ON PRINCIPAL FREQUENCIES AND INRADIUS IN CONVEX SETS

Transcription

1 ON PRINCIPAL FREQUENCIES AND INRADIUS IN CONVEX SES LORENZO BRASCO o Michelino Brasco, master craftsman and father, on the occasion of his 7th birthday Abstract We generalize to the case of the Lalacian an old result by Hersch and Protter Namely, we show that it is ossible to estimate from below the first eigenvalue of the Dirichlet Lalacian of a convex set in terms of its inradius We also rove a lower bound in terms of isoerimetric ratios and we briefly discuss the more general case of Poincaré-Sobolev embedding constants Eventually, we highlight an oen roblem Contents Introduction Overview he results of this aer 3 3 Plan of the aer 5 Preliminaries 5 Notation 5 A geometric lemma 6 3 Eigenvalues of secial sets 6 3 A divertissement on Hardy s inequality 9 4 Proof of heorem 5 A further lower bound 3 6 More general rincial frequencies 5 Aendix A π and π 7 References Introduction Overview For every oen set R N, we consider its rincial frequency or first eigenvalue of the Lalacian with Dirichlet conditions, defined by u dx λ) = inf u C )\{} u dx We recall that, whenever the comletion D, ) of C ) with resect to the norm u L ) is comactly embedded into L ), the number λ) coincides with the smallest λ R such that Mathematics Subject Classification 35P5, 49J4, 35J7 Key words and hrases Convex sets, Lalacian, nonlinear eigenvalue roblems, inradius, Cheeger constant For examle, this haens if is bounded or has finite N dimensional Lebesgue measure

2 BRASCO the boundary value roblem u = λ u, in, u =, on, does admit a nontrivial solution u D, ) For general sets, the exlicit determination of λ) can be a challenging task It is thus imortant to look for shar estimates on λ) in terms on simler quantities, tyically of geometric flavour he most celebrated instance of such an estimate is the so-called Faber-Krahn inequality his asserts that λ) can be estimated from below by a negative ower of the N dimensional measure of Precisely, we have ) ) λ) B N λb), N where B is any N dimensional ball Equality ) is shar in the sense that the dimensional constant B N λb) is attained whenever is itself a ball actually, this is the only ossibility, u to sets of zero caacity) In desite of its elegance, sharness and simlicity, the lower bound dictated by ) loses its interest for oen sets such that = + and λ) > his haens for examle for the infinite slab = R N, ) For such cases, it could be natural to ask whether a lower bound on λ) can be given in terms of the inradius R, ie the radius of the largest oen ball contained in In other words, we can ask whether we can have an inequality like C ) λ) R he ower on R is imosed by scale invariance, once it is observed that λ) has the hysical dimensions length to the ower However, an estimate like ) can not be true for general oen sets, in dimension N Indeed, it is sufficient to consider the set = R N \ Z N It is easy to see that R < +, while λ) = λr N ) =, since oints have zero caacity in R N, if N However, if we imose further geometric restrictions on the oen set, then it is ossible to rove ) An old result due to Hersch see [9]) shows that for an oen convex set R, it holds π ) 3) λ) R he inequality is shar and it is strict among bounded convex sets he roof by Hersch is based on a method that he called évaluation ar défaut Later on, Protter generalized this result to higher dimensions by using the same technique, see [7, age 68] We also oint out that the Hersch-Protter estimate has been recently generalized in [4, heorem 5] to the anisotroic case, ie to the case of λ H ) = inf u C )\{} H u) dx u dx,

