AN EXAMPLE OF FUNCTIONAL WHICH IS WEAKLY LOWER SEMICONTINUOUS ON W,p FOR EVERY p > BUT NOT ON H FERNANDO FARRONI, RAFFAELLA GIOVA AND FRANÇOIS MURAT Abstract. In this note we give an example of functional which is defined and coercive on H, which is sequentially weakly lower semicontinuous on W,p for every p >, but which is not sequentially lower semicontinuous on H. This functional is non local.. Results and comments Let be a bounded open subset of, with if N, and =,R if N =. In this note, we give an example of functional which is defined and coercive on H, which has quadratic growth with respect to Dv = Dv L N, which is sequentially weakly lower semicontinuous on W,p for every p >, but which is not sequentially weakly lower semicontinuous on H. More precisely, when N 3, we recall the Hardy-Sobolev inequality see e.g. Theorems.7 and.8 in [5], Lemma 7. in [6], and also 4. in the Appendix below RN. m N x dx v H RN, where m N denotes the best possible constant in the inequality, i.e.. m N = inf R N v H RN. x dx It is well known that m N is given by see the references above m N N =. 4 We consider a function ϕ which is defined and continuous on [, ], which is non negative, non increasing and which satisfies.3 ϕ > m N and ϕ < m N. Finally we define the functional J by.4 Jv = Our main result is the following ϕ Dv x dx v H. Theorem.. Let N 3 and let be a bounded open subset of, with. Assume that ϕ is a continuous, non negative and non increasing function on [, ] satisfying.3, where m N is given by.; then the functional J defined by.4 satisfies Mathematics Subject Classification. 49J45, 6D. Key words and phrases. Lower semicontinuity, Hardy-Sobolev inequalities.
FERNANDO FARRONI, RAFFAELLA GIOVA AND FRANÇOIS MURAT i there exists a constant C > such that.5 C + Jv v H ; ii the functional J is sequentially weakly lower semicontinuous on W,p for every p >, i.e..6 Jv lim inf Jv n if v n v in W,p weakly; iii the functional J is not sequentially weakly lower semicontinuous on H ; more precisely, there exists a sequence of functions w n H such that w n in H weakly and.7 lim inf Jw n < J. Theorem. is proved in Section below. On the other hand, when N = we consider a bounded open subset of R, with and some R for which B R. We recall the Hardy-Sobolev inequality see e.g. Theorems 4. and 5.4 in [] and Lemma 7.4 in [6].8 m x log x dx v H, R where m denotes the best possible constant in the inequality, i.e..9 m = inf v H. x log x dx R It is well known that m is given by see the references above m = 4. We consider a function ϕ which is defined and continuous on [, ], which is non negative, non increasing and which satisfies. ϕ > m and ϕ < m, and we define the functional J by. Jv = ϕ Dv x log x dx R In this case, we prove the following v H. Theorem.. Let N = and let be a bounded open subset of R, with and B R. Assume that ϕ is a continuous, non negative and non increasing function on [, ] satisfying., where m is given by.9; then the functional J defined by. satisfies the conditions i, ii and iii of Theorem.. Theorem. is proved in Section 3 below. Finally, in the -D case, let be the interval =,R. We recall the Hardy-Sobolev inequality see e.g. Theorem 37 in [3] and Lemma.3 in [5]. m x dx v dx v H,, In this note we denote by BR the open ball of of radius R and center.
