HARDY INEQUALITIES WITH OPTIMAL CONSTANTS AND REMAINDER TERMS

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1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 356, Number 6, Pages S ) Article electronically ublished on December 9, 3 HARDY INEQUALITIES WITH OPTIMAL CONSTANTS AND REMAINDER TERMS FILIPPO GAZZOLA, HANS-CHRISTOPH GRUNAU, AND ENZO MITIDIERI Abstract. We show that the classical Hardy inequalities with otimal constants in the Sobolev saces W 1, and in higher-order Sobolev saces on a bounded domain R n can be refined by adding remainder terms which involve L norms. In the higher-order case further L norms with lower-order singular weights arise. The case 1 <<being more involved requires a different technique and is develoed only in the sace W 1,. 1. Introduction Let n>andlet R n be a bounded domain. A well-known Hardy inequality see [H], [HLP]), which can be considered one of the really classical Sobolev embedding inequalities, reads as: 1) u n ) 1, dx dx for all u W 4 ). Although it is not exlicitly assumed that, here and in what follows we shall always have this articularly interesting case in mind. Related L -versions of 1) are also well known, and the simlest can be stated as follows: ) n ) u u 1, dx dx for all u W ). Here, we are assuming that n> 1. A first higher-order generalization of 1) was roved by Rellich in [R]: 3) u) dx n n 4), dx for all u W 16 4 ), where n>4. Recently some extensions of 3) aeared in [DH], [Mi]. These results allow us to estimate integral terms of the form σ u) dx and consequently to rove Hardy inequalities in higher-order Sobolev saces W k, ), where n>k. If k =m is even, we have m ) 4) m u) n +4m 4l) dx dx, 4 4m l=1 Received by the editors June, and, in revised form, May 8, 3. Mathematics Subject Classification. Primary 46E35; Secondary 355, 35J c 3 American Mathematical Society License or coyright restrictions may aly to redistribution; see htt://

2 15 F. GAZZOLA, H.-C. GRUNAU, AND E. MITIDIERI while if k =m + 1 is odd, m+1 ) 5) m u n +4m + 4l) dx dx. 4 l=1 4m+ In the L -setting further tyes of Rellich inequalities were also roved in [DH,. 5], [Mi]. A simle examle in this direction is given by ) 6) u n ) 1)n u, dx dx for all u W ), where n>. We oint out that all the constants aearing in 3), 4), 5) and 6) are shar. For a more extensive survey, bibliograhy and historical remarks we refer again to [DH]. Refined versions of Hardy inequalities seem to have first aeared in [Ma, Section.1.6, Corollary 3]. This kind of inequalities was alied by rezis and Vazquez [V] to the Gelfand roblem 7) u = λ exu), u, u W 1, ), where λ is a ositive arameter and n>. It is well known that there exists λ > such that for λ λ equation 7) is solvable, while for λ>λ it has no solutions. Further, for λ =n ) and = the unit ball, roblem 7) has the following exlicit singular solution: u sing = log W 1, ). The linearization of equation 7) around u sing leads to the Hardy-tye oerator L lin ϕ = ϕ n ) ϕ. This oerator is studied in [V] for roving, among other results, in which dimensions the singular solution is also extremal in the sense that it corresonds to λ = λ. This is shown to be recisely the case if L lin is ositive semidefinite. In order to decide whether the latter holds true, rezis and Vazquez used the following Hardy inequality with otimal constant and a remainder term: 8) u n ) ) /n dx 4 dx +Λ en dx for all u W 1, ). Here n. Λ denotes the first eigenvalue of the Lalace oerator in the twodimensional unit disk and e n, denote resectively the n-dimensional Lebesgue measure of the unit ball R n and of the domain. Inequality 8) can be roved by means of a reduction of the dimension argument. This technique was imlicitly used also in [Ma, Section.1.6]. For further alications and variants of 8) we refer to [VZ]. Remainder terms aear also in other Sobolev inequalities and in the context of nonlinear eigenvalue roblems; see e.g. [L], [GG]. The goal of this aer is to rove the existence of remainder terms for all Hardytye inequalities ), 3), 4), 5) and 6) mentioned above. We feel that remainder terms in the Rellich inequality 3) may find an interretation for the biharmonic analogue of the Gelfand roblem; 9) u = λ exu), u, u W, ). License or coyright restrictions may aly to redistribution; see htt://

