A shape optimization problem on planar sets with prescribed topology
aa r X i v : . [ m a t h . O C ] J a n A SHAPE OPTIMIZATION PROBLEM ON PLANAR SETS WITHPRESCRIBED TOPOLOGY
LUCA BRIANI, GIUSEPPE BUTTAZZO, AND FRANCESCA PRINARI
Dedicated to Franco Giannessi for his 85th birthday
Abstract.
We consider shape optimization problems involving functionals depend-ing on perimeter, torsional rigidity and Lebesgue measure. The scaling free costfunctionals are of the form P (Ω) T q (Ω) | Ω | − q − / and the class of admissible do-mains consists of two-dimensional open sets Ω satisfying the topological constraintsof having a prescribed number k of bounded connected components of the comple-mentary set. A relaxed procedure is needed to have a well-posed problem and weshow that when q < / q > / q = 1 / k = 0 and k = 1. Keywords: torsional rigidity, shape optimization, perimeter, planar sets, topolog-ical genus.
Introduction
In the present paper we aim to study some particular shape optimization problemsin classes of planar domains having a prescribed topology. The quantities we are goingto consider for a general bounded open set Ω are the distributional perimeter P (Ω)and the torsional rigidity T (Ω). More precisely, we deal with a scaling free functional F q which is expressed as the product of the perimeter, and of a suitable powers of thetorsional rigidity and of the Lebesgue measure of Ω, depending on a positive parameter q . The restriction to the planar case is essential and is not made here for the sakeof simplicity; indeed, in higher dimension stronger topological constraints have to beimposed to make the problems well posed.In a previous paper [1] 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.In the following we consider the optimization problems for F q in the classes A k ofplanar domains having at most k “holes”.While the maximization problems are always ill posed, even in the class of smoothopen sets in A k , it turns out that the minimizing problems are interesting if q ≤ / ∈ A k .In this case, we provide a explicit lower bound for F q in the class of Lipschitz setsin A k , which turns out to be sharp when k = 0 , q = 1 / F q in the class of convex sets, as pointed out by Polya in [2].When q < / F q and our approach isthe one of direct methods of the calculus of variations which consists in the followingsteps:- defining the functional F q only for Lipschitz domains of the class A k ;- relaxing the functional F q on the whole class A k , with respect to a suitabletopology; - showing that the relaxed functional admits an optimal domain in A k ;- proving that such a domain is Lipschitz.The relaxation procedure above is necessary to avoid trivial counterexamples due tothe fact that the perimeter is Lebesgue measure sensitive, while the torsional rigidityis capacity sensitive.As in most of the free boundary problems, the last regularity step presents strongdifficulties and, even if the regularity of optimal domains could be expected, we areunable to give a full proof of this fact. It would be very interesting to establish if anoptimal domain fulfills some kind of regularity, or at least if its perimeter coincideswith the Hausdorff measure of the boundary, which amounts to exclude the presenceof internal fractures.This paper is organized as follows. In Section 2, after recalling the definitions ofperimeter and torsional rigidity, we summarize the main results of this paper. In Sec-tion 3 we describe the key tools necessary to apply the so-called method of interiorparallels , introduced by Makai in [3],[4] and by Polya in [2], to our setting. Section 4contains a review of some basic facts concerning the complementary Hausdorff conver-gence, with respect to which we perform the relaxation procedure. Although Sections3 and 4 may be seen as preliminary, we believe they contain some interesting resultsthat, as far as we know, are new in literature. Finally, in Section 5 we discuss theoptimization problem: we extend a well known inequality due to Polya (Theorem 5.1and Remark 5.2), and we prove the main results (Corollary 5.3 and Theorem 5.9).2. Preliminaries
The shape functionals we consider in this paper are of the form F q (Ω) = P (Ω) T q (Ω) | Ω | q +1 / (2.1)where q >
0, Ω ⊂ R is a general bounded open set and, | Ω | denotes its Lebesguemeasure.For the reader’s convenience, in the following we report the definitions and the basicproperties of the perimeter and of the torsional rigidity. According to the De Giorgiformula, the perimeter is given by P (Ω) = sup (cid:26)Z Ω div φ dx : φ ∈ C c ( R ; R ) , k φ k L ∞ ( R ) ≤ (cid:27) , and satisfies:- the scaling property P ( t Ω) = tP (Ω) for every t > L -convergence, that is the con-vergence of characteristic functions.- the isoperimetric inequality P (Ω) | Ω | / ≥ P ( B ) | B | / (2.2)where B is any disc in R . In addition the inequality above becomes an equalityif and only if Ω is a disc (up to sets of Lebesgue measure zero). SHAPE OPTIMIZATION PROBLEM ON PLANAR SETS WITH PRESCRIBED TOPOLOGY 3
The torsional rigidity T (Ω) is defined as T (Ω) = Z Ω u dx where u is the unique solution of the PDE ( − ∆ u = 1 in Ω ,u ∈ H (Ω) . (2.3)By means of an integration by parts we can equivalently express the torsional rigidityas T (Ω) = max nh Z Ω u dx i h Z Ω |∇ u | dx i − : u ∈ H (Ω) \ { } o . (2.4)The main properties we use for the torsional rigidity are:- the monotonicity with respect to the set inclusionΩ ⊂ Ω = ⇒ T (Ω ) ≤ T (Ω );- the additivity on disjoint families of open sets T (cid:16) [ n Ω n (cid:17) = X n T (Ω n ) whenever Ω n are pairwise disjoint;- the scaling property T ( t Ω) = t T (Ω) , for every t > Saint-Venant inequality ) T (Ω) | Ω | ≤ T ( B ) | B | . (2.5)In addition, the inequality above becomes an equality if and only if Ω is a disc(up to sets of capacity zero).If we denote by B the unitary disc of R , then the solution of (2.3), with Ω = B ,is u ( x ) = 1 − | x | T ( B ) = π . Thanks to the scaling properties of the perimeter and of the torsional rigidity, thefunctional F q defined by (2.1) is scaling free and optimizing it in a suitable class A isequivalent to optimizing the product P (Ω) T q (Ω) over A with the additional measureconstraint | Ω | = m , for a fixed m > F q (in every space dimension) in the classes A all := (cid:8) Ω ⊂ R d : Ω = ∅ (cid:9) A convex := (cid:8) Ω ⊂ R d : Ω = ∅ , Ω convex (cid:9) . We summarize here below the results available in the case of dimension 2:- for every q > (cid:8) F q (Ω) : Ω ∈ A all , Ω smooth (cid:9) = 0;
