Geometrical bounds for the torsion and the first eigenvalue of the Laplacian with Robin boundary condition

Geometrical b ounds for the torsion and the first eigen v alue of the Laplacian with Robin b oundary condition Rosa Barbato, Alba Lia Masiello, Rossano Sannip oli Abstract In this pap er, w e deal with functionals inv olving the torsion and the first eigenv alue of the Lapla- cian with Robin b oundary conditions (to whic h w e refer as Robin T orsion and Robin Eigenv alue), with other geometrical quantities, in the class of con v ex sets. Firstly , w e pro ve an upp er bound for the Robin T orsion in terms of the L 1 and L 2 norms of the distance function from the b oundary , whic h allows us to prov e a generalization of the Makai inequalit y inv olving the Robin T orsion, the Leb easgue measure, and the inradius of a conv ex set. Subsequen tly , we pro ve quan titative esti- mates for the Robin Makai functional and for the Robin Póly a functionals, whic h link the Lebesgue measure and the p erimeter with the Robin T orsion and the Robin Eigen v alue resp ectively . In par- ticular, we pro ve that the optimal v alues of all these shap e functionals are ac hieved b y slab domains. MSC 2020: 35P15, 49Q10, 35J05, 35J25. Keywords: P ólya estimates, Makai estimates, quan titative inequalities, Robin b oundary condi- tions. Con ten ts 1 In tro duction 1 2 Preliminaries 7 2.1 Notations and basic facts . . . . . . . . . . . . . . . . . . 7 2.2 Distance to the b oundary and inner parallel sets . . . . . 8 2.3 Useful results ab out the remainder terms . . . . . . . . . 9 2.4 A one dimensional Laplacian eigen v alue problem . . . . . 10 3 Upp er b ound for the Robin T orsion 11 3.1 On the optimal sets . . . . . . . . . . . . . . . . . . . . . 13 4 Quan titative estimates for the Robin T orsion 14 5 Quan titative estimate for the Robin eigen v alue 17 6 Quan titative inequalities inv olving A (Ω) 21 1 In tro duction Let Ω ⊂ R n b e an op en, b ounded set with Lipsc hitz b oundary . Let β > 0 b e a positive parameter and let us consider the follo wing problems (TP)    − ∆ u = 1 in Ω ∂ u ∂ ν + β u = 0 on ∂ Ω , (EP)    − ∆ u = λu in Ω ∂ u ∂ ν + β u = 0 on ∂ Ω , (1) 1 1 INTR ODUCTION 2 whic h are kno wn in literature as the T orsion problem and the Eigen v alue problem with Robin b oundary conditions, resp ectiv ely . In particular, the L 1 -norm of the unique solution to problem (TP) is called Robin T orsion of the set Ω , which has also the follo wing v ariational c haracterization: T β (Ω) = sup v ∈ H 1 (Ω)  ˆ Ω v dx  2 ˆ Ω |∇ v | 2 dx + β ˆ ∂ Ω v 2 d H n − 1 . (2) The spectrum of the Laplacian for the problem (EP) is discrete, and comp osed by a sequence of p ositiv e and p ositively diverging eigenv alues. The first eigenv alue of this sequence is called first Robin eigen v alue and has the following v ariational characterization λ β (Ω) = inf v ∈ H 1 (Ω) ˆ Ω |∇ v | 2 dx + β ˆ ∂ Ω v 2 d H n − 1 ˆ Ω v 2 dx . (3) In the present paper, w e establish qualitativ e and quan titative relations in the class of op en, b ounded con vex sets for shape functionals th at inv olve T β (Ω) and λ β (Ω) with the Leb esgue measure, the p erimeter and the inradius of Ω . The interest in such topic dates bac k to [ 24 , 26 ], where geometric inequalities for the Diric hlet T orsion and the Dirichlet eigenv alue are studied. W e start b y listing some results that will help clarify the framew ork and the quantities inv olved. Some state of art on the Dirichlet case Let us fo cus, for the momen t, in the case of Dirichlet b oundary conditions, whic h corresp onds to β = + ∞ . Let us denote b y T (Ω) and λ (Ω) the Dirichlet T orsion and the first Diric hlet Eigenv alue, resp ectiv ely . First of all, let us stress that T (Ω) is monotonically increasing and λ (Ω) is monotonically decreasing with respect to the set inclusion, and they satisfy the follo wing scaling prop erties: T ( t Ω) = t n +2 T (Ω) , λ ( t Ω) = t − 2 λ (Ω) , ∀ t > 0 . These tw o quantities, seen as functionals of Ω , ha ve attracted the atten tion of man y mathematicians in the last century due to their physical and engineering applications. F or instance, in the Diric hlet case and for n = 2 , the T orsion of Ω represents the torsional rigidit y of a three-dimensional bar with constan t cross-section Ω , whic h is the resistance of the bar to b e b ended. The first eigenv alue, instead, can b e in terpreted as the fundamental frequency of a bi-dimensional vibrating membrane whose shape is Ω and whic h is fixed to the boundary . Finding the b est shapes that optimize these functionals under some geometrical constrain ts is the goal of shap e optimization problems. T w o classical inequalities iden tify the ball as the optimal set under a v olume constraint, in the class of op en and b ounded sets. Let Ω ⊂ R n b e an open set with finite Leb esgue measure, and let B denote a ball. The first result is the w ell-known Sain t-V enant inequalit y , originally conjectured in [ 13 ], which can b e written in the scale-in v arian t form | Ω | − n +2 n T (Ω) ≤ | B | − n +2 n T ( B ) , where | Ω | stands for the Lebesgue measure of Ω . The second is the F ab er–Krahn inequalit y , stating that | Ω | 2 n λ (Ω) ≥ | B | 2 n λ ( B ) . 