3 HARDY INEQUALITIES 151 In [AGGM], we started the investigation of 9) when is a ball, since in this case comarison rinciles for the biharmonic oerator under Dirichlet boundary conditions are available. This roblem is rather involved, and u to now, we did not succeed in making the connection between 9) and the imroved Hardy inequalities exlicit. In Section we shall study W 1, -Hardy tye inequalities corresonding to the - Lalace oerator. Carefully exloiting the convexity roerties of the function ξ ξ, we show that an extra L -norm may be added to the right hand side of ). U to urely -deendent factors, the constant in front of this remainder term behaves like n ) as n, and it vanishes when 1. Although the roblem of finding otimal constants for the remainder terms in the inequalities is very interesting, our choice is not to develo this kind of study in this aer. However, we believe that the asymtotic behaviour of such constants as n is recisely as stated above. Closely related questions were studied simultaneously and indeendently by Adimurthi, Chaudhuri and Ramaswamy [ACR]. In Sections 3 and 4 we show that remainder terms of the form dx,..., dx k can be added to the right hand sides of 4) and 5). In Section 3 we shall consider functions satisfying Navier boundary conditions, that is = u = u = on, while in Section 4 we deal with functions satisfying Dirichlet boundary conditions =u = u = u = on. In Section 3 we also resent a refinement of the Rellich inequality 6) in the L -setting for. For 1 << we exect a similar henomenon as in W 1,, concerning the behaviour of the constants aearing in the remainder terms. See Theorem and the subsequent remark for more details on this oint. Since the class of functions considered in Section 3 is larger than the class considered in Section 4, the constants in front of the remainder terms are smaller than those in Section 4. Moreover, the constants which aear in the latter seem to be more natural when the order k of the Sobolev sace becomes large. Next we observe that the roofs of the results in Section 3 are easily reduced to the sherically symmetric case, while it aears that this reduction when dealing with functions satisfying Dirichlet boundary conditions is more involved. This is due to the fact that the symmetrization argument used in Section 3 fails, and the iterative rocedures we use, seem to roduce only relatively small constants. For this reason it may be also of interest in its own to study how the result in W k, ) for radial functions can be extended to nonsymmetric functions.. An L -remainder term for the Hardy inequality in W 1, ) In this section we shall rove that a remainder term can be added to the right hand side of ). In what follows we shall always suose that 1 <<n. We set X := {v C 1 [, 1]) : v ) = v1) = }\{}, and define the constants λ =inf X r 1 v r) dr r 1 vr) dr, Λ =inf X r vr) v r)) dr r 1 vr) dr License or coyright restrictions may aly to redistribution; see htt://

4 15 F. GAZZOLA, H.-C. GRUNAU, AND E. MITIDIERI and { ) } n Γ=Γn, ) =max λ, 1) Λ. Remark. For all >1wehaveλ >, and if =,thenγ=λ = λ.moreover, if N then λ is the first eigenvalue of in the unit ball contained in R. In addition, a simle integration by arts argument and an alication of Hölder s inequality show that λ, Λ 1. Therefore, Γ c n as n. Theorem 1. Let R n be a bounded domain and 1,n). Let e n = 1 ) and let be the n-dimensional Lebesgue measure of the unit ball and of the domain resectively. a) If, then for every u W 1, ) we have 1) ) n u dx u dx +Γ ) /n en u dx. b) If 1 <<, then there exists a constant C = Cn, ) > such that for every u W 1, ) the following inequality holds: ) n 11) u u ) /n dx dx + C en u dx. Remarks. i) The exlicit form of the constant Cn, ) aearing in 11) is given in 19) below. We oint out that the asymtotic behaviour of this constant as n is given, u to an n-indeendent factor, by n. Although we do not claim that the constants Γn, ) andcn, ) areshar, we believe that they reflect the recise asymtotic behaviour for n. ii) Following the generalizations in [V] and [VZ] we exect that also the -th ower of any W 1,q -norm with q<or of the L r -norm of u with r< n n may serve as a remainder term in 1) and 11). In this case the constants aearing in front of these remainder terms will be not so simle looking even for. After the resent aer was submitted for ublication, someresultsinthisdirectionwereobtainedin[acr],[ft]. Forimroved Hardy inequalities involving a whole series of remainder terms see [FT]. iii) If =1and= 1 ) the Hardy constant u dx n 1= inf u W 1,1 u 1))\{} dx is achieved by any ositive smooth and radially decreasing function u with u =1 =. This means that for = 1 no remainder term may be added to the corresonding Hardy inequality and shows that necessarily Cn, ) as 1. iv) For related inequalities with weights being the inverse of the distance function δ, i.e. δx) =dx, ), where is a bounded domain contained in R n, we refer to [M], [MS], [MS]. A unified aroach to this kind of inequalities has been given recently in [FT], where the distance function from any k-codimensional manifold, 1 k n, is considered. License or coyright restrictions may aly to redistribution; see htt://