A FUNCTIONAL WHICH IS WEAKLY LOWER SEMICONTINUOUS ON W,p BUT NOT ON H 3 where m denotes the best possible constant in the inequality, i.e..3 m = inf v H, v dx dx. x It is well known that m is given by see the references above m = 4. We consider a function ϕ which is defined and continuous on [, ], which is non negative, non increasing and which satisfies.4 ϕ > m and ϕ < m, and we define the functional J by.5 Jv = R In this case we prove the following v dx ϕ v R x dx v H,R. Theorem.3. In the -D case, let be the interval =,R. Assume that ϕ is a continuous, non negative and non increasing function on [, ] satisfying.4, where m is given by.3; then the functional J defined by.5 satisfies the conditions i, ii and iii of Theorem.. The proof of Theorem.3 is the same as the one of Theorem. and is left to the reader. Remark.. Observe that, in contrast with the case N, the functions v H,R vanish in in the -D case. Remark.. Consider a functional of the integral form.6 Jv = Fx,v,Dvdx v W,p, where F : R R is a Carathéodory function satisfying a x + c ξ p Fx,s,ξ a x + b s p + c ξ p for a.e. x, s,ξ R, where p >, c > and a,a L. It is well known see e.g. Theorems 3. and 3.4 in [] and Theorem.4 in [4] that the functional J is sequentially weakly lower semicontinuous on W,p if and only if Fx,s, is a convex function for a.e. x and for every s R; moreover, in this case, the functional J is sequentially weakly lower semicontinuous on W,q for every q >. It is therefore impossible to write the functionals defined by.4,. and.5 in the integral form.6. Remark.3. Using the result 4.3 of the Appendix below, we can prove an assertion which is stronger than assertion ii, namely: if N 3 then Jv lim inf Jv n if v n v in H weakly with Dv n equi-integrable in L. The same result continues to hold for N = and N =. Assertion ii of Theorems.,. and.3 is a special case of this assertion since is assumed to be bounded. Remark.4. Actually in dimension N 3, Theorem. continues to hold with the same proof if the Hardy-Sobolev inequality. is replaced by the Sobolev inequality.7 m dx v HR N,
4 FERNANDO FARRONI, RAFFAELLA GIOVA AND FRANÇOIS MURAT where is the Sobolev s exponent defined by = N/N and where m is the best possible constant in.7, and if in the definition.4 of the functional J the integral x dx is replaced by dx. More than that, Theorem., Theorem. and Theorem.3 still continue to hold with the same proof if the inequalities.,.8,. and.7 are replaced by an inequality of the type m X v X Dv, where X is a Banach space such that the embedding H X is not compact while the embedding W,p X is compact for any p >. The non compactness of the embedding H L ;ωxdx and the compactness of the embedding W,p L ;ωxdx for p >, where ωx = x if N = or N 3, if N =, x log x R see 4. and 4.3 in the Appendix below, are indeed at the root of the proofs of iii and ii. This explains why Theorem. continues to hold by replacing the Hardy-Sobolev inequality by the Sobolev inequality. In contrast, if the embedding H X is compact e.g. in the case X = L for bounded, it is straightforward to prove that the functional Jv = ϕ Dv v X v H is sequentially weakly lower semicontinuous on H whenever ϕ is non increasing: just take a sequence v n such that v n v in H weakly, and observe that in this framework lim inf Dv n dx, ϕ Dv lim inf ϕ Dv n, lim v n X = v X. Remark.5. Observe finally that in the proof of Theorem. below for N 3 it is not necessary to know the explicit value of the best constant m N in the inequality., whenever the function ϕ is chosen such that.3 holds. In contrast, the proof of Theorem. below where N = uses the fact that the constant m coincides with m. If one does not want to use the fact that m = m, it would be sufficient to assume in. that ϕ > m in place of ϕ > m see also the proof of iii in Theorem.. Also it should be observed that the fact that the best constant m X is attainable or not does not play any role in the proofs below, in contrast with the fact that the embedding H X is not compact for bounded, which is crucial.. Proof of Theorem. Proof of i. By the definition of Jv we have Jv, since ϕ is non negative.