1 INTR ODUCTION 3 Moreo ver, since the latter half of the t wen tieth cen tury , sev eral further inequalities connecting T (Ω) and λ (Ω) ha ve b een explored (see, for example, [ 21 , 22 , 27 ]). Bounds of the torsion and the first eigen v alue ha ve b een in v estigated in terms of other geometrical quan tities, in the class of conv ex sets. The pioneers in this direction hav e b een Póly a, Makai and Hersc h. Denoting by P (Ω) and r (Ω) the perimeter and the inradius of Ω , resp ectiv ely (see Section 2 for the precise definitions), it holds that (I) 1 3 ≤ T (Ω) P 2 (Ω) | Ω | 3 ≤ 2 3 (I I) π 2 4 n 2 ≤ λ (Ω) | Ω | 2 P 2 (Ω) ≤ π 2 4 (I I I) 1 n ( n + 2) ≤ T (Ω) r (Ω) 2 | Ω | ≤ 1 3 (IV) π 2 4 ≤ λ (Ω) r (Ω) 2 ≤ λ 1 ( B 1 ) . (4) W e here briefly discuss the literature b ehind these b ounds. The b ounds in (I) wer e prov ed, in the planar setting, b y Makai and P óly a in [ 24 , 26 ]. Moreov er they pro ved that these estimates cannot b e impro ved. In particular, the lo w er b ound is asymptotically achiev ed b y a sequence of thinning rectan- gles, while the upp er one by a sequence of thinning triangles. The left-hand inequalit y in (I) was later extended to any dimension in [ 16 ], where it w as established for all open, b ounded, conv ex subsets of R n , again with optimalit y ac hieved by a family of thinning cylinders. A dditional generalizations are presen ted in [ 4 , 36 ]. Regarding (I I) , the upper estimate was first established in [ 26 ] in the planar case, with sharpness attained b y sequences of thinning rectangles. This result w as subsequently extended in [ 16 ] to higher dimensions and, more generally , to the first eigenv alue of the anisotropic p -Laplacian. The low er bound w as obtained b y Makai in dimension t wo [ 24 ], and later generalized to all dimensions in [ 8 ] and to the anisotropic case in [ 15 ]. T urning to (I I I) , Makai prov ed the upper estimate in t wo dimensions (see [ 24 ]), also showing that it is sharp, with extremal b eha vior exhibited b y a sequences of thinning rectangles. The lo wer estimate originates in [ 27 ], where equality holds only for disks. Later, b oth b ounds w ere extended to arbitrary dimensions and to more general op erators in [ 17 ], where it is further shown that the upp er b ound is achiev ed by a suitable family of thinning cylinders. Finally , concerning the functional in (IV) , the righ t-hand inequality follo ws immediately from monotonicit y under set inclusion. The left-hand inequality is the classical Hersc h–Protter inequality: it was first pro ved b y Hersch in t w o dimensions [ 20 ] and subsequen tly extended to all dimensions b y Protter [ 29 ]. A dditional dev elopments can b e found in [ 9 , 10 , 28 , 25 ]. Only recen tly , the lo wer b ound in (I) and (IV) , and the upper bound in (I I) and (II I) , hav e b een impro ved, b y pro ving con tin uity inequalities and quan titative v ersions in the papers [ 3 , 4 ]. Here, the authors managed to add geometrical remainder terms which allow ed to b etter understand the nature of the optimal sequences. Bac k to the Robin case One of our aim is to extend this analysis to the Robin case, so let now b e β ∈ (0 , + ∞ ) . The presence of Robin b oundary conditions drastically c hanges the problems studied. The first big difference with the Diric hlet case is that there is no monotonicit y with resp ect to the set inclusion. Moreov er, the scaling prop erties do not hold in a standard w a y; indeed, it holds T β /t ( t Ω) = t n +2 T β (Ω) , λ β /t ( t Ω) = t − 2 λ β (Ω) , ∀ t > 0 . 1 INTR ODUCTION 4 Regarding the shap e optimization issue, it is kno wn that the Robin counterpart of the Sain t-V enant and F ab er-Krahn inequalities holds. Concerning the Robin T orsion, the authors in [ 11 ], prov ed in the class of bounded Lipsc hitz sets in R n , that T β (Ω) ≤ T β (Ω ♯ ) where Ω ♯ is the ball cen tered at the origin having the same measure as Ω . Although the result is analogous to the Diric hlet case, the tec hniques used to pro v e it are completely different. This is due to the fact that the solution to (TP) is not constan t in general on ∂ Ω and therefore all rearrangemen ts argumen ts or T alen ti inequalities fail. Ab out the first Robin eigen v alue, the corresp onding F ab er-Krahn inequalit y has b een prov ed b y Bossel [ 7 ] in dimension 2 and subsequen tly by Daners [ 12 ] in higher dimensions. No wada ys, this inequality is kno wn as the Bossel-Daners inequality and guaran tees that in the class of b ounded Lipsc hitz sets in R n w e ha ve λ β (Ω) ≥ λ β (Ω ♯ ) , where again Ω ♯ is the ball cen tered at the origin having the same measure as Ω . Surprisingly , a rearrangemen t tec hnique in this case is useful to pro ve the ab o v e inequalit y . As already men tioned, one of the aims of this pap er is to obtain estimates for λ β (Ω) and T β (Ω) in terms of geometric quantities related to Ω , in some case impro ving in a quan titative wa y the results obtained in [ 14 , 18 , 31 , 35 ]. F or a smo oth, b ounded, and mean con vex domain Ω , in [ 18 ] the authors prov e a low er b ound for the first Robin eigenv alue in terms of the first eigenv alue of a one-dimensional problem, and, as a consequence, a lo wer estimate in terms of the inradius r (Ω) of the set, λ β (Ω) ≥  π 2  2 1  r (Ω) + π 2 β − 1  2 . (5) This inequalit y , for β → + ∞ , reduces to the Hersh-Protter inequality , whic h is the lo wer b ound in (IV) . In [ 14 ], the author studies the problem of bounding the Robin T orsion and the Robin Eigenv alue in terms of p erimeter and measure of the set Ω . The pro of of these results rely on a dimensional reduction argument and on the study of the one-dimensional problem for the Robin eigen v alue and the Robin T orsion. The one-dimensional eigenv alue problem tak en in to account is the follo wing        X ′′ + ν X = 0 in (0 , s 0 ) , X ′ (0) = 0 , X ′ ( s 0 ) + β X ( s 0 ) = 0 . (6) If w e denote by ν 1 ( β , s 0 ) the first eigen v alue of ( 6 ) and w e choose s 0 = | Ω | /P (Ω) , then the author pro ves that among open and b ounded conv ex sets in R n it holds λ β (Ω) ≤ ν 1  β , | Ω | P (Ω)  , (7) b eing this inequalit y asymptotically sharp on slabs, whic h are the cartesian pro duct b et w een an in terv al and R n − 1 (see Section 2 for the precise definition). In this case, the dep endence on the geometry of the upp er b ound is not explicit as in (I I) , but it is p ossible, b y exploiting the property of ν 1 ( β , · ) (see subsection 2.4 ), to prov e an explicit b ound 