5 HARDY INEQUALITIES 153 To rove Theorem 1 we need the following elementary inequalities, which give a quantitative estimate from below for the convexity behaviour of ower-like functions. Similar inequalities in a more general context but with smaller constants were obtained in [L] and several subsequent aers. Lemma 1. Let 1 and ξ,η be real numbers such that ξ and ξ η. Then max { 1)η ξ, η }, if, ξ η) + ξ 1 η ξ 1 1) η, if 1. ξ + η ) The roof can be obtained as an alication of Taylor s formula. The interested reader may refer to the aendix of this aer for further details. Proof of Theorem 1. Using Schwarz symmetrization and a rescaling argument, we may assume that is the unit ball = 1 ), and that u W 1, ) is nonnegative, radially symmetric and nonincreasing. We recall that symmetrization leaves L q -norms of functions invariant, it increases L -norms with the singular weight see e.g. [AL, Theorem.]), and decreases L -norms of the gradient see [AL, Theorem.7]). y density, we may further assume that u is as smooth as needed. So, in what follows we may write ur), u r) = d dr ur). Clearly, we have ux) = u r) with r =. Our goal is to find a lower bound for ) n 1 I := r n 1 u r) dr r n 1 ur) dr. Similarly as in [V] a suitable transformation reduces the dimension and also uncovers a remainder term. We set 1) vr) :=r n/) 1 ur), ur) =r 1 n/) vr), so that u r) = n r n/ vr)+r 1 n/) v r). Since u is radially nonincreasing, for r, 1] we have 13) and I becomes n vr) r v r), I = r 1 n vr) r ) n 1 v r)) vr) dr dr. r Since 13) holds, we can aly Lemma 1 with the choice ξ = n vr) r and η = v r). License or coyright restrictions may aly to redistribution; see htt://

6 154 F. GAZZOLA, H.-C. GRUNAU, AND E. MITIDIERI We first consider. Using the boundary condition v) = v1) =, we obtain ) 1 n 1 I vr) 1 v r) dr { ) n 1 +max 1) rvr) v r)) dr, { ) } n 1 max 1) Λ,λ r 1 vr) dr = Γ r n 1 ur) dr r 1 v r) dr } and 1) follows. Thecase1<< requires greater effort. y Lemma 1 and v 1 v dr =, it follows that 14) I 1 1) n r 1 v r) ) dr. + v r) vr) r In order to estimate the right hand side of 14) from below we introduce a further regularizing factor r ε. Here for the moment we just require that ε>. We will see below that a good choice for the value of ε is given by ε =. As a consequence we will miss λ as a coefficient for the remainder term. Alication of Hölder s inequality yields 15) = ) / r ε+ 1 v r) dr n r 1)/ v r) vr) r + v r) ) )/ n r ε+ 1) )/ vr) r r 1 v r) ) dr + v r) n vr) r ) ) ) )/ + v r) dr n r ε/ ) r 1 vr) ) + v r) ) dr. r License or coyright restrictions may aly to redistribution; see htt://

7 HARDY INEQUALITIES 155 Proceeding further in estimating the second integral of 15) by means of a Hardy inequality in dimension + ε), we find n r ε/ ) r 1 vr) + v r) ) dr r ) n 1 ) vr) 1 1 r ε+ 1 dr + 1 r ε+ 1 v r) dr r ) n 1 +1) 1 r ε+ 1 v r) dr. ε Here we have used the inequality a + b) 1 a + b ) a, b ) and the fact that r ε/ ) <r ε for all r, 1). Combining the above estimate with 15), inserting the result into 14), and choosing 16) ε :=, we arrive at 17) I 1 ) )/ n 1 +1) 1) 1) )/ r v r) dr. On the other hand, by the two-dimensional embedding W 1, L / ) L 3 ), we may define the constant 18) C ) :=inf r v r) dr X ) 1 1/3 ), r vr) 3 ) dr and thus we obtain r n 1 ur) dr = r 1 vr) dr = r ) /3 ) r 1/3 ) vr) ) dr ) )/3 ) ) 1/3 ) r dr rvr) 3 ) dr ) )/3 ) 1 C ) 1 1 Inserting this estimate into 17), we finally deduce with 19) ) n u dx u dx + Cn, ) Cn, ) = 1 1)1+ )/3 ) 1) )/ n The roof of 11) is now comlete. r v r) dr. u dx, ) ) )/ +1 C ). Remark. Alying the a riori estimates for the -Lalace oerator of [To] and [E] to the Euler-Lagrange equations corresonding to 18) it follows that C ) Λ as. Then, by 19), we deduce Cn, ) Λ =Γn, ) as. Moreover, C ) is uniformly bounded from below for 1, ]. Indeed, a rough estimate of C ) isgivenbyc ) π) 1/8.8. License or coyright restrictions may aly to redistribution; see htt://

