Optimal Sobolev and Hardy-Rellich constants under Navier boundary conditions

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1 Optimal Sobolev and Hady-Rellich constants unde Navie bounday conditions Filippo Gazzola, Hans-Chistoph Gunau, Guido Swees Abstact We pove that the best constant fo the citical embedding of highe ode Sobolev spaces does not depend on all the taces. The poof uses a compaison pinciple due to Talenti [9] and an extension agument which enables us to extend adial functions fom the ball to the whole space with no incease of the Diichlet nom. Simila aguments may also be used to pove the vey same esult fo Hady-Rellich inequalities. AMS Classification: pimay 46E35, seconday 6D, 35J55 Keywods: optimal constant, Sobolev embedding, Hady-Rellich inequality Intoduction and esults Let n N, p (, ) and m N with m < n/p. Then, it is well-known [] that the Sobolev space (R n ) embeds continuously into L np/(n mp) (R n ). Since p := np/(n mp) is the lagest exponent fo which embeddings into L q (R n ) spaces hold, it is called the citical Sobolev exponent. In fact, moe efined embeddings hold tue. On the space Cc (R n ) of smooth compactly suppoted functions, define the nom k u L u m,p,r n := p (R n ) if m = k, () ( k u) L p (R n ) := ( k u) L p (R n ) if m = k +, and denote by D m,p (R n ) the closue of Cc (R n ) with espect to the nom (). Then, also the lage space D m,p (R n ) embeds continuously into L p (R n ) and a best Sobolev constant fo the coesponding embedding is defined, see [, 8] and also pevious esults in []. Moe pecisely, let then S m,p > and S m,p := D m,p (R n )\} u p m,p,r n u p L p (R n ) S m,p u p L p (R n ) u p m,p,r n fo all u D m,p (R n )., () It can be shown that the imum in () is achieved and, when m = o p =, the constant S m,p can be explicitly computed, see [, 7, 8, ]. Simila esults ae also available in bounded domains. Let Ω R n be a bounded domain and on the space Cc (Ω) of smooth compactly suppoted functions, define the nom k u L u m,p,ω := p (Ω) if m = k, (3) ( k u) L p (Ω) := ( k u) L p (Ω) if m = k +, Dipatimento di Matematica, Politecnico di Milano, Italy Institut fü Analysis und Numeik, Otto-von-Gueicke-Univesität Magdebug, Gemany Mathematisches Institut, Univesität zu Köln, Gemany

2 and denote by (Ω) the closue of C c (Ω) with espect to the nom m,p,ω fom (3). Again, one is inteested in the optimal constant fo the embedding inequality. By taking advantage of scaling aguments and the concentation-compactness pinciple, it is shown in [] that (Ω)\} u p m,p,ω u p L p (Ω) = S m,p, (4) whee S m,p is the same constant as in (). In othe wods, the best Sobolev constant fo the citical embedding is independent of the domain. Howeve, contay to the case addessed in (), the imum in (4) is not achieved. The space (Ω) may also be seen as the closed subspace of (Ω) of functions with vanishing taces up to ode m. A natual question which aises is to find out whethe the best embedding constant (4) depends on all the taces. The space of paticula inteest that appeas, is (Ω) := v (Ω); j v Ω = in the sense of taces fo j < m }. (5) Indeed, when p =, the space W m, (Ω) is the space whee vaiational solutions to polyhamonic elliptic pde s ae sought when complemented with the so-called homogenous Navie bounday conditions on Ω. With these bounday conditions, the polyhamonic equation may be ewitten as a second ode system. Note that (Ω) stictly contains (Ω) so that the just mentioned question is of inteest. The fist step to answe this question is to define a suitable nom and, in this case, some smoothness of the bounday is equied. Fo simplicity, we take Ω C m although much less egulaity is needed, see []. In such case, m,p,ω fom (3) is a complete nom also on (Ω). Then, the best embedding constant is defined by S m,p, (Ω) := (Ω)\} u p m,p,ω u p L p (Ω). (6) In view of the just mentioned inclusion (Ω) (Ω), it is clea that S m,p, (Ω) S m,p fo any bounded smooth domain Ω. One then wondes whethe this inequality is even an equality and, subsequently, whethe S m,p, (Ω) is independent of the domain Ω. When m = p =, this question was answeed positively in a pape by van de Vost []. Subsequently, Ge [9] gave a positive answe when p = fo any m N. As fa as we ae awae, the geneal case p > has not been peviously consideed. Both papes [9, ] ae based on the concentation-compactness pinciple by Lions []. Van de Vost [, p.59] claims that the concentation-compactness lemma is a vitual tansciption of the pinciple due to Lions although a cucial pat of the poof is not caied out in full detail. Howeve, it is not clea to us how the esult in [, Lemma A] can be poved with an extension agument as in [] since a) the space Cc (Ω) is not dense in (Ω) and b) functions in (Ω) cannot be tivially extended to R n. The fist pupose of the pesent pape is to povide a complete poof to the following statement: Theoem. Let n N, p (, ) and m N with m < n/p. Let Ω R n be a bounded domain with Ω C m. Then, S m,p, (Ω) = S m,p. This esult is in stiking contast with subcitical embeddings fo which the best embedding constant does depend on the taces, see [7]. Moeove, accoding to Talenti [], if p = the Sobolev inequality behaves in a slightly diffeent manne and Cassani-Ruf-Tasi [5] show that Theoem becomes false fo p =.