1 INTR ODUCTION 5 λ β (Ω) ≤ π 2 4   1 1 + 2 r (Ω) β   P 2 (Ω) | Ω | 2 . (8) The price to pa y in order to ha ve an explicit geometrical upp er b ound is the loss of the sharpness of the estimate. Nev ertheless, the latter inequalit y is the generalization to the Robin setting of the upp er b ound in (II) , and it reduces to it when β div erges. Lastly , for the Robin T orsion, a Póly a-t yp e low er b ound was pro ved in [ 14 ] T β (Ω) P 2 (Ω) | Ω | 3 ≥  1 3 + 1 r (Ω) β  , (9) and, once again, one can see that for β large we recov er the lo w er b ound in (I) , but in this case, the minimizing sequences w ere not c haracterized. Our first aim is to obtain b ounds for the Robin T orsion in terms of the measure and the inradius of the set, obtaining a Robin counterpart of (II I) . In order to do that, w e need to bound the Robin T orsion in terms of the L 1 and L 2 -norms of the distance function from the boun dary of Ω (see Section 2 for the definition and prop erties). In the case of the Dirichlet T orsion, it was prov ed by Makai (see [ 24 ]) in t wo dimensions and in [ 28 ] in higher dimensions and in a more general setting. The result is the following Theorem 1.1. L et Ω b e a b ounde d, op en, nonempty and c onvex set of R n . Then T β (Ω) ≤ ˆ Ω d ( x, ∂ Ω) 2 dx + 2 β ˆ Ω d ( x, ∂ Ω) dx. (10) W e highlight that to prov e Theorem 1.1 w e use the demonstration tec hnique implemented in [ 24 ]. As already p oin ted out b efore, this estimate allo ws us to pro ve a Makai-type inequality in the Robin case and it is contained in the next corollary . Corollary 1.2. L et Ω b e a b ounde d, op en, nonempty and c onvex set of R n . Then T β (Ω) r (Ω) 2 | Ω | ≤ 1 3 + 1 r (Ω) β . (11) W e stress that w e did not mention anything ab out the sharpness of inequalities ( 9 ) and ( 11 ), and the reason is that we need one inequalit y to pro ve the sharpness of the other. Hence, our second aim is to study the equalit y case of inequalities ( 9 ) and ( 11 ) and, in order to do that, we actually fo cus on the stability issues. All the P ólya-t yp e inequalities are asymptotically sharp along a sequence of slabs. The lac k of an optimal set in these inequalities might app ear to be an obstacle when studying the corresp onding stabilit y problem. T o o vercome this issue, in [ 1 , 2 , 3 , 4 ] t wo new remainder terms w ere in tro duced. The one w e will fo cus on is denoted b y R (Ω) , and it is a quan tity b etw een 0 and n , defined as R (Ω) := P (Ω) r (Ω) | Ω | − 1 . The infimum 0 is ac hieved, for instance, b y sequences of thinning cylinders or slabs (see Section 2 , Proposition 2.4 ) and, for this reason, the remainder term R (Ω) will pla y a cen tral role in the next quan titativ e results. The first one w e in tro duce is a quan titative result ab out the Makai Robin inequalit y , and it is the following 1 INTR ODUCTION 6 Theorem 1.3. L et Ω ⊂ R n b e an op en, b ounde d, nonempty and c onvex. Then it holds 2  1 3 + 1 r (Ω) β  R (Ω) ≥  1 3 + 1 β r (Ω)  − T β (Ω) | Ω | r (Ω) 2 ≥ C 1 R (Ω) , (12) with C 1 = n + 1 3 n (2 n − 1) . A s a c onse quenc e, se quenc e of slab-like domain ar e sharp for ine quality ( 11 ) . Regarding the P ólya inequality for the Robin T orsion, w e get Theorem 1.4. L et Ω ⊂ R n b e an op en, b ounde d., nonempty and c onvex set. Then, it holds ( n + 1)  1 3 + 1 r (Ω) β  R (Ω) ≥ T β (Ω) P 2 (Ω) | Ω | 3 −  1 3 + 1 β r (Ω)  ≥ C 2 R 3 (Ω) , (13) with C 2 = 1 2 3 · 3 4 n 3 . A s a c onse quenc e, se quenc e of slab-like domain ar e sharp for ine quality ( 9 ) . The last main Theorem is the quantitativ e version of the inequalit y ( 7 ). Theorem 1.5. L et Ω ⊂ R n b e an op en, b ounde d, nonempty and c onvex set. L et s 0 = | Ω | P (Ω) , and let ν 1 ( β , s 0 ) b e the eigenvalue define d in ( 2.7 ) . Then, it holds K 1 R (Ω) ≥ ν 1 ( β , s 0 ) | Ω | 2 P 2 (Ω) − λ β (Ω) | Ω | 2 P 2 (Ω) ≥ K 2 R (Ω) 4 , (14) wher e K 1 = π 2 2 s 1 + 4 β 2 π 2 ( r (Ω) 2 + π 2 4 β r (Ω)) , K 2 = 1 2 · 3 4 π (2 n − 1) n 3   π 2 4 1 1 + π 2 n 4 β r (Ω)   4 . As a corollary of Theorem 1.5 and the b ound ( 8 ), we can obtain the following Corollary 1.6. L et Ω ⊂ R n b e an op en, b ounde d and c onvex set. Then, it holds π 2 4   1 1 + 2 r (Ω) β   − λ β (Ω) | Ω | 2 P 2 (Ω) ≥ C 3   π 2 4 1 1 + π 2 n 4 β r (Ω)   2 R (Ω) 4 , (15) with C 3 = 1 2 · 3 4 π (2 n − 1) n 3 . Finally , let us note that another remainder term was in tro duced in [ 3 ], which we denote b y A (Ω) , and it is a measure of the "thinness" of the set A (Ω) = w Ω d (Ω) , where w Ω and d (Ω) represen t the minimal width and diameter of the set (for the definition see Sec- tion 2 ). As prov ed in [ 3 ], there exists a p ositiv e dimensional constan t K = K ( n ) > 0 , suc h that R (Ω) ≥ K A (Ω) , and the rev erse inequalit y is not true. F or this reason, the low er b ound in ( 12 ), ( 13 ) and ( 15 ) can b e rewritten also in terms of A (Ω) , where the sharp exponent is ensured only in equalit y 2 PRELIMINARIES 7 ( 12 ). F or completeness, in Section 6 , we manage to prov e that the exp onen t is sharp ev en for the other tw o inequalities, as prov ed in Proposition 6.1 and in Prop osition 6.2 . Plan of the pap er: In Section 2 we recall some basic notions and definitions, and we recall some classical results, fo cusing in particular on the class of conv ex sets. In Section 3 w e pro ve the upp er b ounds for the Robin T orsion, that are Theorem 1.1 and Corollary 1.2 . In Section 4 and 5 we pro ve the quan titative estimates inv olving R (Ω) , whic h are the conten t of Theorems 1.3 - 1.4 and 1.5 , giving also some remark ab out the nature of the optimal sequences. Lastly , in section 6 w e pro v e quantitativ e estimates inv olving the remainder term A (Ω) . 