L. BRIANI, G. BUTTAZZO, AND F. PRINARI - for every q > (cid:8) F q (Ω) : Ω ∈ A all , Ω smooth (cid:9) = + ∞ ;- for every q > / ( inf (cid:8) F q (Ω) : Ω ∈ A convex (cid:9) = 0max (cid:8) F q (Ω) : Ω ∈ A convex (cid:9) is attained;- for every q < / ( sup (cid:8) F q (Ω) : Ω ∈ A convex (cid:9) = + ∞ min (cid:8) F q (Ω) : Ω ∈ A convex (cid:9) is attained;- for q = 1 / ( inf (cid:8) F / (Ω) : Ω ∈ A convex (cid:9) = (1 / / sup (cid:8) F / (Ω) : Ω ∈ A convex (cid:9) = (2 / / , asymptotically attained, respectively, when Ω is a long thin rectangle and whenΩ is a long thin triangle.Here we discuss the optimization problems for F q on the classes of planar domains A k := (cid:8) Ω ⊂ R : Ω = ∅ , Ω bounded, c ≤ k (cid:9) , where, for every set E , we denote by E the number of bounded connected compo-nents of E and Ω c = R \ Ω. In particular A denotes the class of simply connecteddomains (not necessarily connected).From what seen above the only interesting cases to consider are: ( the maximum problem for F q on A k when q ≥ / F q on A k when q ≤ / q > k ≥ (cid:8) F q (Ω) : Ω smooth , Ω ∈ A k (cid:9) = + ∞ . Indeed, it is enough to take as Ω n a smooth perturbation of the unit disc B such that B / ⊂ Ω n ⊂ B and P (Ω n ) → + ∞ . All the domains Ω n are simply connected, so belong to A k for every k ≥
0, and | Ω n | ≤ | B | , T (Ω n ) ≥ T ( B / ) , where we used the monotonicity of the torsional rigidity. Therefore F q (Ω n ) ≥ P (Ω n ) T q ( B / ) | B | q +1 / → + ∞ . Moreover inf (cid:8) F q (Ω) : Ω ∈ A k (cid:9) = 0 , as we can easily see by taking as Ω n the unit disk of R where we remove the n segments (in polar coordinates r, θ ) S i = (cid:8) θ = 2 πi/n, r ∈ [1 /n, (cid:9) i = 1 , . . . , n. We have that all the Ω n are simply connected, and | Ω n | = π, P (Ω n ) = 2 π, T (Ω n ) → , SHAPE OPTIMIZATION PROBLEM ON PLANAR SETS WITH PRESCRIBED TOPOLOGY 5 providing then F q (Ω n ) → (cid:8) F q (Ω) : Ω ∈ A k , Ω Lipschitz } , when q ≤ / k ∈ N . Denoting by m q,k the infimum above we summarize herebelow our main results.- For every q ≤ / m q,k are decreasing with respect to k andlim k →∞ m q,k = 0 . - When k = 0 , m / , = m / , = 3 − / = inf (cid:8) F / (Ω) : Ω convex (cid:9) ;in particular, for q = 1 / F / between the classes A convex , A , A , and the infimum is asymptotically reached by a sequence oflong and thin rectangles.- For every q ≤ / k ∈ N , we have m q,k ≥ ( (8 π ) / − q − / if k = 0 , , (8 π ) / − q (3 / k ) − if k > . - For q < /
2, we define a relaxed functional F q,k , which coincides with F q in the class of the sets Ω ∈ A k satisfying P (Ω) = H ( ∂ Ω), being H the 1-dimensional Hausdorff measure. We also prove that F q,k admits an optimaldomain Ω ⋆ ∈ A k with H ( ∂ Ω ⋆ ) < ∞ .3. Approximation by interior parallel sets
For a given bounded nonempty open set Ω we denote by ρ (Ω) its inradius , definedas ρ (Ω) := sup (cid:8) d ( x, ∂ Ω) : x ∈ Ω (cid:9) , where, as usual, d ( x, E ) := inf (cid:8) d ( x, y ) : y ∈ E (cid:9) . For every t ≥
0, we denote by Ω( t )the interior parallel set at distance t from ∂ Ω, i.e.Ω( t ) := (cid:8) x ∈ Ω : d ( x, ∂ Ω) > t (cid:9) , and by A ( t ) := | Ω( t ) | . Moreover we denote by L ( t ) the length of the interior parallel ,that is the set of the points in Ω whose distance from ∂ Ω is equal to t . More preciselywe set L ( t ) := H ( { x ∈ Ω : d ( x, ∂ Ω) = t } ) . Notice that ∂ Ω( t ) ⊆ { x ∈ Ω : d ( x, ∂ Ω) = t } . Using coarea formula (see [5] Theorem3.13) we can write the following identity: A ( t ) = Z ρ (Ω) t L ( s ) ds ∀ t ∈ (0 , ρ (Ω)) . (3.1)As a consequence it is easy to verify that for a.e. t ∈ (0 , ρ (Ω)) there exists thederivative A ′ ( t ) and it coincides with − L ( t ). The interior parallel sets Ω( t ) belong to A k as soon as Ω ∈ A k , as next elementary argument shows. Lemma 3.1.
Let Ω ∈ A k . Then Ω( t ) ∈ A k for every t ∈ [0 , ρ (Ω)) . L. BRIANI, G. BUTTAZZO, AND F. PRINARI
Proof.
Let α := c ( ≤ k ), and C , C , · · · C α be the (closed) bounded connectedcomponents of Ω c and C the unbounded one. Define C i ( t ) := (cid:8) x ∈ R : d ( x, C i ) ≤ t (cid:9) . Since C i is connected, then C i ( t ) is connected and the set S αi =0 C i ( t ) has at most α + 1connected components. Since we have Ω c ( t ) = S αi =0 C i ( t ), the lemma is proved. (cid:3) In the planar case, even without any regularity assumptions on ∂ Ω, the sets Ω( t ) area slightly smoothed version of Ω. In particular the following result (see [6]), that welimit to report in the two dimensional case, proves that Ω( t ) has a Lipschitz boundaryfor a.e. t ∈ (0 , ρ (Ω)). Theorem 3.2 (Fu) . Let K ⊆ R be a compact set. There exists a compact set C = C ( K ) ⊆ [0 , − / diam ( K )] such that | C | = 0 and if t / ∈ C then the boundary of { x ∈ R : d ( x, K ) > t } is a Lipschitz manifold. We recall now some general facts of geometric measure theory. Let E ⊂ R , wedenote by E ( t ) the set of the points where the density of E is t ∈ [0 , E ( t ) := { x ∈ R : lim r → + ( πr ) − | E ∩ B r ( x ) | = t } . It is well known (see [7] Theorem 3.61) that if E is a set of finite perimeter, then P ( E ) = H ( E / ) and E has density either 0 or 1 / H -a.e x ∈ R . Inparticular it holds H ( ∂E ) = H ( ∂E ∩ E (0) ) + H ( ∂E ∩ E (1) ) + P ( E ) , (3.2)which implies P ( E ) + 2 H ( ∂E ∩ E (1) ) ≤ H ( ∂E ) − P ( E ) . (3.3)The Minkowski content and the outer Minkowski content of E are, respectively, definedas M ( E ) := lim t → |{ x ∈ R : d ( x, E ) ≤ t }| t , and SM ( E ) := lim t → |{ x ∈ R : d ( x, E ) ≤ t } \ E | t , whenever the limits above exist.We say that a compact set E ⊂ R is 1-rectifiable if there exists a compact set K ⊂ R and a Lipschitz map f : R → R such that f ( K ) = E . Any compactconnected set of R , namely a continuum , with finite H -measure is 1-rectifiable (see,for instance, Theorem 4.4 in [8]). Finally, if E is 1-rectifiable then M ( E ) = H ( E ) (3.4)(see Theorem 2 .