3 PRINCIPAL FREQUENCIES AND INRADIUS 3 where H : R N [, + ) is any norm In this case, the definition of inradius has to be suitably adated, in order to take into account the anisotroy H Remark More general sets I) We have already observed that ) can not be true in general However, the lanar case N = is eculiar and well-studied: in this case, if is simly connected, then it is ossible to rove ), but the main oen issue in this case is the determination of the shar constant C he first result in this direction is due to Hayman [8] We refer to [] for a review of this kind of results Actually, Osserman in [4] showed that ) still holds for lanar sets with finite connectivity, the constant C deending on the connectivity k and degenerating as k goes to this is in erfect accordance with the above examle of R \ Z ) he result by Osserman has then been imroved by Croke in [5] For the higher dimensional case N 3, some results for classes of oen sets more general than convex ones have been given by Hayman [8, heorem ] and aylor [8, heorem 3] he results of this aer We now fix an exonent < < +, then for an oen set R N, we introduce the quantity u dx λ ) = inf u C )\{} u dx As in the quadratic case =, whenever the comletion D, ) of C ) with resect to the norm u L ) is comactly embedded into L ), the number λ ) coincides with the smallest λ R such that the boundary value roblem u = λ u u, in, u =, on, does admit a nontrivial solution u D, ) Here is the quasilinear oerator u = div u u), known as Lalacian For this reason, λ ) is called first eigenvalue of the Lalacian with Dirichlet conditions on In this case as well, we have the shar lower bound ) λ ) B N λ B), N which generalizes ) to he main goal of this aer is to generalize the Hersch-Protter estimate 3) to the case of λ At this aim, we introduce the one-dimensional Poincaré constant { ϕ } L [,]) π = : ϕ) = ϕ) = We will rove the following inf ϕ C [,])\{} ϕ L [,]) heorem Let R N be an oen convex set hen we have π ) 4) λ ) R he estimate is shar, equality being attained for examle:

4 4 BRASCO by an infinite slab, ie a set of the form { } x R N : a < x, ω < b, for some a < b and ω S N ; asymtotically by the family of collasing yramids ) C α = convex hull, ) N {,,, α)}, in the sense that lim R π ) α + C α λ C α ) = ; more generally, asymtotically by the family of infinite slabs with section given by a k dimensional collasing yramid, ie R N k C α, for N 3 and k N Remark 3 After the comletion of this aer, we have been informed by Vladimir Bobkov that the same result is contained [, heorem ] In turn, more recently this result has been generalized in [6, heorem 5], to cover the anisotroic case However, in both references the roof of the lower bound is different, as they do not use the original idea by Hersch In [], the so-called method of interior arallels is used see [, Lemmas 45 & 46]), while [6] exloits a method based on maximum rinciles and the so-called P functions We also oint out that our result contains a finer analysis of the equality cases, since in [6, ] the sequence of collasing yramids is not identified Remark 4 More general sets II) For, the case of more general sets has been investigated by Poliquin in [5] In [5, heorem 4] it is roved that for > N and R N oen bounded set, one has λ ) C R, for a constant C = CN, ) > hen in [5, heorem 4] the same estimate is roved, for > N and having a connected boundary In both cases, the constant C is not exlicit In [6, Proosition 35] the same author roved such a lower bound for convex sets with an exlicit constant, which is however not shar As already observed by Makai in the case = N = see [3]), the estimate of heorem in turn imlies another interesting lower bound on λ ), this time in terms of the quantity P ), where P ) is the erimeter of he resulting estimate, which seems to be new for N 3 and, is contained in Corollary 5 below Remark 5 Uer bound) U to now, we never mentioned the ossibility of having an uer bound of the tye λ ) C R he reason is simle: such an estimate is indeed true and very simle to obtain in a shar form, without any assumtion on the set Indeed, by definition of λ it is easy to see that this is a