2 Preliminaries 2.1 Notations and basic facts Throughout this article, | · | will denote the Euclidean norm in R n , while · is the standard Euclidean scalar pro duct for n ≥ 2 . By H k ( · ) , for k ∈ [0 , n ) , we denote the k − dimensional Hausdorff measure in R n . The p erimeter of Ω in R n will b e denoted by P (Ω) and, if P (Ω) < ∞ , we sa y that Ω is a set of finite perimeter. In our case, Ω is a b ounded, op en and conv ex set; this ensures us that Ω is a set of finite perimeter and that P (Ω) = H n − 1 ( ∂ Ω) . Some references for results relativ e to the sets of finite p erimeter and for the coarea formula are, for instance, [ 5 , 23 ]. W e give no w the minimal width (or thickness) of a con v ex set. Definition 2.1. Let Ω b e a b ounded, open and con vex set of R n . The supp ort function of Ω is defined as h Ω ( y ) = sup x ∈ Ω ( x · y ) , y ∈ R n . The width of Ω in the direction y ∈ R is defined as ω Ω ( y ) = h Ω ( y ) + h Ω ( − y ) and the minimal width of Ω as w Ω = min { ω Ω ( y ) | y ∈ S n − 1 } . W e will denote b y r (Ω) the inradius of Ω , i.e. r (Ω) = sup { r ∈ R : B r ( x ) ⊂ Ω , x ∈ Ω } , (16) and by diam (Ω) the diameter of Ω , that is diam (Ω) = sup x,y ∈ Ω | x − y | . Definition 2.2. Let Ω ℓ b e a sequence of non-empt y , b ounded, op en and con vex sets of R n . Ω ℓ is a sequence of thinning domains if w Ω ℓ diam (Ω ℓ ) ℓ → 0 − − → 0 . A particular sequence of thinning domains are what w e call a slab-like domain. 2 PRELIMINARIES 8 Definition 2.3. Let K ⊂ R n − 1 b e a bounded conv ex set, we define a sequence of slab-like domains as Ω ℓ = ( − a ℓ , a ℓ ) × 1 ℓ K, ℓ → 0 , where a ℓ is a positive conv erging sequence. On the other hand, we define the sequence of thinning cylinders as sequence C ℓ =  − ℓ 2 , ℓ 2  × K. (17) W e recall in the follo wing Prop osition the relation b et w een the inradius and the minimal width (see for example [ 30 , 33 , 34 , 6 ]). Prop osition 2.1. L et Ω b e a b ounde d, op en, and c onvex set of R n . Then, the fol lowing estimates hold: w Ω 2 ≥ r (Ω) ≥              w Ω √ n + 2 2 n + 2 n even w Ω 1 2 √ n n o dd , (18) Moreo ver, the following estimate inv olving the p erimeter and the diameter holds P (Ω) ≤ nω n  diam (Ω) 2  n − 1 . (19) 2.2 Distance to the b oundary and inner parallel sets Let Ω ⊂ R n b e a nonempty , b ounded, open and con vex set. W e introduce the distance from the b oundary d ( x, ∂ Ω) = inf y ∈ ∂ Ω | x − y | , x ∈ Ω . As a consequence of the con vexit y of Ω , the function d ( · , ∂ Ω) is concav e in Ω . F or ev ery t ∈ [0 , r (Ω)] , where r (Ω) denotes the inradius of Ω , w e define the asso ciated inner parallel sets by Ω t := { x ∈ Ω : d ( x, ∂ Ω) > t } . W e use the notation µ ( t ) := | Ω t | , P ( t ) := P (Ω t ) , t ∈ [0 , r (Ω)] . Being |∇ d | = 1 a.e. in Ω , the Coarea form ula giv es µ ( t ) = ˆ { d>t } dx = ˆ r (Ω) t ˆ { d = s } d H n − 1 ds = ˆ r (Ω) t P ( s ) ds. As a consequence, the function µ is absolutely con tinuous and decreasing in [0 , r (Ω)] , and satisfies µ ′ ( t ) = − P ( t ) for a.e. t ∈ (0 , r (Ω)) . (20) The conca vity of the distance function, combined with the Brunn–Mink owski inequality for the p erimeter (see [ 32 , Theorem 7.4.5]), implies that the map t 7− → P ( t ) 1 n − 1 2 PRELIMINARIES 9 is concav e in [0 , r (Ω)] . In particular, it is absolutely con tinuous in (0 , r (Ω)) and admits a finite right deriv ativ e at t = 0 . Since this function is strictly decreasing, it follows that P ( t ) is strictly decreasing as well. Moreov er, concavit y ensures that  P ( t ) 1 n − 1  ′′ ≤ 0 in the sense of distributions. In tegrating ( 20 ) from 0 to t and using the fact that P ( s ) ≤ P (Ω) for conv ex sets, w e obtain the estimate µ ( t ) ≥ | Ω | − P (Ω) t for a.e. t ∈ [0 , r (Ω)] . (21) W e recall some b ound on the p erimeter and the measure of the inner parallel sets pro ved in [ 3 ]. Lemma 2.2. L et Ω b e a non-empty, b ounde d, op en and c onvex set of R n . Then, ther e exists a p ositive dimensional c onstant c n , such that P ( t ) ≤ P (Ω) − c n | Ω | − µ ( t ) P 1 n − 1 (Ω) . (22) Lemma 2.3. L et Ω b e a non-empty, b ounde d, op en and c onvex set of R n . Then µ ( t ) ≤ P (Ω)( r (Ω) − t ) + ( r (Ω) − t ) 2 2( n − 1) P ′ ( t ) . (23) 2.3 Useful results ab out the remainder terms W e define the follo wing t wo measures of asymmetry A (Ω) := w Ω diam (Ω) , and R (Ω) := P (Ω) r (Ω) | Ω | − 1 . W e recall the following b ounds, prov ed in [ 6 ] and then generalized in any dimension and for more general op erators (see for instance [ 15 ]. Prop osition 2.4. L et Ω b e a non-empty, b ounde d, op en and c onvex set of R n . Then, 1 < P (Ω) r (Ω) | Ω | ≤ n. (24) The lower b ound is sharp on a se quenc e of thinning cylinders, while the upp er b ound is achieve d for instanc e if Ω (1 − t ) R Ω = t Ω for t ∈ (0 , 1) or on a se quenc e of thinning pyr amids. W e also recall a low er b ound in ( 24 ) in terms of A (Ω) pro ved in [ 3 ]. Prop osition 2.5. L et Ω b e a non-empty, b ounde d, op en and c onvex set of R n . Then, ther e exists a p ositive c onstant K = K ( n ) dep ending only on the dimension of the sp ac e, such that R (Ω) ≥ K ( n ) A (Ω) (25) The exp onent of the quantity A (Ω) is sharp. A r everse ine quality c annot b e true, sinc e ther e ar e se quenc es of thinning domains for which the functional R (Ω) is not c onver ging to zer o (for instanc e a se quenc e of thinning triangles in dimension 2). 