106 in [7]) and by Proposition 4.1 of [9], if E is a Borel set and ∂E is1-rectifiable it holds SM ( E ) = P ( E ) + 2 H ( ∂E ∩ E (0) ) . (3.5)Next two results are easy consequence of (3.4) and (3.5). Theorem 3.3.
Let Ω be a bounded open set with H ( ∂ Ω) < ∞ and ∂ Ω < + ∞ .Then M ( ∂ Ω) = H ( ∂ Ω) and SM (Ω) = P (Ω) + 2 H ( ∂ Ω ∩ Ω (0) ) . Proof.
Since H ( ∂ Ω) < ∞ , each connected component of ∂ Ω is 1-rectifiable. Beingthe connected components pairwise disjoint and compact, we easily prove that theirfinite union is 1-rectifiable. Then, applying (3.4) and (3.5), we get the thesis. (cid:3)
SHAPE OPTIMIZATION PROBLEM ON PLANAR SETS WITH PRESCRIBED TOPOLOGY 7
Corollary 3.4.
Let Ω be an open set such that H ( ∂ Ω) < ∞ and ∂ Ω < + ∞ . Thenthere exists lim r → + r Z r L ( t ) dt = P (Ω) + 2 H ( ∂ Ω ∩ Ω (1) ) . Proof.
We denote by L c ( t ) the following quantity L c ( t ) := H ( { x ∈ Ω c : d ( x, ∂ Ω) = t } ) . By applying coarea formula and Theorem 3.3, it holdslim r → + r Z r L c ( t ) dt = SM (Ω) = P (Ω) + 2 H ( ∂ Ω ∩ Ω (0) ) . (3.6)and lim r → + r Z r [ L ( t ) + L c ( t )] dt = 2 M ( ∂ Ω) = 2 H ( ∂ Ω) . (3.7)Combining (3.2), (3.6) and (3.7) we getlim r → + r Z r L ( t ) dt = lim r → + (cid:18) r Z r L ( t ) dt + 1 r Z r L c ( t ) dt − r Z r L c ( t ) dt (cid:19) = 2 H ( ∂ Ω) − P (Ω) − H ( ∂ Ω ∩ Ω (0) ) = P (Ω) + 2 H ( ∂ Ω ∩ Ω (1) )and the thesis is achieved. (cid:3) Most of the results we present rely on a geometrical theorem proved by Sz. Nagy in[10], concerning the behavior of the function t → A ( t ) = | Ω( t ) | for a given set Ω ∈ A k . Theorem 3.5 (Sz. Nagy) . Let Ω ∈ A k and let α := c . Then the function t
7→ − A ( t ) − ( α − πt is concave in [0 , ρ (Ω)) . As a consequence of Corollary 3.4 and Theorem 3.5 we have the following result.
Theorem 3.6.
Let Ω ∈ A k with H ( ∂ Ω) < ∞ and < + ∞ . Then, for a.e. t ∈ (0 , ρ (Ω)) , it holds: L ( t ) ≤ P (Ω) + 2 H ( ∂ Ω ∩ Ω (1) ) + 2 π ( k − t ; (3.8) A ( t ) ≤ ( P (Ω) + 2 H ( ∂ Ω ∩ Ω (1) ))( ρ (Ω) − t ) + π ( k − ρ (Ω) − t ) . (3.9) In particular A ∈ W , ∞ (0 , ρ (Ω)) .Proof. We denote by g ( t ) the right derivative of the function t
7→ − A ( t ) − ( α − πt where α := c ( ≤ k ). By Theorem 3.5, g is a decreasing function in (0 , ρ (Ω)) andan easy computation through (3.1) shows that g ( t ) = L ( t ) − π ( α − t for a.e. t ∈ (0 , ρ (Ω)) . (3.10)Thus, lim r → + r Z r L ( t ) dt = lim r → + r Z r g ( t ) dt = sup (0 ,ρ (Ω)) g ( t ) . Since Ω ∈ A k and < ∞ we have also ∂ Ω < ∞ . Hence we can apply Corollary3.4 to get P (Ω) + 2 H ( ∂ Ω ∩ Ω (1) ) = sup (0 ,ρ (Ω)) g ( t ) . (3.11)By using (3.10) and (3.11), inequality (3.8) easily follows. Finally, by applying (3.1),we get both A ∈ W , ∞ (0 , ρ (Ω)) and formula (3.9). (cid:3) L. BRIANI, G. BUTTAZZO, AND F. PRINARI
The following lemma can be easily proved by lower semicontinuity property of theperimeter.
Lemma 3.7.
Let Ω ⊂ R be an open set. Let (Ω i ) be its connected components and Ω n := S ni =1 Ω i . Then we have:(i) ∂ Ω n = S ni =1 ∂ Ω i ⊆ ∂ Ω and H ( ∂ Ω n ) ≤ H ( ∂ Ω) ;(ii) P (Ω) ≤ lim inf n →∞ P (Ω n ) ≤ lim sup n →∞ P (Ω n ) ≤ lim sup n →∞ H ( ∂ Ω n ) ≤ H ( ∂ Ω) . We are now in a position to prove the main results of this section. In Theorem 1.1of [11] it is shown that, given any set Ω of finite perimeter satisfying H ( ∂ Ω) = P (Ω),it is possible to approximate P (Ω) with the perimeters of smooth open sets compactlycontained in Ω. Here we show that, if we assume the further hypothesis Ω ∈ A k , thenwe can construct an approximation sequence made up of Lipschitz sets in A k . Theorem 3.8.