5 PRINCIPAL FREQUENCIES AND INRADIUS 5 monotone decreasing quantity, with resect to set inclusion hus, if R N is an oen set with R < +, there exists a ball B R ξ) and we have λ ) λ B R ξ)) If we now use the scaling roerties of λ, the revious can be rewritten as λ ) λ B )) R Observe that this estimate is shar, equality being uniquely) attained by balls 3 Plan of the aer In Section we introduce the notation used throughout the whole aer and the technical facts needed to handle the roof of heorem Section 3 contains a rougher version of our main result, based on Hardy s inequality for convex sets his is a sort of divertissement, that we think to be interesting in its own he roof of heorem is then contained in Section 4 We combine this result with a geometric estimate, to obtain a further lower bound on λ of geometric nature: this is Section 5, which also contains a lower bound on the Cheeger constant Finally, in the last Section 6 we consider the same tye of lower bound in terms of the inradius, with λ relaced by a general Poincaré-Sobolev shar constant he aer ends with an oen roblem Acknowledgements We thank Berardo Ruffini for some comments on a reliminary version of this aer and for ointing out the reference [5] We also thank Vladimir Bobkov and Francesco Della Pietra for some useful bibliograhical references his aer evolved from a set of handwritten notes for a talk delivered during the conferences Variational and PDE roblems in Geometric Analysis and Recent advances in Geometric Analysis held in June 8 in Bologna and Pisa, resectively he organizers Chiara Guidi & Vittorio Martino and Andrea Malchiodi & Luciano Mari are kindly acknowledged Preliminaries Notation For an oen set R N, we indicate by its N dimensional Lebesgue measure For an oen bounded set R N with Lischitz boundary, we define the distance function d x) = inf x y, x y hen we recall that the inradius R of coincides with R = su d x) x We will set ν x) to be the outer normal versor at, whenever this is well-defined Definition We say that R N is an oen olyhedral convex set if there exists a finite number of oen half-saces H,, H k R N such that k = H i i= If is an oen olyhedral convex set, we say that F is a face of if the following hold: F ; F H i, for some i =,, k;

6 6 BRASCO for any E H i such that F E, we have E = F If R N is an oen convex set with R < +, we know that there exists ξ such that B R ξ) Accordingly, we define the contact set Finally, we recall the definition π = It is not difficult to see that inf ϕ C [,])\{} C,ξ = B R ξ) { ϕ L,)) ϕ L,)) ) π = π = and π = π, see Lemmas A and A } : ϕ) = ϕ) = A geometric lemma he following geometric result is one of the building blocks of the roof of heorem It is a higher-dimensional analogue of a simle two-dimensional fact used by Hersch in [9] his is the same as [4, Lemmas 5 & 53], to which we refer for the roof Lemma Let R N be an oen bounded convex set Let ξ be such that B R ξ) hen there exists m and {P,, P m } C,ξ distinct oints such that the oen olyhedral convex domain m = {x R N : x P i, ν P i ) < }, has the following roerties: ; R = R ; i= every face of touches B R ξ) Remark 3 he revious result is similar to an analogous geometric lemma contained in Protter s aer, see [7, age 68] Such a result in [7] is credited to a rivate communication by David Gale, without giving a roof It should be noticed that the statement in [7] is slightly more recise, since it is said that m can be chosen to be smaller than or equal to N + However, in the statement contained [7] the crucial feature that all the faces of touches the internal ball B R ξ) seems to have been accidentally omitted For this reason we refer to refer to the result roved in [4] 3 Eigenvalues of secial sets Lemma 4 Product sets) Let < < + and k {,, N } We take the oen set = R N k ω, with ω R k oen bounded set hen we have λ ) = λ ω) Proof he roof is standard, we include it for comleteness We use the notation x, y) R N k R k, for a oint in R N We first rove that ) λ ) λ ω) For every ε >, we take u ε C ω) to be an almost otimal function for the roblem on ω, ie y u ε dy < λ ω) + ε and u ε dy = ω ω