3 We pove Theoem with the aid of two basic tools. Fist, we use a compaison pinciple due to Talenti [9] to educe it to the case whee Ω is a ball and the imum in (6) is taken among positive adially symmetic functions, see Section. Second, we show how such functions may be extended to R n with no incease of the -nom and with incease of the L p -nom, see Section 3. In fact, these tools may be used to pove simila esults also fo the Hady-Rellich inequalities which can be consideed as classical Sobolev-type embedding inequalities. The fist vesion of these inequalities appeas in [, ] wheeas a highe ode genealization has been poved by Rellich [6]. Moe ecently, futhe extensions have appeaed in [6, 4], see also [8] fo inequalities with emainde tems. Although it is not explicitly assumed that Ω, hee we have this paticulaly inteesting case in mind. Assume again that p > and m < n/p. Then, thee exists a constant H m,p >, independent of Ω, such that (Ω)\} Ω u p m,p,ω u(x) p dx x mp = H m,p. Fo this esult and the exact value of H m,p we efe to [6, Coollay 4]. Then, by aguing as in the poof of Theoem one obtains Theoem. Let n N, p (, ) and m N with m < n/p. Let Ω R n be a bounded domain with Ω C m. Then, (Ω)\} Ω u p m,p,ω u(x) p dx x mp = H m,p. Thee ae only two small diffeences between poving Theoems and. Fist, contay to (), the best Hady-Rellich constant H m,p is not attained on R n. Howeve, this gives no futhe complication in the poof. Second, in ode to educe to the adial situation, beside Talenti s pinciple one should also ecall that symmetization inceases L p -noms with the singula weight x mp, see [3, Theoem.]. This pape is oganized as follows. In Section we ecall a compaison pinciple due to Talenti [9] and we explain how it may be used in the poof of Theoem. In Section 3 we explain and comment the main ideas of the poof of Theoem ; to this end, we give a simple poof in the case m =. In Section 4 we give the complete poof of Theoem fo any m. The iteated Talenti pinciple Hee and in the sequel, we denote by B the unit ball, by e n = B its measue and by f L p (B) the spheical eaangement of f L p (Ω) accoding to [9, p. 7] when Ω = B and p >. We emak that in paticula f = f. A cucial tool fo the poof of Theoem is the following compaison pinciple due to Talenti [9, Theoem ]. Poposition 3. Let Ω R n (n ) be a C m -smooth bounded domain such that Ω = B = e n. Let q n/(n + ) and let m = k be an even numbe. Let f L q (Ω) and let u W m,q (Ω) be the unique stong solution to ( ) k u = f in Ω, j (7) u = on Ω, j =,..., k. 3