Let Ω ∈ A k be a set of finite perimeter. Then there exists an increasingsequence ( A n ) ⊂ A k such that:(i) A n ⊂ Ω ;(ii) S n A n = Ω ;(iii) A n is a Lipschitz set;(iv) P (Ω) ≤ lim inf n →∞ P ( A n ) ≤ lim sup n →∞ P ( A n ) ≤ H ( ∂ Ω) − P (Ω) .In addition, if < ∞ , then lim n →∞ P ( A n ) = P (Ω) + 2 H ( ∂ Ω ∩ Ω (1) ) . Proof.
Let Ω n be defined as in Lemma 3.7. Clearly Ω n ∈ A k . Since Ω n ( t ) convergesto Ω n in L when t → + , it follows that, for every n ,lim inf t → + P (Ω n ( t )) ≥ P (Ω n ) . Then there exists 0 < δ n < /n ∧ ρ (Ω n ) such that P (Ω n ( t )) ≥ P (Ω n ) − n ∀ t < δ n . (3.12)Since n ≤ n , by applying Theorem 3.2, Lemma 3.1 and Theorem 3.6 to the setΩ n , we can choose a decreasing sequence ( t n ) with 0 < t n < δ n such that the set A n := Ω n ( t n ) is in A k , has Lipschitz boundary, and H ( { x ∈ Ω n : d ( x, ∂ Ω n ) = t n } ) ≤ P (Ω n ) + 2 H ( ∂ Ω n ∩ Ω (1) n ) + 2 π ( k − t n . (3.13)It is easy to prove that the sequence ( A n ) is increasing and satisfies (i) and (ii). Byputting together (3.12) and (3.13), we get P (Ω n ) − n ≤ P ( A n ) ≤ P (Ω n ) + 2 H ( ∂ Ω n ∩ Ω (1) n ) + 2 π ( k − t n . By Lemma 3.7, taking also into account (3.3), the previous inequality implies P (Ω) ≤ lim inf n P ( A n ) ≤ lim sup n P ( A n ) ≤ H ( ∂ Ω) − P (Ω)which proves ( iv ).To conclude consider the case < + ∞ . We can choose n big enough such thatΩ n = Ω, A n = Ω( t n ) and α := c = A cn . For simplicity we denote ρ n := ρ ( A n )and ρ := ρ (Ω). By applying equality (3.11) to the Lipschitz set A n , we get P ( A n ) = sup (0 ,ρ n ) g n ( t ) (3.14) SHAPE OPTIMIZATION PROBLEM ON PLANAR SETS WITH PRESCRIBED TOPOLOGY 9 where g n is the right derivative of the function t
7→ −| A n ( t ) | − ( α − πt . Now,exploiting the equality A n ( t ) = Ω( t + t n ), we obtain g n ( t ) = g ( t + t n ) + 2 π ( α − t n for all 0 < t < ( ρ − t n ) ∧ ρ n . Thus, as t → + and applying (3.14), we can concludethat, for every n , it holdslim t → + g ( t + t n ) + 2 π ( α − t n = sup (0 ,ρ n ) g n ( t ) = P ( A n ) . Passing to the limit as n → ∞ in the equality above and taking into account (3.11)we achieve the thesis. (cid:3) Continuity of volume for co-Hausdorff convergence
The Hausdorff distance between closed sets C , C of R is defined by d H ( C , C ) := sup x ∈ C d ( x, C ) ∨ sup x ∈ C d ( x, C ) . Through d H we can define the so called co-Hausdorff distance d H c between a pair ofbounded open subsets Ω , Ω of R d H c (Ω , Ω ) := d H (Ω c , Ω c ) . We say that a sequence of compact sets ( K n ) converges in the sense of Hausdorffto some compact set K , if ( d H ( K n , K )) converges to zero. In this case we write K n H → K . Similarly we say that a sequence of open sets (Ω n ) converges in thesense of co-Hausdorff to some open set Ω, if ( d H c (Ω n , Ω)) converges to zero, andwe write Ω n H c → Ω. In the rest of the paper we use some elementary properties ofHausdorff distance and co-Hausdorff distance for which we refer to [12] and [13], (see,for instance, Proposition 4.6.1 of [12]). In particular we recall that if (Ω n ) is a sequenceof equi-bounded sets in A k and Ω n H c → Ω, then Ω still belongs to A k (see Remark 2.2.20of [13]).The introduction of co-Hausdorff convergence is motivated by Sver´ak’s Theorem(see [14]) which ensures the continuity of the torsional rigidity in the class A k . Actuallythe result is stronger and gives the continuity with respect to the γ -convergence (werefer to [12] for its precise definition and the related details). Theorem 4.1 (Sver´ak) . Let (Ω n ) ⊂ A k be a sequence of equi-bounded open sets. If Ω n H c → Ω , then Ω n → Ω in the γ -convergence. In particular T (Ω n ) → T (Ω) . Combining Sver´ak theorem and Theorem 3.8, we prove that we can equivalentlyminimize the functional F q either in the class of Lipschitz set in A k or in the largerclass of those sets Ω ∈ A k satisfying P (Ω) = H ( ∂ Ω).
Proposition 4.2.
The following identity holds: m q,k = inf { F q (Ω) : Ω ∈ A k , P (Ω) = H ( ∂ Ω) } Proof.
By Theorem 3.8, for every Ω ∈ A k such that P (Ω) = H ( ∂ Ω) < ∞ , there existsa sequence ( A n ) ⊂ A k of Lipschitz sets satisfying lim n P ( A n ) = P (Ω). By construction( A n ) is an equi-bounded sequence which converges both in the co-Hausdorff and inthe L sense. By Theorem 4.1 we havelim n →∞ F q (Ω n ) = F q (Ω) , so that m q,k ≤ inf { F q (Ω) : Ω ∈ A k , P (Ω) = H ( ∂ Ω) } . The thesis is then achieved since the opposite inequality is trivial. (cid:3)
In general the volume is only lower semicontinuous with respect to the H c -convergenceas simple counterexamples may show. In this section we prove that L -convergenceis guaranteed in the class A k under some further hypotheses, see Theorem 4.7. Theproof of this result requires several lemma and relies on the classical Go lab’s semicon-tinuity theorem, which deals with the lower semicontinuity of the Hausdorff measure H (see, for instance, [8], [15]). Theorem 4.3 (Go lab) . Let X be a complete metric space and k ∈ N let C k := { K : K ⊂ X, K is closed , K ≤ k } . Then the function K
7→ H ( K ) is lower semicontinuous on C k endowed with theHausdorff distance. Lemma 4.4.