A FUNCTIONAL WHICH IS WEAKLY LOWER SEMICONTINUOUS ON W,p BUT NOT ON H 5 It remains to prove the inequality on the left-hand side of i. Since ϕ is continuous and satisfies.3, there exists t > such that ϕt = m N /. If Dv t then ϕ Dv m N /. Therefore Jv m N x dx, and the inequality on the left-hand side of i holds. On the other hand, if Dv t, then Jv ϕ ϕ ϕ m N m N ϕ m t, N in view of.3. If we choose a constant C such that x dx ϕ m t C, N we have Jv t ϕ m t C, N and the inequality on the left-hand side of i is again proved. This proves i. Proof of ii. Let p >. Assume that v n v in W,p weakly. Since is bounded, v n v in H weakly and there exists α such that. lim inf Dv n = Dv + α. Since ϕ is continuous and non increasing, there esists β such that. lim inf ϕ Dv n = ϕ Dv + β. Moreover, by the compactness of the embedding W,p L ; in the Appendix below, we get.3 lim v n x dx = x dx. x dx for p > see 4.3 Note that.3 continues to hold if we assume 4.3 in place of v n v in W,p weakly. This allows one to prove the assertion of Remark.3. Combining.,. and.3, we obtain lim inf Jv n Jv + α + β x dx Jv, which proves ii. Proof of iii. Let λ be such that m N < λ < ϕ such a λ exists in view of.3. Recalling
6 FERNANDO FARRONI, RAFFAELLA GIOVA AND FRANÇOIS MURAT the definition. of m N, there exists a function ψ C RN such that RN ψ λ x dx > Dψ dx. Since ϕ is continuous and satisfies.3, there exists t > such that ϕt = λ. Take s such that < s Dψ t. The function w = sψ belongs to C RN and satisfies.4 ϕ Dw λ, as well as RN w.5 λ x dx > Dw dx. Define the sequence w n by then w n x = n N wnx; Dw n x = n N Dwnx. For n sufficiently large, the function w n belongs to H and Dw n dx = Dw w n RN dx and x dx = w x dx. Therefore, for n sufficiently large, the sequence w n is bounded in H with w n in H weakly, and RN Jw n = Dw dx ϕ Dw w x dx. Therefore Jw n < in view of.4 and.5. This proves iii. 3. Proof of Theorem. The proof of Theorem. is similar to the one of Theorem., but differs by technical aspects. Proof of i. Condition.5 is proved exactly as in the proof of Theorem.. Proof of ii. Let p >. Assume that v n v in W,p weakly. Then, as in the proof of Theorem., we have for some α and β 3. lim inf Dv n = Dv + α, { 3. lim inf ϕ Dvn } = ϕ Dv + β. Moreover, since p > N =, we have that v n v uniformly in, and, since we have 3.3 lim Combining 3., 3. and 3.3, we obtain x log x L, R v n x log x dx = R lim inf Jv n = Jv + α + β x log x dx. R v x log x R dx Jv,
A FUNCTIONAL WHICH IS WEAKLY LOWER SEMICONTINUOUS ON W,p BUT NOT ON H 7 which proves ii. Proof of iii. Let λ be such that m < λ, where m is the best constant defined by.3 in the -D Hardy-Sobolev inequality.. Then there exists ψ C, such that λ ψt t dt > ψ t dt. Since ϕ is a continuous and satisfies., and since the best constant m defined by.9 in the -D Hardy-Sobolev inequality.8 coincides with m, we can choose λ such that m = m < λ < ϕ if we do not want to use the property m = m, it would be sufficient to assume in. that ϕ > m in place of ϕ > m. Then, there exists t > such that ϕt = λ. Take s such that < πs ψ t. The function w = sψ belongs to C, and satisfies 3.4 ϕ π w t dt λ, as well as 3.5 λ Define the sequence w n by then w n x = Dw n x = { wt t dt > { n w n log x R w t dt. if x R, if x R, nw n log x R x x if x R, if x R. For n sufficiently large, the function w n belongs to H and R Dw n dx = π w n log r n dr = π R r while w n x log x R dx = π R w n log r nr log r R R dr = π w t dt, wt dt. Therefore, for n sufficiently large, the sequence w n is bounded in H with w n in H weakly, and Jw n = π w t dt πϕ π w t wt dt t dt. Therefore Jw n < in view of 3.4 and 3.5. This proves iii. 4. Appendix In this Appendix we recall some facts about the Hardy-Sobolev inequality in dimension N 3, some of whom are well known. 4.. A classical proof of. is to write, for every v C RN x Dv + cv x dx = t Dv + cv x Dv x + c v x Integrating by parts the second term, one gets v x Dv RN dx = N x x dx, dx.