4 Let f L q (B) and u W,q and let v W m,q Then, v u a.e. in B. (B) denote espectively the spheical eaangements of f and u, (B) be the unique stong solution to ( ) k v = f in B, j v = on B, j =,..., k. Remak 4. In this poposition we assumed that Ω C m. In ode to use the esult of Talenti it is sufficient that Ω C,. Indeed, this condition guaantees that the solution of the Diichlet Laplacian with ight hand side in L p (Ω) exists uniquely in W,p (Ω). Fo such a bounday howeve, the solution of (7) in geneal does not lie in (Ω) but in anothe space, which can be defined fo even m as (Ω) := v W,p (Ω); j v W,p (Ω) fo j < m }. (9) (8) If one dops the smoothness assumption even futhe, the esult does not hold tue any longe with the same geneality. Fo q = m = thee is still a unique weak solution of u = f in the sense of a weak system solution, namely u, u W, (Ω) satisfying weakly u = f and u = u but in geneal u W, (Ω). It shows that on nonsmooth domains Ω the map u u,,ω is not a nom in W, (Ω). Moeove, thee might be a weak equation solution ũ u with ũ W, (Ω) and nevetheless ũ = pointwise almost eveywhee on Ω, see [5]. Poof. When k = Poposition 3 is pecisely [9, Theoem ]. Fo k we poceed by finite induction. We may ewite (7) and (8) as the following systems: u = f in Ω, u i = u i in Ω, i =,..., k; () u = on Ω, u i = on Ω, v = f in B, v i = v i in B, i =,..., k. () v = on B, v i = on B, Note that u k = u and v k = v. By Talenti s pinciple [9, Theoem ] applied fo i =, we know that v u a.e. in B. Assume that the inequality v i u i a.e. in B has been poved fo some i =,..., k. Then, by () and () we e u i+ = u i in Ω v i+ = v i u i in B u i+ = on Ω, v i+ = on B. By combining the maximum pinciple fo in B with a futhe application of Talenti s pinciple, we obtain v i+ u i+ a.e. in B. This finite induction shows that v k u k and poves the statement. Let us now explain how we intend to apply Poposition 3. Assume that m is even, m = k fo some k N, and note that S m,p, (Ω) is invaiant unde scaling so that we may assume that Ω has the same measue as the unit ball, Ω = B = e n. In ode to apply the iteated Talenti pinciple we futhe fix q := n n +. If p q we know that (Ω) W m,q (Ω), conside v (Ω) W m,q (Ω) but if p < q this inclusion fails. Fo any u (B) W m,q (B) given by ( ) k v = (( ) k u) in B j v = on B, j =,..., k. 4 ()

5 Then, ( ) j v is positive, adially symmetic and adially deceasing fo all j =,..., k. Moeove, Poposition 3 yields v u so that v L p (B) u L p (B) = u L p (Ω), (3) whee the last equality follows fom standad popeties of symmetic eaangements, see e.g. [3]. Fo the same eason, we also have k ( v = k u) L = k u L p (B) p (B) and we obtain the following statement: L p (Ω) Lemma 5. Let p > and let m = k be even. Then, fo any u (Ω) W m,q (Ω) thee exists a positive adial function v (B) W m,q (B) (i.e. v = v(), = x ) such that ( ) j v() is positive and adially deceasing fo all j k (4) and u p m,p,ω u p L p (Ω) v p m,p,b v p. (5) L p (B) When m is odd, m = k + fo some k, the same esult may be obtained with slightly moe wok. Fo any u (Ω) conside again v (B) defined by (). Then, ( ) j v is positive, adially symmetic and adially deceasing fo all j =,..., k. Moeove, we obtain again (3). Finally, by standad popeties of symmetic eaangements [3], we e ( k ( v) = k u) L ( k u) L p (B) p (B) and we obtain: L p (Ω) Lemma 6. Let p > and let m = k + be odd. Then, fo any u (Ω) W m,q (Ω) thee exists a positive adial function v (B) W m,q (B) (i.e. v = v(), = x ) such that (4) and (5) hold. Let p >, m be any intege and conside minimizing sequences fo S m,p, (Ω) in (Ω). Since Ω is a bounded smooth domain, smooth functions can be shown to be dense in (Ω) by efeing to the existence theoy fo stong solutions of Navie bounday value poblems. This means that we may estict ouselves to minimizing sequences fo S m,p, (Ω) in (Ω) (Ω). Lemmas 5 and 6 now show that W m,q p u m,p,b (B)\} u p L p (B) R m,p S m,p, (B) S m,p, (Ω), whee R m,p m,p (B) denotes the positive convex cone of W (B) containing adially symmetic functions v such that (4) holds. Moeove, as aleady mentioned in the pevious section, we have S m,p, (Ω) S m,p fo any domain Ω. Hence, we have p u m,p,b (B)\} u p L p (B) R m,p S m,p, (B) S m,p, (Ω) S m,p fo all Ω and the poof of Theoem is complete (fo any m) if we show that p u m,p,b (B)\} u p L p (B) R m,p S m,p. (6) 5