Let (Ω n ) be a sequence of equi-bounded open sets. If Ω n H c → Ω we havealso ρ (Ω n ) → ρ (Ω) .Proof. For simplicity we denote ρ := ρ (Ω), and ρ n := ρ (Ω n ). First we show that ρ ≤ lim inf n ρ n . (4.1)Indeed, without loss of generality let us assume ρ >
0. Then for any 0 < ε < ρ ,there exists a ball B ε whose radius is ρ − ε and whose closure is contained in Ω. Byelementary properties of co-Hausdorff convergence, there exists ν such that B ε ⊂ Ω n ,for n > ν , which implies ρ n ≥ ρ − ε . Since ε > ε > n k ) such that ρ n k > ρ + ε for every k ∈ N . Then there existsa sequence of balls B n k = B ρ nk ( x n k ) ⊆ Ω n k . Eventually passing to a subsequence, thesequence ( x n k ) converges to a point x ∞ and the sequence of the translated open setΩ n k − x n k converges to Ω − x ∞ . Since B r (0) ⊆ Ω n k − x n k for r = ρ + ε , it turns outthat B r (0) ⊆ Ω − x ∞ , i.e. B r ( x ∞ ) ⊆ Ω which leads to a contradiction. (cid:3)
Lemma 4.5.
Let Ω be a connected bounded open set of R n . There exists a sequenceof connected bounded open sets (Ω n ) such that Ω n ⊂ Ω n +1 and S n Ω n = Ω .Proof. We construct the sequence by induction. First of all we notice that there existsan integer ν > ν − ) contains at least one connected component of Ωwith Lebesgue measure greater than πν − . Indeed it suffices to choose ν − ≤ min { d ( y, ∂ Ω) : y ∈ ∂B r ( x ) } ∧ r where B r ( x ) is any ball with closure contained in Ω. Now let M be the number ofconnected components of Ω( ν − ) with Lebesgue measure greater than πν − . If M = 1we define Ω := Ω( ν − ). Otherwise, since Ω is pathwise connected, we can connect theclosures of the M connected components with finitely many arcs to define a connectedcompact set K ⊂ Ω. Then, we choose m such that m > ν and m − < inf { d ( x, ∂ Ω) : x ∈ K } and we set Ω := { x ∈ Ω : d ( x, K ) < (2 m ) − } . In both cases Ω is a connected open set which contains all the connected componentsof Ω( ν − ) having Lebesgue measure greater then πν − . Moreover by construction thereexists ν > ν such that Ω ⊆ Ω( ν − ). Replacing ν with ν we can use the previous SHAPE OPTIMIZATION PROBLEM ON PLANAR SETS WITH PRESCRIBED TOPOLOGY 11 argument to define Ω such that Ω ⊂ Ω . Iterating this argument we eventuallydefine an increasing sequence ν n and a sequence of connected open sets (Ω n ) suchthat Ω n ⊂ Ω n +1 ⊂ Ω and Ω n contains all the connected components of Ω( ν − n ) ofLebesgue measure greater than πν − n . Since for any x ∈ Ω there exists r > B r ( x ) ⊂ Ω, choosing ν − n ≤ min { d ( y, ∂ Ω) : y ∈ ∂B r ( x ) } ∧ r , it is easy to showthat x ∈ Ω n . Thus S n Ω n = Ω. (cid:3) In the following lemma we establish a Bonnesen-type inequality for sets Ω ∈ A k satisfying H ( ∂ Ω) < ∞ (see Theorem 2 in [16] when Ω is a simply connected planedomain bounded by a rectifiable Jordan curve). Lemma 4.6.
Let Ω ∈ A k with H ( ∂ Ω) < ∞ . Then | Ω | ≤ [2 H ( ∂ Ω) − P (Ω) + π ( k − ρ (Ω)] ρ (Ω) . (4.2) Proof.
If < ∞ , by Theorem 3.6 and (3.1), | Ω | ≤ (cid:0) P (Ω) + 2 H ( ∂ Ω ∩ Ω (1) ) + π ( k − ρ (Ω) (cid:1) ρ (Ω) , and we conclude by (3.3). To prove the general case we denote by (Ω i ) the connectedcomponents of Ω and we set Ω n := S ni =1 Ω i . By the previous step we have | Ω n | ≤ (cid:0) H ( ∂ Ω n ) − P (Ω n ) + π ( k − ρ (Ω n ) (cid:1) ρ (Ω n ) . Since Ω n H c → Ω and Ω n → Ω in the L -convergence, taking into account Lemma 4.4and Lemma 3.7, we can conclude that | Ω | = lim n →∞ | Ω n | ≤ (cid:0) H ( ∂ Ω) − lim sup n P (Ω n ) + π ( α − ρ (Ω) (cid:1) ρ (Ω) ≤ (cid:0) H ( ∂ Ω) − P (Ω) + π ( k − ρ (Ω) (cid:1) ρ (Ω) , from which the thesis is achieved. (cid:3) Theorem 4.7.
Let (Ω n ) ⊂ A k be a sequence of equi-bounded open sets with sup n H ( ∂ Ω n ) < ∞ . If Ω n H c → Ω then Ω ∈ A k and Ω n → Ω in the L -convergence. If, in addition, either sup n ∂ Ω n < ∞ or < ∞ then H ( ∂ Ω) ≤ lim inf n H ( ∂ Ω n ) . (4.3) Proof.