2 PRELIMINARIES 10 Another useful result contained in [ 3 ] is the follo wing. Lemma 2.6. L et Ω b e a non-empty, b ounde d, op en and c onvex set of R n . Then, µ  | Ω | P (Ω)  ≥ q 1 ( n, Ω) | Ω | , (26) P  | Ω | P (Ω)  ≤ q 2 ( n, Ω) P (Ω) , (27) wher e q 1 ( n, Ω) = R (Ω) 6 n and q 2 ( n, Ω) =  1 + R (Ω) n  − 1 . (28) 2.4 A one dimensional Laplacian eigenv alue problem In this Section, w e study a one-dimensional problem that will b e useful to estimate λ β (Ω) . W e refer to [ 14 , 18 , 35 ] for further details. W e consider the eigen v alue problem in the unkno wn X = X ( s ) :        X ′′ + ν X = 0 in (0 , s 0 ) , X ′ (0) = 0 , X ′ ( s 0 ) + β X ( s 0 ) = 0 , (29) where s 0 is a giv en p ositive num b er. Theorem 2.7. L et β ≥ 0 . Then ther e exists the smal lest eigenvalue ν of ( 29 ) , which has the fol lowing variational char acterization: ν 1 ( β , s 0 ) = inf v ∈ W 1 , 2 (0 ,s 0 ) v ′ (0)=0 ´ s 0 0 | v ′ ( s ) | 2 ds + β v ( s 0 ) 2 ´ s 0 0 | v ( s ) | 2 ds . Mor e over, the c orr esp onding eigenfunctions ar e unique up to a multiplic ative c onstant and have c on- stant sign. The first eigenvalue ν 1 ( β , s 0 ) is p ositive. Mor e over, the first eigenfunction is X ( s ) = cos  q ν 1 ( β , s 0 ) s  , s ∈ (0 , s 0 ) , and the eigenvalue ν 1 ( β , s 0 ) is the first p ositive value that satisfies ν = β 2 tan 2 ( √ ν s 0 ) . (30) Remark 2.1. It is p ossible to pro ve that X is decreasing when β > 0 , so 0 ≤ X ≤ 1 . Hence 0 ≤ arccos ( X ( s )) ≤ π 2 . Therefore, we ha ve 0 ≤ √ ν 1 s ≤ π 2 , and then ν 1 ( β , s 0 ) ≤  π 2 s 0  2 . It is p ossible to obtain sharp er b ounds if one consider the follo wing elementary b ounds on the tan( x ) 3 UPPER BOUND F OR THE R OBIN TORSION 11 2 x π 2 4 − x 2 ≤ tan( x ) ≤ π 2 4 x π 2 4 − x 2 . Indeed, one obtain π 2 4 s 2 0 1 1 + π 2 4 β s 0 ≤ ν 1 ( β , s 0 ) ≤ π 2 4 s 2 0 1 1 + 2 β s 0 . (31) 3 Upp er b ound for the Robin T orsion Concerning the upper b ound in Theorem 1.1 , we prov e an estimate of the Robin torsion in terms of the L 1 and L 2 -norms of the distance function from the boundary . Pr o of of The or em 1.1 . W e will giv e the proof in the case of Ω b eing a con vex p olytope, whic h is defined as the conv ex h ull of finitely man y p oin ts in R n . The pro of for a general op en, b ounded con vex set will follow by appro ximation arguments (see [ 32 , Theorem 1 . 8 . 16 ]). Let us assume that the p olytop e Ω has N ∈ N facets, denoted b y F i , i = 1 , ..., N , and let us decomp ose Ω = N [ i =1 E i , where E i is a connected comp onen t of Ω , where the map x → d ( x, ∂ Ω) is differentiable. If v is the Robin torsion function in Ω , then denoting by ∇ x ′ v = ( ∂ 1 v , ..., ∂ n − 1 v ) , where x ′ = ( x 1 , ..., x n − 1 ) and b y ∂ k v = ∂ v ∂ x k , k = 1 , ..., n , by Cauc hy’s inequalit y , w e obtain T β (Ω) = N X i =1 ˆ E i v dx ! 2 N X i =1  ˆ E i ( |∇ x ′ v | 2 + ( ∂ n v ) 2 ) dx + β ˆ F i v 2 d H n − 1  ≤ N X i =1 ˆ E i v dx ! 2 ˆ E i ( |∇ x ′ v | 2 + ( ∂ n v ) 2 ) dx + β ˆ F i v 2 d H n − 1 . Without loss of generality , we can assume that F i lies on the R n − 1 h yp erplane, so that E i can b e espressed as follo ws E i = { ( x ′ , y ) : x ′ ∈ F i , 0 < y < f i ( x ′ ) } , i = 1 , ..., N , 3 UPPER BOUND F OR THE R OBIN TORSION 12 where f i : x ′ ∈ F i → R parametrizes ∂ E i ∩ Ω . Clearly F i = E i ∩ { y = 0 } . In this w ay T β ,i : = ˆ E i v dx ! 2 ˆ E i ( |∇ x ′ v | 2 + ( ∂ n v ) 2 ) dx + β ˆ F i v 2 dx ′ ≤ ˆ E i v dx ! 2 ˆ E i ( ∂ n v ) 2 dx + β ˆ F i v 2 dx ′ = ˆ F i dx ′ ˆ f i ( x ′ ) 0 v dx n ! 2 ˆ F i  ˆ f i ( x ′ ) 0 ( ∂ n v ) 2 dx n + β v 2  dx ′ . ≤ ˆ F i ˆ f i ( x ′ ) 0 v dx n ! 2 ˆ f i ( x ′ ) 0 ( ∂ n v ) 2 dx n + β v 2 dx ′ , (32) where, in the last line, w e hav e used again the Cauch y-Sch w arz inequality . The term in the last in tegral is such that ˆ f i ( x ′ ) 0 v dx n ! 2 ˆ f i ( x ′ ) 0 ( ∂ n v ) 2 dx n + β v 2 ≤ T β ([0 , s ]) , (33) with s = f i ( x ′ ) and T β ([0 , s ]) is the 1D Robin T orsion in the segment [0 , s ] , that is to sa y T β ([0 , s ]) = sup g ∈ H 1 ([0 ,s ])  ˆ s 0 g ( t ) dt  2 ˆ s 0 ( g ′ ( t )) 2 dt + β g 2 (0) . The Robin T orsion is the L 1 -norm of the unique solution to the follo wing ODE        − φ ′′ ( t ) = 1 in [0 , s ] φ ′ ( s ) = 0 φ ′ (0) = β φ (0) . (34) In tegrating ( 34 ) 1 , we get φ ( t ) = − t 2 2 + c 1 t + c 2 , and imp osing the boundary conditions we arriv e to φ ( t ) = − t 2 2 + st + s β . Therefore T β ([0 , s ]) = ˆ s 0 φ ( t ) dt = − s 3 6 + s 3 2 + s 2 β = s 3 3 + s 2 β . 3 UPPER BOUND F OR THE R OBIN TORSION 13 Com bining this with ( 32 ), w e get T β ,i ≤ ˆ F i  f i ( x ′ ) 3 3 + f i ( x ′ ) 2 β  dx ′ = ˆ F i ˆ f i ( x ′ ) 0  y 2 + 2 y β  dy dx ′ = ˆ E i d ( x, ∂ Ω) 2 dx + 2 β ˆ E i d ( x, ∂ Ω) dx. Ev entually , T β (Ω) ≤ N X i =1 T β ,i ≤ ˆ Ω d ( x, ∂ Ω) 2 dx + 2 β ˆ Ω d ( x, ∂ Ω) dx. F rom Theorem 1.1 , w e obtain a generalization of the Makai inequality for the Robin T orsion as follo ws: Pr o of of Cor ol lary 1.2 . F rom the pro of of Theorem 1.1 we kno w that T β ,i ≤ ˆ F i  f i ( x ′ ) 3 3 + f i ( x ′ ) 2 β  dx ′ ≤ r (Ω) 2 3 ˆ F i f i ( x ′ ) dx ′ + r (Ω) β ˆ F i f i ( x ′ ) dx ′ . But recalling that ˆ F i f i ( x ′ ) dx ′ = ˆ F i ˆ f i ( x ′ ) 0 dy dx ′ = ˆ E i dx = | E i | , w e ha ve T β ,i ≤ r (Ω) 2 | E i | 3 + r (Ω) | E i | β . Summing up for i = 1 , ..., n and dividing by r (Ω) 2 | Ω | we get the thesis. 3.1 On the optimal sets In this subsection w e discuss about the optimal sequences of sets that achiev es the equality case in inequalities ( 9 ), ( 10 ) and ( 11 ).Let us stress that inequality ( 11 ) is a direct consequence of inequality ( 10 ), meaning that they share the same optimal sequences. Inequalities ( 9 ) and ( 11 ) are b oth optimal on slabs, and one can use either inequalities to prov e the sharpness of the other. Indeed, if one wan ts to prov e that ( 11 ) is sharp along a sequence of slab-like domain, we can comp ose ( 11 ) with ( 9 ) 1 3 + 1 r (Ω ℓ ) β ≥ T β (Ω ℓ ) r (Ω ℓ ) 2 | Ω ℓ | ≥  1 3 + 1 r (Ω ℓ ) β  | Ω ℓ | 2 P 2 (Ω ℓ ) r 2 (Ω ℓ ) . (35) and if w e send ℓ → 0 , b y Prop osition 2.4 w e ha v e | Ω ℓ | 2 P 2 (Ω ℓ ) r 2 (Ω ℓ ) → 1 , pro ving the sharpness. Analogously , if we wan t to prov e that ( 9 ) is sharp on a family of slab-lik e domains, we can use ( 11 ), obtaining 1 3 + 1 r (Ω ℓ ) β ≤ T β (Ω ℓ ) P 2 (Ω ℓ ) | Ω ℓ | 3 ≤  1 3 + 1 r (Ω ℓ ) β  P 2 (Ω ℓ ) r (Ω ℓ ) 2 | Ω ℓ | 2 . (36) 4 QUANTIT A TIVE ESTIMA TES F OR THE R OBIN TORSION 14 Instead, if w e consider sequences of thinning cylinders, it is easily seen that lim ℓ → + ∞ T β (Ω ℓ ) r (Ω ℓ ) 2 | Ω ℓ | = lim ℓ → + ∞ T β (Ω ℓ ) P 2 (Ω ℓ ) | Ω ℓ | 3 = + ∞ . While ( 35 ) naturally highlights slab-lik e domains as optimal candidates, it seems that it do es not immediately rev eal the role of thinning cylinders as in the Dirichlet case. This is due to the fact that the Robin T orsion do es not scale in the standard wa y , but it b ecomes visible once we normalize b oth functionals by 1 3 + 1 r (Ω ℓ ) β . In fact w e hav e that 1 ≥ T β (Ω ℓ ) r (Ω ℓ ) 2 | Ω ℓ |  1 3 + 1 r (Ω ℓ ) β  − 1 ≥ | Ω ℓ | 2 P 2 (Ω ℓ ) r 2 (Ω ℓ ) , and 1 ≤ T β (Ω ℓ ) P 2 (Ω ℓ ) | Ω ℓ | 3  1 3 + 1 r (Ω ℓ ) β  − 1 ≤ P 2 (Ω ℓ ) r (Ω ℓ ) 2 | Ω ℓ | 2 . The righ t-hand side of b oth previous inequalies tends to 1 as ℓ → ∞ , implying that thinning cylinders are also optimal sequences. 