6 3 The extension agument and comments In this section we will give a poof of Theoem fo the simplest case. Poof fo m =. As aleady mentioned, Theoem follows if we pove (6). We assume that m = and, fo contadiction, also that thee exists u R,p (B) \ } such that u p m,p,b u p L p (B) < S,p. (7) Note that necessaily u () < since u () = would imply u W, (B) contadicting (4). Conside the adial entie function defined by w() := u() + n u () fo (, ], n n u () fo [, ), and note that w C, (, ) and w D,p (R n ). Moeove, w L p (R n ) > u L p (B) wheeas w L p (R n ) = u L p (B). Hence, by (7), w p m,p,r n w p L p (R n ) < u p m,p,b u p L p (B) < S,p, which contadicts (). In the emaining of this section we comment the above poof in the simplest Hilbetian case p =. It consists of thee steps. Fist, Talenti s compaison pinciple enables us to estict ou attention to the case whee Ω = B and u is adially symmetic. Then, by contadiction, if (7) holds we may build an entie function by inceasing the L -nom and maintaining the L -nom of its Laplacian, since the modification is pefomed by adding a constant in B and a multiple of the fundamental solution outside B. This extension is possible fo any u R, (B) \ } and the incease of its L -nom may be estimated as follows: u w L (R n ) = + Cn u () + L (B) w ( ) < u L (B)... + C n u () + C n u (), > u L (B) + C n u (), L (R n \B)... whee C n ae positive constants which may diffe also within the same line. The two above inequalities show that u () measues the incease of the L -nom and lage values of u () coespond to lage inceases of the nom. Fo any ε > conside the entie functions u ε (x) := ε n 4 (ε + x ) n 4 which all satisfy u ε () =. It is known (see e.g. []) that they achieve the best constant in (), that is, We lowe u ε by setting U ε (x) := u ε (x) S, = u ε,,r n u ε L (R n ) ε n 4 (ε + ) n 4 = 6 fo all ε >. ε n 4 (ε + x ) n 4 ε n 4 (ε + ) n 4

7 so that U ε W, (B) and one can show that (see [4]) lim ε U ε L (B) U ε L (B) = S,. In this espect, U ε() = (n 4)εn 4 (+ε ) (n )/ is a measue of concentation. If it is small then ε is small, which coesponds to high concentation (Sobolev atio close to S, ). In this case, we have seen above that the L -nom of the extension of U ε is small. Summaizing, not only u () weights the incease of the L -nom of the extension of u but also its degee of concentation. A simila but slightly moe delicate intepetation can be given fo all m and p. As we shall see in next section, in this case thee ae moe extension paametes. Since the nom of the extension is elated to the degee of concentation, it seems that all these paametes could possibly educe to only one o at least just a few. This gives ise to a futhe question. Does the optimal Sobolev constant depend only on the fist tace(s)? In othe wods, is S m,p also the best constant fo the embedding B (Ω) = (Ω); B (u) = } L p (Ω)? Hee B (u) is a set of at least [ ] m+ bounday conditions which guaantee that m,p,ω is a nom on B (Ω). Maz ya [3, Section.6.6] poves that fo (Ω) W l,p (Ω) with l m embeddings only need the boundedness of Ω. If B (Ω) (Ω) W l,p (Ω) fo some l m, that is, if not all fist [ ] m+ -taces ae contained in B, then the egulaity of the domain Ω plays a ole. 4 Poof of Theoem The full poof of Theoem needs to distinguish between even and odd m. Even m, m = k fo some k. Fo g : [, ] R let us define ρ ( ) s n (G g) () := g(s)dsdρ. ρ Hence, G is the solution opeato fo the adially symmetic Poisson poblem in the unit ball of R n, that is, it satisfies (G g) ( x ) = g ( x ) fo x <, (G g) ( x ) = fo x =. Let us also define fo g : [, ) R with appopiate integability conditions ρ ( ) s n (Gg) () := g(s)dsdρ. ρ If g also goes to fast enough fo (e.g. like γ with γ > ), then an integation by pats gives (Gg) () = n n s n g(s)ds + sg(s)ds, (8) n and (Gg) ( x ) = g ( x ) fo x R n. Note that g = Gg G g in B. (9) We now descibe the inductive pocedue which we will use in ode to suitably extend functions in R m,p (B). 7