7 PRINCIPAL FREQUENCIES AND INRADIUS 7 We take η C R) such that η, η on [, ], η on R \ [, ], then for every R >, we choose ϕx, y) = η R x ) u ε y), where η R t) = R k N ) t η R By using Fubini s heorem, we obtain x η R x ) u ε x N ) + y u ε y) η R x ) ) dx dy B λ ) R ) ω, η R x ) dx B R ) where B R ) = {x R N k : x < R} We now use the definition of η R and the change of variables x = R x, so to get [R k N) η x ) u ε y) + R k N) y u ε y) η x ) ] R N k dx dy B λ ) ) ω η x ) dx = B ) B ) [ ] R η x ) u ε y) + y u ε y) η x ) dx dy η x ) dx B ) By taking the limit as R goes to + and using the Dominated Convergence heorem, from the revious estimate we get y u ε y) η x ) dx dy B λ ) ) ω = y u ε dy < λ ω) + ε η x ) dx B ) he arbitrariness of ε > imlies ) We now rove the reverse inequality 3) λ ) λ ω) For every ε >, we take ϕ ε C ) \ {} such that ϕ ε dx dy < λ ) + ε ϕ ε dx dy

8 8 BRASCO Observe that ϕ ε dx dy R N k λ ω) ω R N k ) y ϕ ε dy dx ) ϕ ε dy ω dx = λ ω) ϕ ε dx dy, where we used that y ϕ ε x, y) is admissible for the one-dimensional roblem, for every x We thus obtained λ ω) λ ) + ε he arbitrariness of ε > imlies 3) he following technical result is the core of the roof of heorem It enables to estimate from below an eigenvalue with mixed boundary conditions, when the set is a yramid-like one We have to ay attention to ossibly unbounded sets In what follows W, ) is the usual Sobolev sace of L ) functions, having their distributional gradient in L ), as well Lemma 5 Let Σ R N be an oen olyhedral convex set Let ξ = ξ,, ξ N ) R N be a oint whose rojection on R N belongs to Σ and such that ξ N > We consider the N dimensional olyhedral convex set = convex hull Σ {ξ} ), and define hen we have µ ) = inf u C ) W, )\{} µ ) u dx : u = on Σ u dx π ) ξ N ) Proof By recalling the definition of π, we have that for a > and for every ϕ C [, a]) such that ϕ) = it holds a 4) ϕ t) π ) a a ϕt) dt, see [, Lemma A] We now take a function u C ) W, ) which is admissible for the roblem defining µ ) By hyothesis, there exists an affine function Ψ : Σ [, ξ N ] such that = { x, x N ) : R N R : x Σ, < x N < Ψx ) } hus by Fubini s heorem and 4) we have ) Ψx ) u dx u xn dx = u xn dx N dx Σ Σ Ψx ) ) π ) Ψx ) u dx N dx ) Ψx ) u dx N dx = π ) ξ N Σ π ) ξ N By taking the infimum over admissible functions u, we get the desired conclusion u dx

9 PRINCIPAL FREQUENCIES AND INRADIUS 9 3 A divertissement on Hardy s inequality Before roving the shar estimate à la Hersch-Protter 4), we resent a rougher estimate his is a consequence of Hardy s inequality for convex sets Even if the resulting estimate is not shar, we believe that the roof has its own interest and we reroduce it for the reader s convenience Proosition 3 Let < < and let R N be an oen bounded convex set hen we have ) λ ) Proof We recall that the following Hardy s inequality holds for a convex set ) 3) u dx u dx, for every u C ) d By using this inequality, it is easy to obtain the claimed estimate By recalling that R R = d L ), from 3) we get ) R u dx < u dx By taking the infimum over admissible test functions, we finally obtain the lower bound on λ ) For comleteness, we now recall how to rove 3) Let us consider the distance function d x) = min x y, x y his is a Lischitz function, which is concave on, due to the convexity of his imlies that d is weakly suerharmonic, ie d, ϕ dx, for every nonnegative ϕ C ) By observing that 3) d =, almost everywhere in, from the revious inequality we also get 33) d d, ϕ dx, for every nonnegative ϕ C ), ie d is weakly suerharmonic as well By a standard density argument, we easily see that we can enlarge the class of test functions u to ϕ W, ), ie the closure of C ) in W, ) We now insert in 33) the test function ϕ = where u C ) and ε > We thus obtain ) d d + ε u dx + that is d d + ε u dx u d + ε), d d, u u u dx d + ε) d d, u u dx d + ε)