8 156 F. GAZZOLA, H.-C. GRUNAU, AND E. MITIDIERI 3. Weakly singular L -remainder terms in Rellich-tye inequalities in higher-order Sobolev saces W k, ) under Navier boundary conditions In this section by modifying the reduction of the dimension technique used in [V], we shall rove higher-order Rellich-tye inequalities in W k, with remainder terms for functions satisfying Navier boundary conditions. Let. Let v = vr) be a smooth radially symmetric function and denote by ϑ the oerator defined by ϑ v := v + ϑ 1 v, where ϑ = ϑn, ) =4+ r n ). Throughout this section we shall set X = {v C [, 1]) : v ) = v1) =, v } and γ =inf r v r) { dr 1) 1 n ) n, Γ=max X λ ϑ =inf X rvr) dr r 1 ϑ vr) dr r 1 vr) dr, Λ ϑ =inf X ) Λ ϑ,λ ϑ r3 vr) ϑ vr)) dr r 1 vr) dr These notations should not be confused with those of Section. Remarks. The value of γ does not deend neither on nor on n. Indeed, γ is nothing else than the first eigenvalue of under homogeneous Dirichlet boundary conditions in the unit disk R ; γ = j.4.herej ν denotes the first ositive zero of the essel function J ν. If =,thenλ ϑ = λ ϑ and the first eigenvalue of the biharmonic oerator under Navier boundary conditions in the unit ball in R 4 is given by Λ ϑ = λ ϑ = j With the hel of a tedious calculation, which is outlined in Lemma 4 in the aendix, we find ositive lower bounds for Λ ϑ and λ ϑ also in the case >. These bounds imly that if >, then Γ c n for all n. In the sace W, W 1, we have the following statement. Theorem. Let and let R n n ) be a bounded domain. Denote by its n-dimensional Lebesgue measure and let e n =, where = 1 ) is the unit ball. Then, for any u W, W 1, ) we have ) u n ) 1)n u dx dx ) + 4 1) n ) 1 n 1 γ en +Γ 1 ) /n u dx. ) /n en u dx Proof. Ste 1. y scaling it suffices to consider = e n. First we show that we may restrict ourselves to = and radial suerharmonic functions u. For this },. License or coyright restrictions may aly to redistribution; see htt://

9 HARDY INEQUALITIES 157 urose we assume that ) has already been shown in this setting. Now let be an arbitrary bounded domain with = e n and let u W, W 1, ). Define f = u and let w W, W 1, ) be the radial strong solution of { w = f in, w = on, where f denotes the Schwarz symmetrization of f. y [Ta, Theorem 1] we know that w u. Hence u dx = f dx = f ) dx = w dx, w α dx u α dx u α dx, α {,, }. For the last inequality we refer again to [AL, Theorem.]. Assuming that ) holds in the radial suerharmonic setting, the same follows for any domain and any u W, W 1, ). Ste. Assume now that = and that u is suerharmonic and radially symmetric: u = ur), r =. Let us also recall that in the W, setting, the best Hardy constant is given by C see [DH, Theorem 1], [Mi, Theorem 3.1]), where n ) 1)n C =. Similarly as in 1), we introduce the transformation vr) :=r n/) ur), ur) =r n/) vr), so that ur) =r n/ r ϑ vr)+c vr) ). This allows us to aly Lemma 1 with the choice ξ = C v and η = r ϑ v. Indeed, we have I := = r n 1 u) dr C C 1 r n 1 u dr r 1 [ C v r ϑ v ) C v) ] dr { +max rv 1 v dr 1)C r 3 vr)) ϑ v) dr, } r 1 ϑ v dr, where we have used the boundary condition u1) = v1) =. Integrating by arts and using the identity v v = 4 v / ) gives rv 1 v dr = 1) rv v dr = 4 1) r v / ) dr. License or coyright restrictions may aly to redistribution; see htt://

10 158 F. GAZZOLA, H.-C. GRUNAU, AND E. MITIDIERI Therefore, recalling the definition of γ, λ ϑ and Λ ϑ,weinfer 4 1) 1 I C 1 γ r v dr +max { 1)C } Λ ϑ,λ ϑ r 1 v dr r n 1 4 1) = C 1 γ u dr +Γ r thereby comleting the roof of Theorem. r n 1 u dr Remark. For the case 1 <<, the receding roof shows the existence of a remainder term u with a factor which deends on n like n ) 1).We are confident that, even in this case, methods similar to those used in the roof of Theorem 1 can be emloyed to rove that an additional term of the tye u may be added to the right hand side of 6). In order to avoid tedious calculations, from now on we deal only with the Hilbert sace case =. Thecase> is exected to be similar. To make the statements that follow more recise, we introduce some further notations. Set X = {v C 1 [, 1]) : v ) = v1) =, v } and for n N let Λn) =inf rn 1 v r) dr X 1 rn 1 vr) dr. Again, this notation should not be confused with the revious ones. Proosition 1. Let R n be the unit ball. The first eigenvalue λ of the following Navier boundary value roblem: { u = λu in, u = u =on, satisfies Λn) = λ = inf u W, W 1, )\{} u) dx u dx. Proof. The first equality is due to the fact that the biharmonic oerator u u under the Navier boundary conditions u = u = on is actually the square of the Lalacian under Dirichlet boundary conditions. For the second equality, see e.g. [V, Lemma 3]. A direct consequence of Theorem is the following corollary. Corollary 1. Let R n, n 4, be a bounded domain. Denote by its n- dimensional Lebesgue measure and let e n =, where = 1 ) is the unit ball. Then, for any u W, W 1, ) we have u) dx n n 4) + 16 nn 4) Λ) 4 dx en ) /n ) 4/n en dx +Λ4) dx. License or coyright restrictions may aly to redistribution; see htt://