We first deal with the case when sup n ∂ Ω n < ∞ , already considered in [17]and [12]. By compactness we can suppose that ∂ Ω n converges to some nonemptycompact set K which contains ∂ Ω. Then it is easy to show that ¯Ω n H → Ω ∪ K , whichimplies χ Ω n → χ Ω pointwise in R \ K , where χ E denotes the characteristic functionof a set E . By Theorem 4.3 we have also H ( ∂ Ω) ≤ H ( K ) ≤ lim inf n →∞ H ( ∂ Ω n ) < + ∞ , (4.4)which implies (4.3). In particular, we have | K | = 0, and Ω n → Ω in the L conver-gence.We consider now the general case. Let (Ω i ) be the connected components of Ω and ε >
0. There exists an integer ν ( ε ) such that | Ω | − ε < | ν ( ε ) [ i =1 Ω i | ≤ | Ω | (when < ∞ we simply choose ν ( ε ) = i ≤ ν ( ε ), and for each setΩ i , we consider the sequence (Ω in ) given by Lemma 4.5. By elementary properties ofco-Hausdorff convergence there exists l := l ( n ) such that ν ( ε ) [ i Ω in ⊂ Ω l . Let’s denote by e Ω il the connected component of Ω l which contains Ω in (eventually e Ω hl = e Ω sl ), and define e Ω l := ν ( ε ) [ i =1 e Ω il . By compactness, passing eventually to a subsequence, there exists e Ω ∈ A k such that e Ω l H c → e Ω. Moreover, since e Ω l ∈ A k , sup l e Ω l ≤ ν ( ε ), and by Lemma 3.7 we havesup l H ( ∂ e Ω l ) ≤ sup l H ( ∂ Ω l ) < ∞ , we can apply the first part of the proof to conclude that e Ω l → e Ω in the L -convergence.If < ∞ an easy argument shows that e Ω must be equal to Ω and that (4.4) holdswith K the Hausdorff limit of ( ∂ e Ω l ). In particular (4.3) holds. Otherwise we considerthe set Ω Rl of those connected components of Ω l that have been neglected in thedefinition of e Ω l , that is Ω Rl := Ω l \ e Ω l . Passing to a subsequence we can suppose that Ω
Rl H c → Ω R , for some open set Ω R ∈ A k .Moreover since | e Ω | > | Ω | − ε , Ω R ∩ ˜Ω = ∅ and Ω R ⊂ Ω we have also | Ω R | ≤ ε . Thisimplies ρ (Ω R ) ≤ √ π − ε and by Lemma 4.4,lim l →∞ ρ (Ω Rl ) ≤ √ π − ε. Finally, by Lemma 4.6, we have | Ω | ≤ lim inf n →∞ | Ω n | ≤ lim sup l →∞ ( | e Ω l | + | Ω Rl | ) = | e Ω | + lim sup l →∞ | Ω Rl | ≤ | Ω | + o ( ε ) . Since ε was arbitrary this shows thatlim inf n →∞ | Ω n | = | Ω | , and the thesis is easily achieved. (cid:3) As an application of the previous theorem we prove the following fact.
Corollary 4.8.
Let Ω ∈ A k with H ( ∂ Ω) < ∞ and < ∞ . Then it holds H ( ∂ Ω ∩ Ω (0) ) ≤ H ( ∂ Ω ∩ Ω (1) ) . Proof.
By Theorem 3.8 we can consider a sequence ( A n ) ∈ A k of Lipschitz sets suchthat A n H c → Ω and P ( A n ) → P (Ω) + 2 H ( ∂ Ω ∩ Ω (1) ) < ∞ . Then, by Theorem 4.7, weconclude H ( ∂ Ω) ≤ lim n →∞ P ( A n ) ≤ P (Ω) + 2 H ( ∂ Ω ∩ Ω (1) ) , which easily implies the thesis, using (3.2). (cid:3) SHAPE OPTIMIZATION PROBLEM ON PLANAR SETS WITH PRESCRIBED TOPOLOGY 13
Remark 4.9.
We remark the fact that the inequalitylim n →∞ P ( A n ) ≥ H ( ∂ Ω)is not in general satisfied when ∞ , see also Remark 5.6.5. Existence of relaxed solutions
Our next result generalizes the estimate F / (Ω) ≥ − / , proved in [2] for the class A convex , to the class A k . Theorem 5.1.
For every Ω ∈ A k set of finite perimeter we have T / (Ω) | Ω | / ≥ − / (2 H ( ∂ Ω) − P (Ω) + 2 π ( k − ρ (Ω)) . (5.1) Proof.
Without loss of generality we may assume that H ( ∂ Ω) < ∞ and we set ρ := ρ (Ω). First we consider the case < ∞ . We define G ( t ) := Z t A ( t ) L ( t ) dt, u ( x ) := G ( d ( x, ∂ Ω)) . Notice that, since for any t ∈ (0 , ρ ) it holds L ( t ) ≥ H ( ∂ Ω( t )) ≥ P (Ω( t )), by isoperi-metric inequality (2.2) we have A ( t ) L ( t ) = | Ω( t ) | / L ( t ) A / ( t ) ≤ | Ω( t ) | / P (Ω( t )) A / ( t ) ≤ | B | / P ( B ) A / ( t ) . In particular, since A is bounded, we get that L − A is summable on (0 , ρ ) and G isa Lipschitz function on in the interval (0 , ρ ). Thus u ∈ H (Ω). Using (2.4) and (3.8)we have T (Ω) ≥ (cid:0)R Ω udx (cid:1) R Ω |∇ u | dx ≥ (cid:0)R ρ G ( t ) L ( t ) dt (cid:1) R ρ ( G ′ ( t )) L ( t ) dt ≥ Z ρ ( A ( t )) L ( t ) dt = Z ρ A ( t ) L ( t ) L ( t ) dt ≥ P (Ω) + 2 H ( ∂ Ω ∩ Ω (1) ) + 2 π ( k − ρ ) Z ρ A ( t ) L ( t ) dt. Since A ∈ W , ∞ (0 , ρ (Ω)) by Corollary 3.6 then, set ψ ( s ) = s , we have that thefunction ψ ◦ A ∈ W , ∞ (0 , ρ (Ω)), so that Z ρ A ( t ) L ( t ) dt = − Z ρ A ( t ) A ′ ( t ) dt = − (cid:2) A ( t ) (cid:3) ρ (Ω)0 = 13 | Ω | . Thus T (Ω) | Ω | ≥ P (Ω) + 2 H ( ∂ Ω ∩ Ω (1) ) + 2 π ( k − ρ ) . (5.2)Taking into account (3.3) we get T (Ω) | Ω | ≥ H ( ∂ Ω) − P (Ω) + 2 π ( k − ρ ) . To prove the general case, let Ω n be defined as in Lemma 3.7. Since n < ∞ andΩ n ∈ A k , by the first part of this proof we have that T (Ω n ) | Ω n | (cid:0) H ( ∂ Ω n ) − P (Ω n ) + 2 π ( k − ρ n (cid:1) ≥ , where ρ n := ρ (Ω n ). When n → ∞ we have | Ω n | → | Ω | , ρ n → ρ by Lemma 4.4and T (Ω n ) → T (Ω) by Theorem 4.1. Hence, passing to the lim sup in the previousinequality and using Lemma 3.7, we get (5.1). (cid:3) Remark 5.2.
Note that, in the special case of Ω ∈ A k and < ∞ , we have theimproved estimate (5.2). Moreover, if k = 0 ,
1, (5.1) implies F / (Ω) ≥ − / P (Ω)2 H ( ∂ Ω) − P (Ω) , (5.3)while, if k >
1, we can use the inequality 2 πρ (Ω) ≤ P (Ω) (which can be easily derivedfrom (2.2)), to obtain F / (Ω) ≥ − / P (Ω)2 H ( ∂ Ω) + ( k − P (Ω) . (5.4)As a consequence of Theorem 5.1, and using the well known fact that for a Lipschitzopen set Ω it holds P (Ω) = H ( ∂ Ω), we have the following main results.
Corollary 5.3.