4 Quan titativ e estimates for the Robin T orsion W e can no w prov e the quan titative improv emen t of the Makai-type inequality for the Robin T orsion. Pr o of of The or em 1.3 . Let us recall the upp er bound ( 10 ), T β (Ω) ≤ ˆ Ω d ( x, ∂ Ω) 2 dx + 2 β ˆ Ω d ( x, ∂ Ω) dx, and let us start b y considering the first term on the righ t-hand side. W e stress that, for this term, the computations can b e found in [ 3 ] and w e rewrite them for the sake of completeness. Applying Coarea F orm ula, in tegrating by parts and using estimate ( 23 ), w e get ˆ Ω d ( x, ∂ Ω) 2 dx = ˆ r (Ω) 0 t 2 P ( t ) dt = 2 ˆ r (Ω) 0 tµ ( t ) dt ≤ 2 ˆ r (Ω) 0 t ( r (Ω) − t ) P ( t ) dt + 1 n − 1 ˆ r (Ω) 0 t ( r (Ω) − t ) 2 P ′ ( t ) dt. (37) If we in tegrate b y parts the second in tegral on the right-hand side of ( 37 ), we get ˆ r (Ω) 0 t ( r (Ω) − t ) 2 P ′ ( t ) dt = t ( r (Ω) − t ) 2 P ( t )     r (Ω) 0 − ˆ r (Ω) 0 [( r (Ω) − t ) 2 − 2 t ( r (Ω) − t )] P ( t ) dt = 2 ˆ r (Ω) 0 t ( r (Ω) − t ) P ( t ) dt − ˆ r (Ω) 0 ( r (Ω) − t ) 2 P ( t ) dt. (38) W e notice that one of the tw o integrals in ( 38 ) is equal to the one in ( 37 ), therefore ˆ r (Ω) 0 t 2 P ( t ) dt ≤ 2 n n − 1 ˆ r (Ω) 0 t ( r (Ω) − t ) P ( t ) dt − 1 n − 1 ˆ r (Ω) 0 ( r (Ω) − t ) 2 P ( t ) dt = 2 n + 1 n − 1 r (Ω) ˆ r (Ω) 0 tP ( t ) dt − 2 n + 1 n − 1 ˆ r (Ω) 0 t 2 P ( t ) dt − r (Ω) 2 | Ω | n − 1 . 4 QUANTIT A TIVE ESTIMA TES F OR THE R OBIN TORSION 15 Summing up the same terms, w e ha v e ˆ r (Ω) 0 t 2 P ( t ) dt ≤ 2( n + 1) 3 n r (Ω) ˆ r (Ω) 0 tP ( t ) dt − r (Ω) 2 | Ω | 3 n . (39) W e no w estimate the integral on the righ t-hand side of ( 39 ). Integrating by parts and using again ( 20 ) and ( 23 ) ˆ r (Ω) 0 tP ( t ) dt = ˆ r (Ω) 0 µ ( t ) dt ≤ ˆ r (Ω) 0 ( r (Ω) − t ) P ( t ) dt + 1 2( n − 1) ˆ r (Ω) 0 ( r (Ω) − t ) 2 P ′ ( t ) dt = n n − 1 ˆ r (Ω) 0 ( r (Ω) − t ) P ( t ) dt − r (Ω) 2 P (Ω) 2( n − 1) = n n − 1 r (Ω) | Ω | − n n − 1 ˆ r (Ω) 0 tP ( t ) dt − r (Ω) 2 P (Ω) 2( n − 1) . Therefore ˆ r (Ω) 0 tP ( t ) dt ≤ n 2 n − 1 r (Ω) | Ω | − r (Ω) 2 P (Ω) 2(2 n − 1) (40) Com bining ( 39 ) and ( 40 ), we get ˆ r (Ω) 0 t 2 P ( t ) dt ≤ 2( n + 1) 3(2 n − 1) r (Ω) 2 | Ω | − n + 1 3 n (2 n − 1) r (Ω) 3 P (Ω) − r (Ω) 2 | Ω | 3 n = r (Ω) 2 | Ω | 3 + n + 1 3 n (2 n − 1)  r (Ω) 2 | Ω | − r (Ω) 3 P (Ω)  = r (Ω) 2 | Ω | 3 − n + 1 3 n (2 n − 1) r (Ω) 2 | Ω |R (Ω) . With regards to the second in tegral, applying Coarea formula and integrating by parts, w e get ˆ Ω d ( x, ∂ Ω) dx = ˆ r (Ω) 0 tP ( t ) dt = ˆ r (Ω) 0 µ ( t ) dt. Recalling that µ ( t ) ≤ P ( t )( r (Ω) − t ) , we ha ve that ˆ r (Ω) 0 tP ( t ) dt = ˆ r (Ω) 0 µ ( t ) dt ≤ r (Ω) | Ω | − ˆ r (Ω) 0 tP ( t ) dt, so that ˆ r (Ω) 0 tP ( t ) dt ≤ r (Ω) | Ω | 2 . Therefore 2 β ˆ Ω d ( x, ∂ Ω) dx ≤ r (Ω) | Ω | β , and  1 3 + 1 r (Ω) β  − T β (Ω) r (Ω) 2 | Ω | ≥ n + 1 3 n (2 n − 1) R (Ω) . 4 QUANTIT A TIVE ESTIMA TES F OR THE R OBIN TORSION 16 Let us now prov e the upp er b ound. If we m ultiply and divide the functional by P 2 (Ω) / | Ω | 2 and use the lo wer b ound for the Póly a functional ( 9 ), we get 1 3 + 1 r (Ω) β − T β (Ω) r (Ω) 2 | Ω | = 1 3 + 1 r (Ω) β − T β (Ω) P 2 (Ω) | Ω | 3 | Ω | 2 r (Ω) 2 P 2 (Ω) ≤  1 3 + 1 r (Ω) β   1 − | Ω | 2 r (Ω) 2 P 2 (Ω)  =  1 3 + 1 r (Ω) β   1 + | Ω | r (Ω) P (Ω)  1 − | Ω | r (Ω) P (Ω)  ≤ 2  1 3 + 1 r (Ω) β   P (Ω) r (Ω) | Ω | − 1  . W e pro ceed to the pro of of Theorem 1.4 , starting with the low er b ound. Pr o of of The or em 1.4 . W e start by proving the low er b ound in ( 13 ). Let us consider as a test function in problem ( 2 ) f ( x ) = g ( d ( x, ∂ Ω)) , where g is suitably c hosen. A t this p oin t, coarea form ula and an in tegration b y parts allo w us to write ˆ Ω f ( x ) dx = ˆ r (Ω) 0 g ( t ) P ( t ) dt = ˆ r (Ω) 0 g ′ ( t ) µ ( t ) dt + | Ω | g (0) , and β ˆ ∂ Ω f 2 d H n − 1 = β g 2 (0) P (Ω) , ˆ Ω |∇ f | 2 dx = ˆ r (Ω) 0 g ′ 2 ( t ) P ( t ) dt. In this w ay , we get T β (Ω) ≥  ˆ Ω f ( x ) dx  2 ˆ Ω |∇ f | 2 dx + β ˆ ∂ Ω f 2 d H n − 1 = ˆ r (Ω) 0 g ′ ( t ) µ ( t ) dt + | Ω | g (0) ! 2 ˆ r (Ω) 0 g ′ 2 ( t ) P ( t ) dt + β g 2 (0) P (Ω) , and choosing g ( t ) = ˆ t 0 µ ( s ) P ( s ) ds + | Ω | β P (Ω) , we finally hav e T β (Ω) ≥ ˆ r (Ω) 0 µ 2 ( t ) P ( t ) dt + | Ω | 2 β P (Ω) . (41) T o obtain our quantitativ e result, w e m ust split the integral on the right hand-side of ( 41 ) into the t wo in terv als [ 0 , ¯ t ] and [ ¯ t, r (Ω)] , where ¯ t = | Ω | /P (Ω) < r (Ω) . By ( 21 ) we get ˆ r (Ω) 0 µ 2 ( t ) P ( t ) dt ≥ 1 P 2 (Ω) ˆ | Ω | P (Ω) 0 ( | Ω | − P (Ω) t ) 2 P (Ω) dt + 1 P 2 (Ω) ˆ r (Ω) | Ω | P (Ω) µ 2 ( t )( − µ ′ ( t )) dt = | Ω | 3 3 P 2 (Ω) + 1 P 2 (Ω) µ 3  | Ω | P (Ω)  3 , 5 QUANTIT A TIVE ESTIMA TE F OR THE R OBIN EIGENV ALUE 17 Therefore, multiplying b y P 2 (Ω) / | Ω | 3 , and using the fact that P (Ω) / | Ω | > r (Ω) − 1 , we get T β (Ω) P 2 (Ω) | Ω | 3 ≥ 1 3 + P (Ω) β | Ω | + 1 3 µ 3  | Ω | P (Ω)  | Ω | 3 > 1 3 + 1 r (Ω) β + 1 3 µ 3  | Ω | P (Ω)  | Ω | 3 . Rearranging the terms and applying Lemma 2.6 , w e obtain T β (Ω) P 2 (Ω) | Ω | 3 −  1 3 + 1 r (Ω) β  ≥ 1 3 µ 3  | Ω | P (Ω)  | Ω | 3 ≥ q 1 ( n, Ω) 3 3 = 1 2 3 · 3 4 n 3  P (Ω) r (Ω) | Ω | − 1  3 , and this pro ves the low er b ound in ( 13 ). Now, regarding the upp er b ound, w e just m ultiply and divide b y r (Ω) 2 and use the Robin Makai inequalit y ( 11 ), having T β (Ω) P 2 (Ω) | Ω | 3 −  1 3 + 1 r (Ω) β  = T β (Ω) r (Ω) 2 | Ω | P 2 (Ω) r (Ω) 2 | Ω | 2 −  1 3 + 1 r (Ω) β  ≤  1 3 + 1 r (Ω) β  P 2 (Ω) r (Ω) 2 | Ω | 2 −  1 3 + 1 r (Ω) β  =  1 3 + 1 r (Ω) β  P 2 (Ω) r (Ω) 2 | Ω | 2 − 1  =  1 3 + 1 r (Ω) β  P (Ω) r (Ω) | Ω | + 1  R (Ω) ≤ ( n + 1)  1 3 + 1 r (Ω) β  R (Ω) . 