11 HARDY INEQUALITIES 159 Combining this result with the Hardy inequality 8) of rezis-vazquez, for third order Sobolev saces we have Theorem 3. Let R n, n 6, be a bounded domain. Then for any u W 3, W 1, ) with u =on, i.e. u W 1, ), we have: u dx n +) n ) n 6) 64 6 dx + 1 3n ) 4 +8n ) +16 ) ) /n en Λ) 16 ) ) n ) 4/n + Λ4) nn 4) + Λ) en 4 ) 6/n + Λ) Λ4) en dx. 4 dx dx Proof. First, by combining the argument from the roof of Theorem and [AL, Theorem.7], we reduce the assertion to the case = in the radial setting. An alication of the rezis-vazquez inequality 8) reduces the order of terms by one. Indeed, we have r n 1 u n ) dr r n 3 u) dr +Λ) r n 1 u) dr. 4 The second term of this inequality can be estimated with the hel of Corollary 1, while the singular term r n 3 u) dr can be handled by introducing the transformation vr) :=r n/) 3 ur), ur) =r 3 n/) vr). We obtain r n 3 u) dr n +) n 6) r n 7 dr 16 = r 3 4 v + ) n +)n 6) 1 r v dr = r 3 4 v) dr +4 r 4 v) v dr r v + 5r ) v vdr +4 r v ) n +)n 6) dr + r v) vdr = r 3 4 v) dr + [ r v r) ] n ) +8 ) r v) vdr Λ4) r 3 v dr + 1 n ) +8 ) Λ) rv dr = Λ4) r n 3 dr + 1 n ) +8 ) Λ) r n 5 dr. Note that only the boundary condition u1) = v1) = was used. Collecting terms comletes the roof. License or coyright restrictions may aly to redistribution; see htt://

12 16 F. GAZZOLA, H.-C. GRUNAU, AND E. MITIDIERI In order to iterate further we quote from [DH, Theorem 1], [Mi, Theorem 3.3] the following Lemma. Let R n be a sufficiently smooth bounded domain and σ<n 4. Then for every u C ) with u =we have u) σ dx n 4 σ) n + σ) dx. 16 σ+4 We emhasize that for the validity of the above inequality it is enough to assume that u =. With the hel of this result it is not difficult to rove the existence of remainder terms in Hardy inequalities of arbitrary order. Indeed, we have Corollary. Let R n be a sufficiently smooth bounded domain and let k N, k n. Then there exist constants c 1,...,c k, deending only on k, n and, such that for every u W k, ) with j u =for j N and j <kwe have: a) if k is even, k =m, m N: m u) dx 1 4 m m j=1 n +4m j)) m 4m dx + l=1 b) if k is odd, k =m +1, m N : m+1 m u 1 dx 4 m+1 n +4m + 4j) m+1 + l=1 c l j=1 dx. 4m+ l c l dx; 4m l dx 4m+ The existence of some constants c 1,...,c k as above is easily shown. Of articular interest would be their resective otimal values. In this regard, we oint out that already the roof of Theorem 3 indicates that iterative methods used to rove this kind of inequalities for functions satisfying the Navier boundary conditions = u = u = on, yield constants which aarently cannot be easily obtained in a closed form. One may hoe that things will imrove under the more restrictive Dirichlet boundary conditions, i.e. u W k, ). This will be discussed in the following section. 4. Weakly singular L -remainder terms in Rellich-tye inequalities in higher-order Sobolev saces under Dirichlet boundary conditions Our first goal in this section will be to adat Corollary 1 in the case =,for functions satisfying Dirichlet boundary conditions, i.e. for functions belonging to the sace W, ) instead of W, W 1, ). At least when is a ball or close to a ball, the estimation constants will be considerably larger. On the other hand, the symmetrization argument used in the roof of Theorem fails: the function w used in that roof does not belong to the sace W, ) considered here. We overcome this difficulty by a relatively involved argument based on the ositivity License or coyright restrictions may aly to redistribution; see htt://

13 HARDY INEQUALITIES 161 roerties of Green s functions, which seems to work only in the Hilbert sace case = and if is relaced with the circumscribed ball. Since the technical difficulties for obtaining exlicit large constants in the inequalities that follow increase quickly with the order k of the sace W k, ), we study the imroved Hardy inequality only in W, ). For the general case we formulate a conjecture on which behaviour of the remainder terms we exect when the order of saces becomes arbitrarily large. In this section the eigenvalues of biharmonic and more general olyharmonic oerators will lay an imortant role. Let R n denote the unit ball and let ) Λ ) k,n = inf W k, )\{} inf W k, )\{} m u) dx u dx, k =m, m N; m u dx, k =m +1,m N. u dx The notation Λn) of Section 3 is a secial case of the notation introduced here; indeed we have Λn) =Λ ),n). Remark. Taking Proosition 1 into account we immediately see that Λ ),n) Λ,n). With the hel of qualitative roerties of first eigenfunctions in articular under Navier boundary conditions), it can be shown that also the strict inequality holds. Moreover, the ratio of these eigenvalues is rather large; by elementary calculations one finds e.g. Λ ), 1) = while Λ ), 1) = π 4 /16 = For n = 4, the case which is needed below, a rough estimate according to [PS,. 57] gives Λ ), 4) j1 j = while Λ ), 4) = j1 4 = Theorem 4. Let n 4 and let R n be such that R ). Then for every u W, ) we have u) dx n n 4) 16 4 dx nn 4) + Λ, ) R u 1) dx +Λ ), 4 ) R 4 dx. Proof. y trivial extension, we have W, ) W, R )). Further, a scaling argument shows that we may assume R =1 R = ). If in addition, u is radially symmetric, we roceed as in Ste of the roof of Theorem with =. In its last conclusion we may instead exloit that u satisfies homogeneous Dirichlet ) boundary conditions and relace Λ ϑ ϑ=4 =Λ4) = Λ, 4) with Λ ), 4, thereby roving 1) for radial u. It remains to extend 1) to arbitrary functions u W, ). To this end, consider for l N the sequence of relaxed minimum roblems: F l v) µ l := inf W, )\{} v dx, License or coyright restrictions may aly to redistribution; see htt://