For every q ≤ / we have m / , = m / , = 3 − / (5.5) and the value − / is asymptotically reached by a sequence of long thin rectangles.More in general, for k ≥ , it holds m q,k ≥ (8 π ) / − q (3 / k ) − (5.6) and the sequence ( m q,k ) decreases to zero as k → ∞ .Proof. By inequality (5.3) we have that m / , , m / , ≥ − / . Moreover the compu-tations made in [1] show that the value 3 − / is asymptotically reached by a sequenceof long thin rectangles, that are clearly in A . Thus, being A ⊂ A , (5.5) holds. Toprove (5.6) it is enough to notice that F q (Ω) = F / (Ω) (cid:18) T (Ω) | Ω | (cid:19) q − / and apply (5.4) together with the Saint-Venant inequality (2.5). Finally to prove that m q,k → k → ∞ , it is enough to consider the sequence (Ω ,n ) defined in Theorem2.1 of [1], taking into account that Ω ,n ∈ A k for k big enough. (cid:3) We now introduce a relaxed functional F q,k . More precisely, for Ω ∈ A k we denoteby O k (Ω) the class of equi-bounded sequences of Lipschitz sets in A k which convergeto Ω in the sense of co-Hausdorff and we define F q,k as follows: F q,k (Ω) := inf n lim inf n →∞ F q (Ω n ) : (Ω n ) ∈ O k (Ω) o . It is straightforward to verify that F q,k is translation invariant and scaling free. Asalready mentioned in the introduction, when q < /
2, we prove the existence of aminimizer for F q,k . We notice this relaxation procedure can be made on the perimeterterm only. More precisely, defining P k (Ω) := inf n lim inf n →∞ P (Ω n ) : (Ω n ) ∈ O k (Ω) o , the following proposition holds. SHAPE OPTIMIZATION PROBLEM ON PLANAR SETS WITH PRESCRIBED TOPOLOGY 15
Proposition 5.4.
For every Ω ∈ A k we have F q,k (Ω) = P k (Ω) T q (Ω) | Ω | q +1 / . Proof.
Fix ε >
0. Suppose that ∞ > P k (Ω) + ε ≥ lim n P (Ω n ), for some (Ω n ) ∈ O k (Ω).By Theorems 4.1 and 4.7, we have( P k (Ω) + ε ) T q (Ω) | Ω | q +1 / ≥ lim n (cid:18) P (Ω n ) T q (Ω n ) | Ω n | q +1 / (cid:19) ≥ F q,k (Ω) , and since ε is arbitrary we obtain the ≤ inequality. Similarly, to prove the oppositeinequality assume lim n F q (Ω n ) ≤ F q,k (Ω) + ε < ∞ , for some sequence (Ω n ) ∈ O k (Ω).Let D be a compact set which contains each Ω n . Thanks to Theorem 4.1, we havethat T (Ω n ) → T (Ω) and, since P (Ω n ) = H (Ω n ), we have alsosup n H ( ∂ Ω n ) = sup n (cid:18) F q (Ω n ) | Ω n | q +1 / T q (Ω n ) (cid:19) ≤ sup n (cid:18) F q (Ω n ) | D | q +1 / T q (Ω n ) (cid:19) < + ∞ . Applying again Theorem 4.7 we have | Ω n | → | Ω | and we can conclude P k (Ω) T q (Ω) | Ω | q +1 / ≤ lim n F q (Ω n ) ≤ F q,k (Ω) + ε, which implies the ≥ inequality as ε → (cid:3) The perimeter P k satisfies the following properties. Proposition 5.5.
For every Ω ∈ A k of finite perimeter we have P (Ω) ≤ P k (Ω) ≤ H ( ∂ Ω) − P (Ω) . (5.7) Moreover if < ∞ and H ( ∂ Ω) < + ∞ it holds H ( ∂ Ω) ≤ P k (Ω) ≤ P (Ω) + 2 H ( ∂ Ω ∩ Ω (1) ) (5.8) and P (Ω) = P k (Ω) if and only if P (Ω) = H ( ∂ Ω) .Proof. Taking into account Theorem 4.7 and lower semicontinuity of the perimeterwith respect to the L -convergence we have P k (Ω) ≥ P (Ω). To prove the right-hand inequalities in (5.7) and (5.8) it is sufficient to take the sequence ( A n ) given byTheorem 3.8. Finally, when < ∞ , the inequality H ( ∂ Ω) ≤ P k (Ω) follows byTheorem 4.7. (cid:3) Remark 5.6.
If we remove the assumption < ∞ , then (5.8) is no longer true.For instance, we can slightly modify the Example 3 .
53 in [7] to define Ω ∈ A suchthat P (Ω) , P (Ω) < ∞ while H ( ∂ Ω) = ∞ . More precisely let ( q n ) be an enumerationof Q ∩ B (0) and ( r n ) ⊂ (0 , ε ) be a decreasing sequence such that P n πr n ≤
1. Werecursively define the following sequence of open sets. LetΩ := B r ( q ) , Ω n +1 := Ω n ∪ B s n ( q h n ) , where h n := inf { k : q k ∈ Ω cn } , s n := r n +1 ∧ sup { r k : B r k ( q h n ) ∩ Ω n = ∅} . Finally let Ω = S n Ω n . By construction Ω n H c → Ω and since Ω n ∈ A for all n , we havealso Ω ∈ A . Moreover we notice that P (Ω) ≤ ∂ Ω is positive, which implies H ( ∂ Ω) = ∞ . Finally,since the sequence (Ω n ) ∈ O (Ω), we have also P (Ω) ≤ Next we prove that the relaxed functional F q,k agrees with F q on the class of Lips-chitz open sets in A k . Corollary 5.7.
For every Ω ∈ A k we have F q,k (Ω) ≥ F q (Ω) . (5.9) If, in addition, P (Ω) = H ( ∂ Ω) then we have F q (Ω) = F q,k (Ω) . (5.10) In particular F q,k and F q coincide on the class of Lipschitz sets and it holds m q,k = inf {F q,k (Ω) : Ω ∈ A k } . (5.11) Proof.
The inequalities (5.9) and (5.10) follow by Proposition 5.4 and (5.7). The lastpart of the theorem follows as a general property of relaxed functionals. (cid:3)
Lemma 5.8.
For every Lipschitz set Ω ∈ A k , there exists a sequence of connectedopen sets (Ω n ) ⊂ A k such that P (Ω n ) = H ( ∂ Ω n ) and lim n →∞ F q (Ω n ) = F q (Ω) . Proof.