5 Quan titativ e estimate for the Robin eigen v alue In this section, we prov e Theorem 1.5 , which ensures that the optima in inequalit y ν 1 ( β , s 0 ) ≥ λ β (Ω) are the same as in the inequality R (Ω) ≥ 0 . Pr o of of The or em 1.5 . The first lines of the proof follow the same argumen t proposed in [ 14 ], whose computations are analogous to the one shown in Subsection 4 . Let us use as a test function in the v ariational c haracterization of λ β (Ω) the function f ( x ) = g ( t ) , where g dep ends only on the distance function from the b oundary of Ω . Then b y coarea form ula w e get λ β (Ω) ≤ ˆ r (Ω) 0 ( g ′ ( t )) 2 P ( t ) dt + β g (0) 2 P (Ω) ˆ r (Ω) 0 g 2 ( t ) P ( t ) dt . (42) The latest, with the change of v ariables s = µ ( t ) P (Ω) , leads to λ β (Ω) ≤ ˆ | Ω | P (Ω) 0 h ′ ( s ) 2 P ( t ( s )) 2 ds + β P 2 (Ω) h 2  | Ω | P (Ω)  P 2 (Ω) ˆ | Ω | P (Ω) 0 h ( s ) 2 ds 5 QUANTIT A TIVE ESTIMA TE F OR THE R OBIN EIGENV ALUE 18 where h ( s ) = g ( t ) . Now, we choose ¯ t = | Ω | P (Ω) and we denote by ¯ s = µ ( ¯ t ) P (Ω) . (43) Hence, we divide the in tegral at the numerator in ( 42 ) at ¯ s , obtaining λ β (Ω) ≤ ˆ ¯ s 0 h ′ ( s ) 2 P ( t ( s )) 2 ds + ˆ | Ω | P (Ω) ¯ s h ′ ( s ) 2 P ( t ( s )) 2 ds + β P 2 (Ω) h 2  | Ω | P (Ω)  P 2 (Ω) ˆ | Ω | P (Ω) 0 h ( s ) 2 ds ≤ P (Ω) ˆ ¯ s 0 h ′ ( s ) 2 P ( t ( s )) ds + P (Ω) 2 ˆ | Ω | P (Ω) ¯ s h ′ ( s ) 2 ds + P 2 (Ω) β h  | Ω | P (Ω)  P 2 (Ω) ˆ | Ω | P (Ω) 0 h ( s ) 2 ds . Let us observ e that if s ∈ [0 , s ] , P ( t ( s )) ≤ P ( t ) thanks to the monotonicity of the p erimeter, w e can apply Lemma 2.6 , obtaining λ β (Ω) ≤ q 2 ( n, Ω) P 2 (Ω) ˆ ¯ s 0 h ′ ( s ) 2 ds + P 2 (Ω) ˆ | Ω | P (Ω) ¯ s h ′ ( s ) 2 ds + P 2 (Ω) β h  | Ω | P (Ω)  P 2 (Ω) ˆ | Ω | P (Ω) 0 h ( s ) 2 ds = ( q 2 ( n, Ω) − 1) ˆ ¯ s 0 h ′ ( s ) 2 ds + ˆ | Ω | P (Ω) 0 h ′ ( s ) 2 ds + β h  | Ω | P (Ω)  ˆ | Ω | P (Ω) 0 h ( s ) 2 ds . (44) No w, let us denote b y s 0 = | Ω | P (Ω) , and let us choose h ( s ) = cos( √ ν 1 s ) , for 0 ≤ s ≤ s 0 , we ha ve ˆ s 0 0 h ( s ) 2 ds = | Ω | 2 P (Ω) + sin  2 √ ν 1 | Ω | P (Ω)  4 √ ν 1 ≤ | Ω | P (Ω) . In this w ay , we get ν 1 ( β , s 0 ) − λ β (Ω) ≥ (1 − q 2 ( n, Ω)) P (Ω) | Ω | ν 1 ( β , s 0 ) ˆ ¯ s 0 sin 2 ( √ ν 1 s ) ds. Using the inequalit y sin ( r ) ≥ 2 π r , whic h is v alid for ev ery r ∈ [0 , π / 2] , we ha ve ν 1 ( β , s 0 ) − λ β (Ω) ≥ 4 3 π 2 (1 − q 2 ( n, Ω)) P (Ω) | Ω | ν 2 1 ¯ s 3 . (45) 5 QUANTIT A TIVE ESTIMA TE F OR THE R OBIN EIGENV ALUE 19 Recalling ( 43 ) and Lemma 2.6 , we ha ve that ¯ s 3 = µ  | Ω | P (Ω)  3 P 3 (Ω) ≥ | Ω | 3 6 3 n 3 P (Ω) 3  P (Ω) r (Ω) | Ω | − 1  3 . (46) Moreo ver, b y the definition of q 2 ( n, Ω) and Prop osition ( 24 ), w e get 1 − q 2 ( n, Ω) = 1 − 1 1 + 1 n  P (Ω) r (Ω) | Ω | − 1  = 1 n  P (Ω) r (Ω) | Ω | − 1  1 + 1 n  P (Ω) r (Ω) | Ω | − 1  ≥ 1 2 n − 1  P (Ω) r (Ω) | Ω | − 1  . (47) Com bining ( 45 ), ( 46 ), and ( 47 ), w e hav e ν 1 ( β , s 0 ) − λ β (Ω) ≥ 1 2 · 3 4 π (2 n − 1) n 3 · | Ω | 2 P (Ω) 2 ν 2 1  P (Ω) r (Ω) | Ω | − 1  4 . No w let us observe that µ is the first p ositiv e ro ot of √ ν 1 tan( √ ν 1 s 0 ) = β , that combined with the inequalit y ( 31 ) gives ν 1 s 2 0 ≥   π 2 4 1 + π 2 P (Ω) 4 | Ω | β   ≥   π 2 4 1 + π 2 n 4 β r (Ω)   ν 1 ( β , s 0 ) | Ω | 2 P 2 (Ω) − λ β (Ω) | Ω | 2 P 2 (Ω) ≥ 1 2 · 3 4 π (2 n − 1) n 3 ·   π 2 4 1 + π 2 n 4 β r (Ω)   2  P (Ω) r (Ω) | Ω | − 1  4 , whic h concludes the pro of of the low er b ound. W e now pro ve the upp er b ound in ( 14 ). Let u Ω b e the solution to ( − ∆ u Ω = 1 in Ω u Ω = 0 on ∂ Ω (48) and let M (Ω) = ∥ u Ω ∥ ∞ . In [ 19 ], it w as pro ved that 2 M (Ω) ≥ 3 T (Ω) | Ω | ≥ | Ω | 2 P 2 (Ω) , hence, if we define t 0 = p 2 M (Ω) , we ha ve t 0 > s 0 . In [ 18 ], it was pro ved that λ β (Ω) ≥ ν 1 ( β , t 0 ) , hence 0 ≤ ν 1 ( β , s 0 ) − λ β (Ω) ≤ ν 1 ( β , s 0 ) − ν 1 ( β , t 0 ) , so we aim to pro ve that the right-hand side is bounded from abv o ve b y a constan t times R (Ω) . W e denote ν 1 ( s 0 ) = ν 1 ( β , s 0 ) , ν 1 ( t 0 ) = ν 1 ( β , t 0 ) . In the v ariational definition of ν 1 ( β , s 0 ) , we use as a test function X = cos( p ν 1 ( t 0 ) s ) , so w e obtain ν 1 ( s 0 ) − ν 1 ( t 0 ) ≤ ν 1 ( t 0 )      ˆ s 0 0 sin 2 ( q ν 1 ( t 0 ) s ) ds ˆ s 0 0 cos 2 ( q ν 1 ( t 0 ) s ) ds − 1      + β cos( q ν 1 ( t 0 ) s 0 ) ´ s 0 0 cos 2 ( p ν 1 ( t 0 ) s ) . 5 QUANTIT A TIVE ESTIMA TE F OR THE R OBIN EIGENV ALUE 20 W e can explicitly compute ˆ s 0 0 sin 2 ( q ν 1 ( t 0 ) s ) = s 0 2 − sin( p ν 1 ( t 0 ) s 0 ) cos( p ν 1 ( t 0 ) s 0 ) 2 p ν 1 ( t 0 ) ˆ s 0 0 cos 2 ( q ν 1 ( t 0 ) s ) = s 0 2 + sin( p ν 1 ( t 0 ) s 0 ) cos( p ν 1 ( t 0 ) s 0 ) 2 p ν 1 ( t 0 ) obtaining ν 1 ( s 0 ) − ν 1 ( t 0 ) ≤ cos( p ν 1 ( t 0 ) s 0 ) h β cos( p ν 1 ( t 0 ) s 0 ) − p ν 1 ( t 0 ) sin( p ν 1 ( t 0 ) s 0 ) i s 0 2 + sin( p ν 1 ( t 0 ) s 0 ) cos( p ν 1 ( t 0 ) s 0 ) 2 p ν 1 ( t 0 ) . (49) Let us m ultiply and divide the n umerator in ( 49 ) by p β 2 + ν 1 ( t 0 ) and let b e γ ∈ R suc h that sin γ = β p β 2 + ν 1 ( t 0 ) , cos γ = p ν 1 ( t 0 ) p β 2 + ν 1 ( t 0 ) . In this w ay , the term in the squared brack ets b ecomes β p β 2 + ν 1 ( t 0 ) cos( q ν 1 ( t 0 ) s 0 ) − s ν 1 ( t 0 ) β 2 + ν 1 ( t 0 ) sin( q ν 1 ( t 0 ) s 0 ) = sin γ cos( q ν 1 ( t 0 ) s 0 ) − cos γ sin( q ν 1 ( t 0 ) s 0 ) = sin( γ − q ν 1 ( t 0 ) s 0 ) . In particular, from ( 30 ), w e can choose γ ∈ [0 , π / 2] so that γ = p ν 1 ( t 0 ) t 0 . Hence, we can rewrite ( 49 ) as follows ν 1 ( s 0 ) − ν 1 ( t 0 ) ≤ p β 2 + ν 1 ( t 0 ) sin( p ν 1 ( t 0 )( t 0 − s 0 )) s 0 2 + sin( p ν 1 ( t 0 ) s 0 ) cos( p ν 1 ( t 0 ) s 0 ) 2 p ν 1 ( t 0 ) ≤ 2 q β 2 + ν 1 ( t 0 ) q ν 1 ( t 0 ) ( t 0 − s 0 ) s 0 . F rom [ 18 ], w e know that t 0 ≤ r (Ω) , hence ν 1 ( s 0 ) − ν 1 ( t 0 ) ≤ 2 q β 2 + ν 1 ( t 0 ) q ν 1 ( t 0 ) R (Ω) . Let us no w m ultiply b y s 2 0 . Since t 0 > s 0 , then ν 1 ( t 0 ) ≤ ν 1 ( s 0 ) . Moreo ver w e kno w that p ν 1 ( s 0 ) s 0 ≤ π/ 2 , therefore 2 q β 2 + ν 1 ( t 0 ) q ν 1 ( t 0 ) s 2 0 ≤ 2 ν 1 ( s 0 ) s 1 + β 2 ν 1 ( s 0 ) s 2 0 ≤ π 2 2 s 1 + β 2 ν 1 ( s 0 ) . No w, using ( 31 ) and the fact that s 0 < r (Ω) , w e hav e that ν 1 ( s 0 ) ≥ π 2 4 1 r (Ω) 2 + π 2 4 β r (Ω) , so that ν 1 ( s 0 ) s 2 0 − ν 1 ( t 0 ) s 2 0 ≤ π 2 2 s 1 + 4 β 2 π 2 ( r (Ω) 2 + π 2 4 β r (Ω)) R (Ω) , completing the proof. 