14 16 F. GAZZOLA, H.-C. GRUNAU, AND E. MITIDIERI where F l v) = v) dx n n 4) nn 4) l 1 1 ) l ) v v 4 dx Λ, ) dx. Since we may already use Corollary 1, we have F l v) 1 l v) dx for every v W, ). This means that for every minimizing sequence v k W, ) with ) v k dx = 1 and lim k F l v k )=µ l, the sequence u k W, ) is bounded. k N After selecting a subsequence we may assume that u k u W, ) andu k u strongly in L ). The bilinear form Φ l v, w) := v wdx n n 4) 1 1 ) v w 16 l 4 dx nn 4) Λ, ) 1 1 ) v w l dx associated with F l v) defines a scalar roduct on W, ), which, by 1 v) dx Φ l v, v) v) dx = v l W, ), defines an equivalent norm. If we consider, just for the following argument, W, ) with Φ l.,.) as scalar roduct, then, the lower semicontinuity of the corresonding norm in the weak toology gives µ l = lim inf F lv k ) F l v). k Further, we have 1 = lim vk dx = v dx. k Hence F l v) =µ l,andv W, ) is an otimal nontrivial) function for µ l. Consequently, v is a weak solution of the Euler-Lagrange equation v = 1 1 ) n n 4) v l 16 4 ) + ) 1 1 nn 4) l Λ, ) v + µ l v in, v = v = on. Next we show that v is of fixed sign. Assume by contradiction that there exist subsets +,, both of ositive measure, such that v>on + and v< on. We may now aly a decomosition method exlained in detail in [GG, Section 3]; see also [Mo]. Let { } K = u W, ) : u be the closed convex cone of nonnegative functions and { } K = u W, ) : u,u) for all u K the dual cone. Here u,u)= u ) u) dx is the standard scalar roduct in W, ) andk is the cone of all weak subsolutions of the biharmonic equation License or coyright restrictions may aly to redistribution; see htt://

15 HARDY INEQUALITIES 163 in under Dirichlet boundary conditions. y a comarison result of oggio [o,. 16] the elements of K are nonositive. The strict ositivity of the biharmonic Green function in even shows that u K \{} u < ; cf. [AGGM, Lemma 1]. According to [Mo], we decomose v = v 1 + v, v 1 K, v K, v 1 v. Since v 1, v <, relacing v = v 1 + v with v 1 v yields F l v 1 + v ) >F l v 1 v ), v 1 + v ) dx < v 1 v ) dx, contradicting the minimality of F l v)/ v dx. Hencewemayassumethat v. Considering olar coordinates x = rξ, r [, 1], ξ = 1 and integrating the Euler-Lagrange equation ) over { ξ =1} shows that wr) := 1 vrξ) dωξ) W, ) ne n ξ =1 is a radial weak solution of ). y virtue of v, we also have w. Since we may already use the Hardy inequality 1) for the radial function w, we conclude that µ l = > F l w) w dx w) dx n n 4) Λ ), 4 ). 16 w dx nn 4) 4 Λ, ) w dx w dx To sum u, we have shown that for any l N and for every u W, ) wehave u) dx 1 1 ) n n 4) l 16 4 dx ) nn 4) Λ, ) l dx +Λ ), 4 ) dx. Sending l in the above inequality, the claim follows. We conclude this section with some remarks concerning general higher-order Sobolev saces. We recall that W k, ) is equied with the norm m u) dx, if k =m, m N; u = W k, ) m u dx, if k =m +1,m N. License or coyright restrictions may aly to redistribution; see htt://