Since Ω is a bounded Lipschitz set we necessarily have < ∞ . If Ω isconnected we can take Ω n to be constantly equal to Ω. Suppose instead that and Ω be the connected components of Ω. Since Ω is Lipschitz there exist x ∈ ∂ Ω , x ∈ ∂ Ω such that0 < d := d ( x , x ) = inf { d ( w, v ) : v ∈ Ω , w ∈ Ω } . Define Ω n := Ω − (cid:18) − n (cid:19) ( x − x ) . Clearly we have Ω n ∩ Ω = ∅ for every n ≥ = Ω . We set x n = x − (cid:18) − n (cid:19) ( x − x ) . Now we can join x and x n through a segment Σ n . By using the fact that the bound-ary of both ∂ Ω and ∂ Ω n are represented as the graph of a Lipschitz functions in aneighborhood of x and x n respectively, then the thin open channel C ε := { x ∈ R \ Ω ∪ Ω n : d ( x, Σ n ) < ε } of thickness ε := ε ( n ) is such that the setΩ n := Ω ∪ Ω n ∪ C ε belongs to A k , it is connected and P (Ω n ) = H ( ∂ Ω n ). The following identities arethen verified | Ω n | → | Ω | , T (Ω n ) → T (Ω) , P (Ω n ) ≈ P (Ω ) + P (Ω ) + 2 εn , so that F q (Ω n ) → F q (Ω) (notice that this does not imply Ω n → Ω). The general caseis achieved by induction on N + 1. Let (Ω i ) bethe connected components of Ω. By induction we have F q (Ω ∪ · · · ∪ Ω N ) = lim n →∞ F q (Ω ′ n ) , SHAPE OPTIMIZATION PROBLEM ON PLANAR SETS WITH PRESCRIBED TOPOLOGY 17 for a sequence (Ω ′ n ) ⊂ A k of connected open sets satisfying P (Ω ′ n ) = H ( ∂ Ω ′ n ). Usingthe fact that, being Ω Lipschitz, the value of F q (Ω) do not change if we translate (pos-sibly in different direction and with different magnitude) each connected componentof Ω, being careful to avoid intersections, we can suppose Ω N +1 to have a positivedistance from Ω ′ n , as n is large enough. We then apply the previous step to define asequence of connected open sets Ω n,m ∈ A k such that P (Ω n,m ) = H ( ∂ Ω n,m ) and F q (Ω n,m ) → F q (Ω ′ n ∪ Ω N +1 ) , as m → ∞ . Using a diagonal argument we achieve the thesis. (cid:3) We finally show the existence of a relaxed solution to the minimization problem of F q,k in A k when q < / Theorem 5.9.
For q < / there exists a nonempty bounded open set Ω ⋆ ∈ A k minimizing the functional F q,k such that H ( ∂ Ω ⋆ ) < ∞ .Proof. Let ( e Ω n ) ⊂ A k be a sequence of Lipschitz sets such thatlim n →∞ F q ( e Ω n ) = m q,k . Applying Lemma 5.8 and (5.10), we can easily replace the sequence ( e Ω n ) with asequence (Ω n ) ⊂ A k of connected (not necessarily Lipschitz) open sets, satisfying H (Ω n ) = P (Ω n ) and such thatlim n →∞ F q (Ω n ) = lim n →∞ F q ( e Ω n ) = m q,k . Eventually using the translation invariance of F q and possibly rescaling the sequence(Ω n ), we can assume that (Ω n ) is equi-bounded and H (Ω n ) = P (Ω n ) = 1 . (5.12)By compactness, up to subsequences, there exists an open sets Ω ⋆ ∈ A k such thatΩ n H c → Ω ⋆ . By (5.11) we have m q,k ≤ F q,k (Ω ⋆ ) . Let us prove the opposite inequality. We notice that, by Theorem 3.8 and (5.12), forevery n there exists a sequence ( A n,m ) m ⊂ A k of Lipschitz sets, such that, as m → ∞ , P ( A n,m ) → P (Ω n ) and | A n,m | → | Ω n | . By Theorem 4.1, we have also T ( A n,m ) → T (Ω n ) as m → ∞ . Thus F q (Ω n ) = lim m →∞ F q ( A n,m ) . A standard diagonal argument allows us to define a subsequence A n,m n ∈ O k (Ω ⋆ ).Then we have F q,k (Ω ⋆ ) ≤ lim n F q ( A n,m n ) = lim n F q (Ω n ) = m q,k . Hence Ω ⋆ is a minimum for F q,k . Moreover, notice that there exists a compact set K containing ∂ Ω ⋆ such that, up to a subsequence, ∂ Ω n H → K . So, being Ω n connected,we have sup n ∂ Ω n < ∞ , and by Theorem 4.3, H ( ∂ Ω ⋆ ) ≤ H ( K ) ≤ lim inf n →∞ H (Ω n ) ≤ . To conclude we have only to show that Ω ∗ is nonempty. Notice that for n big enoughthere exists C > F q (Ω n ) < C . Thus we have C > F q (Ω n ) = T q (Ω n ) | Ω n | q +1 / = (cid:18) T (Ω n ) | Ω n | (cid:19) q | Ω n | q − / ≥ | Ω n | / − q ( √ k ) q , (5.13)where the last inequality follows by (5.3), using (5.12). By (4.2) we have also | Ω n | ≤ (1 + π ( k − ρ (Ω n )) ρ (Ω n ) . (5.14)Combining (5.13), (5.14) and the assumption q < /
2, we conclude that the sequenceof inradius ( ρ (Ω n )) must be bounded from below by some positive constant. ByLemma 4.4, Ω ⋆ is nonempty. (cid:3) Conclusions
We have seen that in the planar case the topological constraint present in classes A k is strong enough to ensure the existence of at least a relaxed optimizer. In higherdimensions this is no longer true and easy examples show that it is possible to constructsequences (Ω n ) in A k with P (Ω n ) bounded and T (Ω n ) →
0. This suggests that inhigher dimensions stronger constraints need to be imposed in order to have well posedoptimization problems.Another interesting issue is the analysis of the same kind of questions when theexponent 2 is replaced by a general p > T (Ω) thenbecomes the p -torsion T p (Ω) and it would be interesting to see how our results dependon the exponent p and if in this case the analysis in dimensions higher than two ispossible.Finally, shape functionals F (Ω) involving quantities other than perimeter and tor-sional rigidity are interesting to be studied: we point out some recent results in [18],[19]and references therein. However, to our knowledge, the study of these shape function-als under topological constraints as the ones of classes A k is still missing. Acknowledgments.
The work of GB is part of the project 2017TEXA3H “Gradientflows, Optimal Transport and Metric Measure Structures” funded by the Italian Min-istry of Research and University. The authors are member of the Gruppo Nazionaleper l’Analisi Matematica, la Probabilit`a e le loro Applicazioni (GNAMPA) of theIstituto Nazionale di Alta Matematica (INdAM).
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