {0} .
Moreover T is increasing with respect to the set inclusion, that is 1 ⊂ 2 ⇒ T (1 ) ≤ T (2 ) and T is additive on disjoint families of open sets. The scaling property of the torsional rigidity is T (t) = t d+2 T (),
for every t > 0,
and the relation between T () and the Lebesgue measure || is the well-known Saint-Venant inequality (see for instance [16,17]): T (B) T () ≤ . ||(d+2)/d |B|(d+2)/d
123
(1.3)
Applied Mathematics & Optimization
Again, the inequality above becomes an equality if and only if is a ball (up to sets of capacity zero). If we denote by B1 the unitary ball of Rd and by ωd its Lebesgue measure, then the solution of (1.2), with = B1 , is u(x) =
1 − |x|2 2d
which provides T (B1 ) =
ωd . d(d + 2)
(1.4)
We are interested in the problem of minimizing or maximizing quantities of the form P α ()T β () on some given cla
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