16 164 F. GAZZOLA, H.-C. GRUNAU, AND E. MITIDIERI We exect the following result: Conjecture 1. Let R n be such that R ). Let k N, n k. Then for all u W k, ) we have: k ) { } u 1 k j W k, ) 4 j n +k 4l)n k 4+4l)) j j= l=1 Λ ) k j, k j ) R j k 3) dx j with the convention that Λ ), ) = 1. In the radial setting, the terms originate from Λ ) k j, k j ) ne n if k j is even, and from ne n dx j r k j 1 k j)/ k j v) dr, r k j 1 k j 1)/ k j v) ) dr, if k j is odd. Here k j is the radial Lalacian in dimension k j. The radial functions u and v are related by means of the transformation vr) =r n/) k ur), ur) =r k n/) vr). In Conjecture 1 as well as in the roof of Theorem 4 we simly used eigenvalue estimates for these terms. One may also wish to use Hardy-like inequalities in order to have largest ossible constants in front of the most singular remainder terms. Since we are assuming that we are dealing with functions satisfying homogeneous Dirichlet boundary conditions, we can aly the following lemma. Lemma 3. a) Let k, j 1, v C [, 1]) with v 1) =. Then: r k j v) k +1 1 j) dr r k v ) dr. 4 b) Let k, v C 1 [, 1]) with v1) =. Then: r k v ) k 1 1) dr r k v dr. 4 To illustrate how the alication of this elementary and well-known lemma shifts less singular remainder terms to more singular ones, we modify the roof of Theorem 4 and obtain: Corollary 3. Let n 4 and let R n be such that R ). Then for every u W, ) we have u) dx n n 4) 16 4 dx + 1 nn 4) + 8) Λ, ) R dx. License or coyright restrictions may aly to redistribution; see htt://

17 HARDY INEQUALITIES 165 Aendix A. Proofs of some technical inequalities First we resent a roof of Lemma 1 from Section. Proof of Lemma 1. y means of integration by arts or Taylor s formula we find 4) ξ η) + ξ 1 η ξ = 1)η 1 t)ξ tη) dt. Assume that. Since ξ η, we have ξ tη 1 t)ξ and the right hand side in 4) becomes 1)η 1 t)ξ tη) dt 1)η ξ 1 t) 1 dt = 1)η ξ. To get the second estimate from below in the case, we first consider nonositive η. Then ξ tη t η and the right hand side in 4) becomes 1)η 1 t)ξ tη) dt 1) η 1 t) t dt = η. For η wegetξ tη 1 t) η and thus 1)η 1 t)ξ tη) dt 1) η 1 t) 1 = 1) η η. Now let 1 <. This imlies ξ tη) ξ + η ), and consequently 1)η 1 t)ξ tη) 1 1)η dt ξ + η ) 1 t) dt 5) 1) η = ξ + η ). This comletes the roof. We now refer to the estimate on Γ and Λ ϑ mentioned in the first remark of Section 3. Clearly we have Λ ϑ. We shall show that even Λ ϑ > andλ ϑ as n whenever >. Lemma 4. Let X = {v C [, 1]) : v ) = v1) =, v }. Assume that >; then we have def Λ ϑ =inf r3 vr) ϑ vr)) dr X 1 4 n ) r 1 vr) dr >, λ ϑ def =inf X r 1 ϑ vr) dr r 1 vr) dr n ) >. Proof. Let v X. A first integration by arts shows that 6) r 1 vr) dr = 1 y means of the Cauchy-Schwarz inequality we obtain 7) r 1 vr) dr 1 4 r vr) vr)v r) dr. r +1 vr) v r)) dr. License or coyright restrictions may aly to redistribution; see htt://

18 166 F. GAZZOLA, H.-C. GRUNAU, AND E. MITIDIERI Recalling that from Section 3 we have ϑ =4+ n ), a further integration by arts gives 8) n ) 1)) = 1) r vr) vr)v r) dr r +1 vr) v r)) dr + Combining 7), 8) and 6) yields 4 r 1 vr) dr r +1 vr) vr) ϑ vr) dr. r +1 vr) v r) dr n ) 1) = r vr) vr)v r) dr 1) 1 r +1 vr) vr) ϑ vr) dr 1 ) n ) 1 = 1) r 1 vr) dr 1 r +1 vr) vr) ϑ vr) dr, 1 from which we infer at once that recall r 1) n ) r 1 vr) dr r +1 vr) vr) ϑ vr) dr r 1/ vr) /) ) r 3/ vr) )/ ϑ vr) r dr ) 1/ 1/ r 1 vr) dr r 3 vr) ϑ vr)) dr), and the first claim follows. In order to obtain the second statement, we only need to modify this last estimate as follows: n ) r 1 vr) dr r +1 vr) vr) ϑ vr) dr r 1) 1)/ vr) 1) ) r 1)/ ϑ vr) r dr ) 1)/ 1/ r 1 vr) dr r 1 ϑ vr) dr). This comletes the roof of Lemma 5. Acknowledgement We are grateful to the referees and to Professor William eckner for useful comments and suggestions on how to imrove the exosition of the resent work. License or coyright restrictions may aly to redistribution; see htt://

