Some Inequalities Involving Perimeter and Torsional Rigidity

Abstract

We consider shape functionals of the form $$F_q(\Omega )=P(\Omega )T^q(\Omega )$$ F q ( Ω ) = P ( Ω ) T q ( Ω ) on the class of open sets of prescribed Lebesgue measure. Here $$q>0$$ q > 0 is fixed, $$P(\Omega )$$ P ( Ω ) denotes the perimeter of $$\Omega $$ Ω and $$T(\Omega )$$ T ( Ω ) is the torsional rigidity of $$\Omega $$ Ω . The minimization and maximization of $$F_q(\Omega )$$ F q ( Ω ) is considered on various classes of admissible domains $$\Omega $$ Ω : in the class $$\mathcal {A}_{all}$$ A all of all domains, in the class $$\mathcal {A}_{convex}$$ A convex of convex domains, and in the class $$\mathcal {A}_{thin}$$ A thin of thin domains.

A Shape Optimization Problem on Planar Sets with Prescribed Topology

“…Hence, by inequality (5.3) we have that m 1/2,0 , m 1/2,1 ≥ 3 −1/2 . Moreover, the computations made in [5] show that the value 3 −1/2 is asymptotically reached by a sequence of long thin rectangles, that are clearly in A 0 . Thus, being A 0 ⊂ A 1 , (5.5) holds.…”

“…In a previous paper [5], we considered the minimum and the maximum problem for F q (in every space dimension) in the classes…”

“…and apply (5.4) together with the Saint-Venant inequality (2.5). Finally, to prove that m q,k → 0 as k → ∞, it is enough to consider the sequence ( 1,n ) defined in Theorem 2.1 of [5], taking into account that 1,n ∈ A k for k big enough.…”

“…In a previous paper [5], we treated the problem above in every space dimension and, after discussing it for general open sets, we focused to the class of convex open sets.…”

An application of the continuous Steiner symmetrization to Blaschke-Santaló diagrams

“…Despite the fact that this is a very specific class, the result is enough to prove that the Blaschke-Santaló diagram corresponding to the pair λ(Ω), T (Ω) is between two graphs. Several other estimates for various kinds of quantities depending on a domain Ω are available in the recent literature; we refer the interested reader to [2,4,8,14,16] and to references therein.…”