10 BRASCO We can now use Young s inequality in the following form a, b δ a his yields d d + ε u dx δ which can be recast into ) δ δ) + δ d d + ε b, for a, b R N, δ > d d + ε u dx + δ u dx Finally, we observe that the quantity δ δ) is maximal for δ =, u dx u dx, thus by taking the limit as ε goes to and recalling 3), by Fatou s Lemma we end u with 3), as desired Remark 3 We observe that the boundedness of can be droed, both in 3) and in the lower bound on λ ) We also oint out that, even if the constant ), is not shar, it only deends on, just like the shar one 4 Proof of heorem We start with a articular case of heorem, when the convex set is olyhedral Its roof heavily relies on Lemma 5 Proosition 4 Let < < + and let R N be an oen olyhedral convex set We suose that R < + and we assume further that there exists a ball B with radius R and such that each face of touches B hen we have π ) λ ) R Proof Let us indicate by F,, F j the faces of We take the center ξ of B and then define i = convex hull F i {ξ} ), i =,, j, see Figures and We now consider i for a fixed i =,, j and estimate from below u dx µ i = inf i : u = on F i u C i) W, i)\{} u dx i U to a rigid motion, we can assume that i satisfies the assumtions of Lemma 5 Observe that in this case, we have ξ N = R,

11 PRINCIPAL FREQUENCIES AND INRADIUS Figure he construction for the roof of Proosition 4, when N = and has j = 3 faces Figure he construction for the roof of Proosition 4, when N = 3 and is an unbounded set with j = 3 faces In this case, the subsets,, 3 not drawn in the icture) are unbounded, as well by construction hus we get 4) µ i π ) ξ N ) = π ) R On the other hand, for every ε >, we take ϕ ε C ) \ {} such that ϕ ε dx λ ) λ ) + ε ϕ ε dx

12 BRASCO We observe that the restriction of ϕ ε to each i is admissible for the roblem defining µ i hen, we obtain j j ϕ ε dx ϕ ε dx µ i ϕ ε i= i i= dx i λ ) + ε = ϕ ε dx j j min µ i i=,,j ϕ ε dx ϕ ε dx i i= By recalling the lower bound 4), we get the the desired conclusion, thanks to the arbitrariness of ε > We eventually come to the roof of heorem Proof of heorem We first rove the inequality and then analyze the equality cases Part : roof of the inequality Let us first assume that is bounded By aealing to Lemma, we know that there exists R N an oen olyhedral convex set such that i= and R = R Moreover, each face of touches a maximal ball B R ξ) By alying Proosition 4 to the set, we get π ) π ) λ ) λ ) R = R his concludes the roof, in the case is bounded If in unbounded, we can suose that R < +, otherwise there is nothing to rove hen we can consider the bounded set R = B R ) for R large enough By alying π ) λ R ) R, R and taking on both sides the limit as R goes to +, we get the conclusion Part : sharness of the inequality It is easy to see that equality is attained on a slab Indeed, by Lemma 4 we have ) λ R N, )) = λ, )) = π and R R N,) = As for the collasing yramids ) C α = convex hull, ) N {,,, α)}, we are going to use a urely variational argument, thus we not need the exlicit determination of λ for these sets We first observe that C α R N, α), thus we have λ C α ) λ R N π ), α)) = α In order to rove the reverse estimate, we observe that for < α < Q α := α), ) N α, α ) α) C α,