19 HARDY INEQUALITIES 167 References [ACR] Adimurthi, N. Chaudhuri, M. Ramaswamy, An imroved Hardy-Sobolev inequality and its alication, Proc. Amer. Math. Soc. 13 ), MR j:353 [AE] Adimurthi, M.J. Esteban, An imroved Hardy-Sobolev inequality in W 1, and its alication to Schrödinger oerator, Nonlinear Differ. Eq. Al. NoDEA, toaear. [AL] F. Almgren, E. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc. 1989), MR 9f:4938 [AGGM] G. Arioli, F. Gazzola, H.-Ch. Grunau, E. Mitidieri, A semilinear fourth order ellitic roblem with exonential nonlinearity, rerint. [FT] G. arbatis, S. Filias, A. Tertikas, A unified aroach to imroved L Hardy inequalities with best constants, Trans. Amer. Math. Soc., this issue. [o] T. oggio, Sulle funzioni di Green d ordine m, Rend. Circ. Mat. Palermo 195), [L] H. rezis, E. Lieb, Sobolev inequalities with remainder terms, J. Funct. Anal ), MR 86i:4633 [M] H. rezis, M. Marcus, Hardy s inequalities revisited, Ann. Sc. Norm. Su. Pisa ), MR 99m:4675 [MS] H. rezis, M. Marcus, I. Shafrir, Extremal functions for Hardy s inequality with weight, J. Funct. Anal. 171 ), MR 1g:4668 [V] H. rezis, J. L. Vazquez, low-u solutions of some nonlinear ellitic roblems, Rev. Mat. Univ. Comlutense Madrid ), MR 99a:3581 [DA] L. D Ambrosio, Some Hardy inequalities on the Heisenberg grou, Differential Equations, in ress. [DH] E.. Davies, A. M. Hinz, Exlicit constants for Rellich inequalities in L ), Math. Z ), MR 99e:58169 [E] H. Egnell, Existence and nonexistence results for m-lalace equations involving critical Sobolev exonents, Arch. Ration. Mech. Anal ), MR 9e:3569 [FT] S. Filias, A. Tertikas, Otimizing imroved Hardy inequalities, J. Funct. Anal. 19 ), MR 3f:4645 [GG] F. Gazzola, H.-Ch. Grunau, Critical dimensions and higher order Sobolev inequalities with remainder terms, Nonlin. Diff. Eq. Al. NoDEA 8 1), MR d:354 [H] G. H. Hardy, Notes on some oints in the integral calculus, Messenger Math ), [HLP] G.H.Hardy,J.E.Littlewood,G.Pólya, Inequalities, Cambridge: University Press, [L] P. Lindqvist, On the equation div u u)+λ u u =,Proc. Amer. Math. Soc ), MR 9h:3588 [MS] M. Marcus, I. Shafrir, An eigenvalue roblem related to Hardy s L inequality, Ann. Scuola Norm. Su. Pisa Ser. IV 9 ), MR c:498 [Ma] V. G. Maz ya, Sobolev saces, Transl. from the Russian by T. O. Shaoshnikova, erlin etc.: Sringer-Verlag, MR 87g:4656 [Mi] E. Mitidieri, A simle aroach to Hardy s inequalities, Math. Notes 67 ), , translation from Mat. Zametki 67 ), MR 1f:6 [Mo] J. J. Moreau, Décomosition orthogonale d un esace hilbertien selon deux cônes mutuellement olaires, C. R. Acad. Sci. Paris ), MR 5:3346 [PS] P. Pucci, J. Serrin, Critical exonents and critical dimensions for olyharmonic oerators, J. Math. Pures Al ), MR 91i:3565 [R] F. Rellich, Halbbeschränkte Differentialoeratoren höherer Ordnung, in: J. C. H. Gerretsen et al. eds.), Proceedings of the International Congress of Mathematicians Amsterdam 1954, Vol. III,. 43-5, Groningen: Nordhoff, MR 19:55c [Ta] G. Talenti, Ellitic equations and rearrangements, Ann. Scuola Norm. Su. Pisa Ser. IV ), MR 58:917 [To] P. Tolksdorf, Regularity for a more general class of quasilinear equations, J. Differ. Equations ), MR 85g:3547 License or coyright restrictions may aly to redistribution; see htt://

20 168 F. GAZZOLA, H.-C. GRUNAU, AND E. MITIDIERI [VZ] [V] J. L. Vazquez, E. Zuazua, The Hardy constant and the asymtotic behaviour of the heat equation with an inverse-square otential, J. Funct. Anal. 173 ), MR 1j:351 R. van der Vorst, est constant for the embedding of the sace H H 1 ) into L N/N 4) ), Diff. Int. Eq ), MR 94b:4653 Diartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 3, I-133 Milano, Italy address: gazzola@mate.olimi.it Fakultät für Mathematik, Otto-von-Guericke-Universität, Postfach 41, D-3916 Magdeburg, Germany address: Hans-Christoh.Grunau@mathematik.uni-magdeburg.de Diartimento di Scienze Matematiche, Via A. Valerio 1/1, Università deglistudidi Trieste, I-341 Trieste, Italy address: mitidier@univ.trieste.it License or coyright restrictions may aly to redistribution; see htt://