8 FERNANDO FARRONI, RAFFAELLA GIOVA AND FRANÇOIS MURAT and therefore N c c RN x dx. The choice c = N / proves. with m N = N /4. 4.. Let us now prove by mean of a counterexample that, when, the embedding H ; L dx is not compact. For that we consider the functions x 4. u n x = n T n G R x, where G R : R is the function defined by 4. G R x = x N R N if x R, if x R, with R > such that the ball B R, and where T n : R R is the truncation at height n, i.e. t if t n, T n t = n t t if t n. Then Du n dx = Du n dx = N S B R n r n r N dr, where S N is the area of the unit sphere of and where r n is defined by Therefore r N n = n. Du n dx = N S N, and u n in H weakly. On the other hand, one has u n x Brn dx u n rn x dx = S N n r N 3 dr = S N N nrn n, and then u n lim x dx S N N. This proves that the embedding H ; L is not compact. x dx In dimension N =, this counterexample continues to hold by defining in 4. the function G R by G R x = log x R if x R. In dimension N =, one uses the continuous piecewise affine functions u n such that u n =, u n R /n = / n and u n R =. 4.3. Let us finally prove that, when 4.3 u n u in H weakly with Du n equi-integrable in L, then u n u in L ; x dx. Note that every sequence satisfying u n u in W,p weakly, with p >, satisfies 4.3 since is bounded; therefore this claim implies that the embedding W,p L ; x dx is compact for p >. Let δ > be small. We write u n u u n u 4.4 x dx = \Bδ u n u x dx + Bδ x dx,
A FUNCTIONAL WHICH IS WEAKLY LOWER SEMICONTINUOUS ON W,p BUT NOT ON H 9 where B δ is the ball of radius δ. Since L \ B x δ and since the embedding H L is compact for bounded, the first term of 4.4 tends to zero when n. Let ψ δ be the radial function defined by ψ δ x = if x δ, x δ if δ x δ, if x δ. For δ sufficiently small, ψ δ has compact support in, and using Hardy-Sobolev inequality. we have m u n u B δ x dx m ψ δ u n u x dx D ψ δ u n u dx Dψ δ u n u dx + D u n u ψ δ dx Dψ δ u n u dx + D u n u dx. B δ For δ fixed, the first term tends to zero when n still because the embedding H L is compact, while the second term is small uniformly in n when δ is small in view of the equi-integrability assumption 4.3. This proves the claim. This also proves the assertion of Remark.3. Acknowledgement This work has been done when the third author was visiting the Dipartimento di Matematica ed Applicazioni Renato Caccioppoli of the Università di Napoli Federico II. It was completed when the first author was visiting the Laboratoire Jacques-Louis Lions of the Université Pierre et Marie Curie Paris VI, with the support of the Università di Napoli Federico II. This support and the hospitality of both institutions are gratefully acknowledged. References [] Barbatis, G., Filippas, S., Tertikas, A., A unified approach to improved L p Hardy inequalities with best constants. Trans. Amer. Math. Soc., 356 4, 69 96. [] Dacorogna, B., Direct Methods in the Calculus of Variations. Springer-Verlag, Berlin 989. [3] Hardy, G.H., Littlewood, J.E., Polya, G., Inequalities. Cambridge University Press, Cambridge 95. [4] Marcellini, P., Sbordone, C., Semicontinuity problems in the calculus of variations. Nonlinear Anal., 4 98, 4 57. [5] Opic, B., Kufner, A., Hardy-type Inequalities. Pitman Research Notes in Mathematics, 9, Longman Scientific and Technical, Harlow 99. [6] Tartar, L., An Introduction to Sobolev Spaces and Interpolation Spaces. Lecture Notes of the Unione Matematica Italiana, 3, Springer-Verlag, Berlin 7. Dipartimento di Matematica e Applicazioni R. Caccioppoli, Università degli studi di Napoli Federico II, via Cintia - 86 Napoli - Italy E-mail address: fernando.farroni@dma.unina.it Dipartimento di Statistica e Matematica, Università degli studi di Napoli Parthenope, via Medina 4-833 Napoli - Italy E-mail address: raffaella.giova@uniparthenope.it Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie Paris VI, Boîte courrier 87-755 Paris cedex 5 - France E-mail address: murat@ann.jussieu.fr