13 PRINCIPAL FREQUENCIES AND INRADIUS 3 thus by monotonicity and scaling By observing that we thus get that λ q C α ) λ Q α ) = lim α + λ In conclusion, we obtained that We are left with observing that α ) α) λ α, ) N, )) α α, ) N ),, )) = λ R N, )) = π α R Cα = π ) λ Q α ), for α + α ) lim α λ C α ) = π α + α + + α α, for α + his concludes the roof of the otimality of the sequence {C α } α Finally, we observe that for the sets R N k C α, for N 3 and k N, it is sufficient to use the comutations above and the fact that by Lemma 4 λ R N k C α ) = λ C α ), together with R R N k C α = R Cα Remark 4 By comaring the shar estimate 3) with the estimate of Proosition 3, we get π > By recalling ), we have that both sides converge to, as goes to + his shows that even if the estimate of Proosition 3 is not shar for every finite, it is asymtotically otimal for + 5 A further lower bound It what follows, we will use the notation P ) to denote the distributional erimeter of a set R N On convex sets, this coincides with the N ) dimensional Hausdorff measure of the boundary We recall that for bounded convex sets, it is ossible to bound λ ) from above in terms of the isoerimetric tye ratio P ) Namely, we have λ ) < π ) ) P ), see [, Main heorem] and [7, heorem 4] he inequality is strict and the estimate is shar

14 4 BRASCO As a straightforward consequence of heorem, we get that the revious estimate can be reverted hus ) P ) λ ) and, are equivalent quantities on oen bounded convex sets For N = =, this result is due to Makai, see [3] For all the other cases, to the best of our knowledge it is new Corollary 5 Let < < + and let R N be an oen bounded convex set hen we have π ) ) P ) 5) λ ) N he inequality is shar, equality being attained asymtotically by the sequence of collasing yramids of heorem Proof In order to rove 5), it is sufficient to recall that for an oen bounded convex set, we have the shar estimate see for examle [, Lemma A]) 5) R N P ) By inserting this in 4), we get the claimed estimate We now come to the sharness issue Observe that 5) has been obtained by joining the two inequalities 4) and 5) We already know that the family of collasing yramids is asymtotically otimal for the first one, thus we only need to verify that the same family is asymtotically otimal for 5), as well Let us set as before ) ) N C α = convex hull, {,,,, α)} We recall that while and hus we get as desired C α = N α R Cα α, z ) N α N dz = α N, N P C α ), ) = N C α P C α ) α N R C α, for α, N We recall the definition of Cheeger constant of an oen bounded set R N, ie { } P E) h ) = inf : E > E E Observe that if P ) < +, then itself is admissible in the revious variational roblem hus we have the trivial estimate P ) h )

15 PRINCIPAL FREQUENCIES AND INRADIUS 5 For convex sets, this estimate can be reverted Indeed, by recalling that see [, Corollary 6]) lim λ ) = h ) and lim π = π =, if we take the limit as goes to in 5), we get the following Corollary 5 Let < < + and let R N be an oen bounded convex set hen we have h ) N P ) Remark 53 he case = + ) he limit as goes to + of 5) is less interesting Indeed, by taking the th root on both sides and recalling that see [, Lemma 5]) ) lim λ ) =, + R from 5) we get again 5) 6 More general rincial frequencies By aealing to its variational characterization, the first eigenvalue λ ) is nothing but the shar constant for the Poincaré inequality C u dx u dx, for every u C ) From a theoretical oint of view, it is thus quite natural to consider more generally the rincial frequencies u dx λ,q ) = inf ) u C )\{}, for q u q q dx Of course, such a quantity is interesting only if q is such that { q <, if N, where = q +, if > N, N N For < N and q =, the quantity λ,q ) does not deend on and is a universal constant, coinciding with the shar constant in the Sobolev inequality ) C u dx u dx, for every u C R N ) R N R N In this section, we briefly investigate the ossibility to have a lower bound of the tye C R β λ,q ), among convex sets, in this case as well Observe that by scale invariance, the only ossibility for the exonent β is β = N + + N q In the case q <, such an estimate is not ossible, as shown in the following