6 QUANTIT A TIVE INEQUALITIES INVOL VING A (Ω) 21 6 Quan titativ e inequalities in v olving A (Ω) W e conclude with the tw o theorem in which w e show that is p ossible to improv e inequalities ( 13 ) and ( 15 ) by adding the sharp p o wer of the asymmetry A (Ω) . Prop osition 6.1. L et Ω b e a non-empty, b ounde d, op en and c onvex set of R n . Then, T β (Ω) P 2 (Ω) | Ω | 3 −  1 3 + 1 β r (Ω)  ≥ C 4 ( n ) A (Ω) , (50) wher e C 4 ( n ) is a p ositive c onstant dep ending only on the dimension of the sp ac e n . Pr o of of Pr op osition 6.1 . Let us recall the low er bound in ( 41 ) T β (Ω) ≥ ˆ r (Ω) 0 µ 2 ( t ) P ( t ) dt + | Ω | 2 β P (Ω) , and let us divide the in tegral abov e at the v alue t defined for some ˜ c ∈ (0 , 1) as µ ( t ) = ˜ c | Ω | . (51) Moreo ver, ( 22 ) and the monotonicity of µ gives, for all t ∈ [ t, r (Ω)] P ( t ) ≤ P (Ω) − c n | Ω | − µ ( t ) P 1 n − 1 (Ω) and together with ( 51 ), w e can write ˆ r (Ω) 0 µ 2 ( t ) P ( t ) dt ≥ ˆ t 0 µ 2 ( t ) P ( t ) dt + ˆ r (Ω) t µ 2 ( t ) P ( t ) dt ≥ 1 P 2 (Ω) ˆ t 0 µ 2 ( t )( − µ ′ ( t )) dt + 1 P (Ω) P (Ω) − c n | Ω | − µ ( t ) P 1 n − 1 (Ω) ! ˆ r (Ω) t µ 2 ( t )( − µ ′ ( t )) dt = 1 P 2 (Ω) | Ω | 3 − µ 3 ( t ) 3 + 1 P (Ω) P (Ω) − c n | Ω | − µ ( t ) P 1 n − 1 (Ω) ! µ 3 ( t ) 3 ≥ 1 P 2 (Ω) | Ω | 3 − µ 3 ( t ) 3 + 1 P 2 (Ω) 1 + c n | Ω | − µ ( t ) P n n − 1 (Ω) ! µ 3 ( t ) 3 = | Ω | 3 3 P 2 (Ω) + c n (1 − ˜ c ) ˜ c 3 P 2+ n n − 1 (Ω) | Ω | 4 3 . No w w e c ho ose ˜ c in order to maximize (1 − ˜ c ) ˜ c 3 . So w e find the maxim um in (0 , 1) of the function f ( x ) = (1 − x ) x 3 , which giv es ˜ c = 3 4 . Hence, we ha ve T β (Ω) P 2 (Ω) | Ω | 3 ≥  1 3 + 1 β r (Ω)  + 27 c n 256 | Ω | P (Ω) 1 P 1 n − 1 (Ω) ≥  1 3 + 1 β r (Ω)  + 27 c n 256 n r (Ω) P 1 n − 1 (Ω) (52) 6 QUANTIT A TIVE INEQUALITIES INVOL VING A (Ω) 22 Com bining ( 52 ) with ( 18 ) and ( 19 ), we get the thesis. Moreo ver, w e managed to prov e a sharp result ev en for the Póly a eigenv alue functional. Prop osition 6.2. L et Ω b e a non-empty, b ounde d, op en and c onvex set of R n . Then, π 2 4 1 1 + 2 β r (Ω) − λ (Ω) | Ω | 2 P 2 (Ω) ≥ C 5 ( n ) A (Ω) , (53) wher e C 5 ( n ) is a p ositive c onstant dep ending only on the dimension of the sp ac e n . Pr o of of Pr op osition 6.2 . W e start from ( 42 ) and we p erform the change of v ariables s = µ ( t ) P (Ω) as in Theorem 1.6 λ β (Ω) ≤ ˆ | Ω | P (Ω) 0 h ′ ( s ) 2 P ( t ( s )) 2 ds + β P 2 (Ω) h 2  | Ω | P (Ω)  P 2 (Ω) ˆ | Ω | P (Ω) 0 h ( s ) 2 ds (54) Let us start b y observing that if s ∈ h 0 , | Ω | 2 P (Ω) i , then t ( s ) ∈ [ t, r (Ω)] , where µ ( t ) = | Ω | / 2 , and so b y ( 22 ) we get P ( t ( s )) ≤ P (Ω) − c n | Ω | 2 P 1 n − 1 (Ω) , ∀ s ∈  0 , | Ω | 2 P (Ω)  . W e can no w split the integral at the n umerator in ( 54 ) at the v alue s 0 2 = | Ω | 2 P (Ω) , obtaining ˆ s 0 0 h ′ ( s ) 2 P ( t ( s )) 2 ds = ˆ s 0 2 0 h ′ ( s ) 2 P ( t ( s )) 2 ds + ˆ s 0 s 0 2 h ′ ( s ) 2 P ( t ( s )) 2 ds ≤ P (Ω) P (Ω) − c n | Ω | 2 P 1 n − 1 (Ω) ! ˆ s 0 2 0 h ′ ( s ) 2 ds + P 2 (Ω) ˆ s 0 s 0 2 h ′ ( s ) 2 ds = P 2 (Ω) ˆ s 0 0 h ′ ( s ) 2 ds − c n P 2 (Ω) | Ω | 2 P n n − 1 (Ω) ˆ s 0 2 0 h ′ ( s ) 2 ds, (55) hence ( 54 ) and ( 55 ) give λ β (Ω) ≤ ´ s 0 0 | h ′ ( s ) | 2 ds + β h ( s 0 ) 2 ´ s 0 0 | h ( s ) | 2 ds − c n | Ω | 2 P n n − 1 (Ω) ´ s 0 2 0 h ′ ( s ) 2 ds ´ s 0 0 h 2 ( s ) ds . 6 QUANTIT A TIVE INEQUALITIES INVOL VING A (Ω) 23 Cho osing h ( s ) = cos ( √ ν 1 s ) , we get ´ | Ω | P (Ω) 0 | h ′ ( s ) | 2 ds + β h  | Ω | P (Ω)  2 ´ | Ω | P (Ω) 0 | h ( s ) | 2 ds = ν 1 ( β , s 0 ) ˆ s 0 0 h ( s ) 2 ds = | Ω | 2 P (Ω) + sin  2 √ ν 1 | Ω | P (Ω)  4 √ ν 1 ≤ | Ω | P (Ω) , ˆ s 0 2 0 h ′ ( s ) 2 = ν 1 ( β , s 0 ) ˆ s 0 2 0 sin 2 ( √ µs ) ds ≥ 4 π 2 µ 2 1 ( β , s 0 ) s 3 0 24 (56) Then equation ( 56 ) and P (Ω) r (Ω) ≤ | Ω | n , giv es ν 1 ( β , s 0 ) − λ β (Ω) ≥ c n 12 nπ 2 r (Ω) P 1 n − 1 (Ω) ν 2 1 ( β , s 0 ) s 2 0 (57) Again, combining ( 57 ) with ( 18 ), ( 19 ), and ( 31 ), w e get ν 1 ( β , s 0 ) − λ β (Ω) ≥ c n 12 nπ 2 r (Ω) P 1 n − 1 (Ω) ν 2 1 ( β , s 0 ) s 2 0 and so m ultiplying by s 2 0 and recalling ( 31 ), we get π 2 4 1 1 + 2 P (Ω) β | Ω | − λ β (Ω) | Ω | 2 P 2 (Ω) ≥ C 5 ( n )   π 2 4 1 1 + π 2 | Ω | 4 β P (Ω)   2 A (Ω) . The thesis follo ws using the upp er b ound in ( 24 ). A c kno wledgemen ts W e would like to thank F rancesco Della Pietra for the v aluable advices. This work has b een partially supp orted by GNAMP A group of INdAM. R. Sannip oli was supp orted by the grant no. 26-21940S of the Czech Science F oundation. Conflicts of in terest and data a v ailabilit y statemen t The authors declare that there is no conflict of interest. Data sharing not applicable to this article as no datasets w ere generated or analyzed during the current study . E-mail addr ess , R. Barbato: rosa.barbato2@unina.it E-mail addr ess , A.L. Masiello: albalia.masiello@unina.it Dip ar timento di Ma tema tica e Applicazioni “R. Ca ccioppoli”, Universit à degli studi di Napoli Federico I I, Via Cintia, Complesso Universit ario Monte S. Angelo, 80126 Napoli, It al y. E-mail addr ess , R. 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