16 6 BRASCO Proosition 6 Sub-homogeneous case) Let < < + and q < hen { } inf λ,q ) : R N oen bounded convex set = R N q N+ Proof By scale invariance, we can imose the further restriction that R = We recall that for q < we have λ,q ) > the embedding D, ) Lq ) is comact, see [3, heorem ] We now observe that for the oen convex set = R N, ) the embedding above can not be comact, due to the translation invariance of the set in the first N coordinate directions hus we get By taking the sequence L = λ,q R N, )) = L, L ) N, ), L >, and using that lim λ,q L ) = λ,q R N, )), L + we get the desired conclusion Before analyzing the case q >, we notice that for the case q <, it is ossible to have a lower bound on λ,q in terms of an integral norm of the distance from the boundary In a sense, this is the natural counterart of the Hersch-Protter estimate Proosition 6 Let < < + and q < hen for every R N oen bounded convex set, we have ) 6) λ,q ) ) q q q q d dx Proof We observe that by Hölder inequality, for every u C ) we have ) u q dx u q ) q q q d dx d dx By taking the ower /q on both sides and using Hardy s inequality 3), we get ) ) u q q ) q dx u q q q dx d dx By taking the infimum over u C ), we get the desired estimate On the contrary, for q > it is ossible to have a lower bound on λ,q in terms of the inradius Proosition 63 Suer-homogeneous case) Let < < and q > such that { q <, if N, q +, if > N

17 PRINCIPAL FREQUENCIES AND INRADIUS 7 hen there exists a constant C = CN,, q) > such that for every R N oen convex set, we have 6) λ,q ) C R N q N+ Proof By using the classical Gagliardo-Nirenberg inequalities, we have for every u C ) ) ) 63) u q q ϑ ϑ dx C u dx u dx), where C = CN,, q) > and ϑ = N q N + For every ε >, we take ϕ C ) such that λ,q ) + ε > ϕ dx ϕ q dx By using 63) to estimate the denominator, we end u with ϑ ϕ dx λ,q ) + ε > ϕ dx ) q λ )) ϑ If we now use heorem and recall the definition of ϑ, we get the desired conclusion he revious roofs very likely do not roduce the shar constants in 6) and 6) Moreover, in the case q >, the Hersch s argument used for the case = q does not seem to work in this case hus, we leave an oen roblem, which is quite interesting in our oinion Oen roblem Find the shar constants in estimates 6) and 6), among oen bounded convex sets Aendix A π and π We observed in Section that π = π = For the reader s convenience, we resent a roof of these facts Lemma A We have π = inf ϕ C [,])\{} ϕ dt ϕ dt : ϕ) = ϕ) = =

18 8 BRASCO Proof We take an admissible test function ϕ, for every t [, /] we have t t ϕt) = ϕt) ϕ) = ϕ τ) dτ ϕ τ) dτ By integrating over [, /] and exchanging the order of integration, we obtain t ) ϕt) dt ϕ ) τ) dτ dt = ϕ τ) dt dτ A) Similarly, for every t [/, ] we have ϕt) = ϕ) ϕt) = t = ) τ ϕ τ) dτ ϕ τ) dτ ϕ τ) dτ By integrating over [/, ] and exchanging again the order of integration, we obtain ) ) τ ϕt) dt ϕ τ) dτ dt = ϕ τ) dt dτ A) If we now sum A) and A), we get ϕt) dt t = τ ϕ τ) dτ t τ τ ) ϕ τ) dτ ϕ τ) dτ his roves that π In order to rove the reverse estimate, we fix < δ < / and take the iecewise affine function, if t < δ, t δ, if δ < t < δ, δ ϕ δ t) =, if δ t δ, δ t, if δ < t < δ, δ, if δ t We take {ϱ ε } ε> a family of standard mollifiers, then for < ε the function ϕ δ ϱ ε is admissible hus, we get π lim ε + ϕ δ ϱ ε ) dt ϕ δ ϱ ε dt = lim ε + ϕ δ) ϱ ε dt = ϕ δ ϱ ε dt By taking the limit as δ goes to, we get the desired conclusion Lemma A We have π = { ϕ L inf [,]) ϕ C [,])\{} ϕ L [,]) ϕ δ dt = ϕ δ dt } : ϕ) = ϕ) = = 3 δ

19 PRINCIPAL FREQUENCIES AND INRADIUS 9 Figure 3 he function ϕ δ for = left) and = right) Proof We take an admissible test function ϕ, then we take t, ) one of the maximum oints of ϕ We obtain and ϕ L [,]) = ϕt ) = ϕt ) ϕ) ϕ L [,]) = ϕt ) = ϕ) ϕt ) t By taking the roduct of the last two estimates, we get t t ) ϕ L [,]) ϕ τ) dτ t ϕ L [,]), t ϕ τ) dτ t ) ϕ L [,]) ϕ L [,]) By observing that t t ), for every t, ), we get that π In order to get the reverse inequality, we fix < δ < / and take the function, if t δ, t ϕ δ =, if δ < t < δ, δ, if δ t

20 BRASCO By taking as above the convolution with the standard mollifiers {ϱ ε } ε>, we get ϕ δ π lim ϱ ε L [,]) ϕ δ lim L [,]) = ϕ δ L [,]) = ε + ϕ δ ϱ ε L [,]) ε + ϕ δ ϱ ε L [,]) ϕ δ L [,]) δ We can now take the limit as δ goes to and obtain that π, as well References [] R Bañuelos, Carroll, Brownian motion and the fundamental frequency of a drum, Duke Math J, ), [] L Brasco, On rincial frequencies and isoerimetric ratios in convex sets, rerint 8), available at htt://cvgmtsnsit/erson/98/ 8, 3, 4 [3] L Brasco, B Ruffini, Comact Sobolev embeddings and torsion functions, Ann Inst H Poincaré Anal Non Linéaire, 34 7), [4] G Buttazzo, S Guarino Lo Bianco, M Marini, Shar estimates for the anisotroic torsional rigidity and the rincial frequency, J Math Anal Al, 457 8), 53 7, 6 [5] C B Croke, he first eigenvalue of the Lalacian for lane domains, Proc Amer Math Soc, 8 98), [6] F Della Pietra, G di Blasio, N Gavitone, Shar estimates on the first Dirichlet eigenvalue of nonlinear ellitic oerators via maximum rincile, rerint 7), available at htts://arxivorg/abs/734 4 [7] F Della Pietra, N Gavitone, Shar bounds for the first eigenvalue and the torsional rigidity related to some anisotroic oerators, Math Nachr, 87 4), [8] W K Hayman, Some bounds for rincial frequency, Alicable Anal, 7 977/78), [9] J Hersch, Sur la fréquence fondamentale d une membrane vibrante: évaluations ar défaut et rincie de maximum, Z Angew Math Phys, 96), , 6 [] P Juutinen, P Lindqvist, J J Manfredi, he eigenvalue roblem, Arch Rational Mech Anal, ), [] R Kajikiya, A riori estimate for the first eigenvalue of the Lalacian, Differential Integral Equations, 8 5), 8 4 [] B Kawohl, V Fridman, Isoerimetric estimates for the first eigenvalue of the Lalace oerator and the Cheeger constant, Comment Math Univ Carolin, 44 3), [3] E Makai, On the rincial frequency of a membrane and the torsional rigidity of a beam In Studies in Math Analysis and Related oics, Stanford Univ Press, Stanford 96, 7 3 4, 4 [4] R Osserman, A note on Hayman s theorem on the bass note of a drum, Comment Math Helvetici, 5 977), [5] G Poliquin, Princial frequency of the Lalacian and the inradius of Euclidean domains, J o Anal, 7 5), , 5 [6] G Poliquin, Bounds on the rincial frequency of the Lalacian Geometric and sectral analysis, , Contem Math, 63, Centre Rech Math Proc, Amer Math Soc, Providence, RI, 4 4 [7] M H Protter, A lower bound for the fundamental frequency of a convex region, Proc Amer Math Soc, 8 98), 65 7, 6 [8] M E aylor, Estimate on the fundamental frequency of a drum, Duke Math J, ), Diartimento di Matematica e Informatica Università degli Studi di Ferrara Via Machiavelli 35, 44 Ferrara, Italy address: lorenzobrasco@unifeit In the second inequality, we use that ϱ ε dt =, for every ε