8 Lemma 7. Let n, γ >, γ n and let l. If f W l,p loc (Rn ) is adially symmetic, positive and such that f (x) c f x γ fo x >, then thee is a unique adially symmetic solution u W l+,p loc (R n ) of u = f in R n, Moeove, u = Gf implies that u is positive and u (x) c x n + Equality holds if f (x) = c f x γ fo x >. lim u (x) =. x c f (γ ) (n γ) x γ fo x >. Poof. In view of the bounday condition at inity, uniqueness follows fom Weyl s lemma and Liouville s theoem. Suppose fist that f is continuous. We have Since u is bounded in we find n ( n u () ) = f (). n u () = and since u goes to at it follows that ρ u() = ρ n s n f(s)dsdρ = ρ [ρ n s n f(s)ds n = n n s n f(s)ds ] s n f(s)ds + n + n sf(s)ds If f is not identically, then u >. Fo > it follows fom () that u () = ( n n s n f(s)ds + c f ( s n f(s)ds n c ) f n + n γ ) s n γ ds + sf(s)ds. () c f (n )(γ ) γ c f (γ ) (n γ) γ. Equality holds if f (x) = c f x γ fo x >. The fomula in () also holds fo f W l,p loc (Rn ). The claim that u W l+,p loc (R n ) is diect. R m,p The second tool is a vaiation of an extension esult which enables us to modify functions in (B) to functions on the whole space with no incease of the Diichlet nom. Lemma 8. Let m = k and let u R m,p (B) \ }. Let w() = ( G k f ) () fo ( ) k u() fo, f() = fo >, then w D m,p (R n ) and 8

9 . w m,p,r n = u m,p,b. w L p (R n ) > u L p (B). Poof. Fom Lemma 7 we find that w () = c n + c 4 n + + c m m n fo > which implies with w loc (Rn ) that w D m,p (R n ). Hee, it is cucial that p > is assumed. Since ( ) f() = ( ) k G k f () = ( ) k w () it even follows that w m,p,r n = k w = f L p (R n L ) p (R n ) = f L p (B) = k u = u L p m,p,b. (Ω) Moeove, by (8) it follows that Gf() > = ( ) ( ) k u () and hence by the maximum pinciple and by (9) ( Gf G f = ( ) k u in B. ) () Since G f() > = ( ) k u () and since () holds, a futhe iteation of the maximum pinciple and (9) implies Repeating this agument we find Hence w L p (R n ) > w L p (B) u L p (B). G f G f = ( ) k u in B. w = G k f G k f = u in B. As aleady mentioned, the poof of Theoem follows if we show that (6) holds. By contadiction, assume that thee exists u R m,p (B) \ } such that u p m,p,b u p L p (B) < S m,p. Let w D m,p (R n ) \ } be the function constucted in Lemma 8. Then, Lemma 8 shows that w p m,p,r n w p L p (R n ) < u p m,p,b u p L p (B) < S m,p which contadicts (). This contadiction completes the poof of Theoem fo even m = k. Odd m, m = k + fo some k. In this case, we take advantage of what has just been poved fo the even exponent k. Since W k+,p (B) W k,p (B), by Lemma 8 we know that any u R k+,p (B) \ } allows to define an entie function w such that In paticula, this implies that also w > u in B, k (w u) = in B, k w = in R n \ B. ( k (w u)) = in B, ( k w) = in R n \ B. The constuction fo the k-case also enables us to conclude that w C k (R n ), a egulaity which is not enough to obtain w D k+,p (R n ), hee we need one moe degee of egulaity. This is obtained by ecalling the exta bounday condition that appeas by going fom W k,p (B) to W k+,p (B), namely k u = on B, and that k w = in R n \ B. 9

10 Remak 9. The inductive pocedue descibed in Lemmas 7 and 8 can be esolved even explicitly. Fo the compaison function w intoduced above one finds afte some tedious calculations that k u() + a (k) l l fo, l= w() = k b (k) l l+ n fo >, l= whee and a (k) l = b (k) l = l l! l s= l l! l s= ( ) l+ (n + s) k l j= (n s) k l j= j! j s)(( )j+l j s= (n u) () ( ) j+ j! j s)(( )j+l j s= (n + u) (). Acknowledgement. We thank Daniele Cassani fo aising the poblem and fo stimulating discussions. We also thank Andea Cianchi fo an inspiing discussion on esults of Maz ya [3] which led to the comments at the end of Section 3. Refeences [] R.A. Adams, Sobolev Spaces, Pue and Applied Mathematics 65, Academic Pess, New Yok-London, 975 [] V. Adolfsson, L -integability of second-ode deivatives fo Poisson s equation in nonsmooth domains, Math. Scand. 7, 99, 46-6 [3] F.J. Almgen, E.H. Lieb, Symmetic deceasing eaangement is sometimes continuous, J. Ame. Math. Soc., 989, [4] E. Bechio, F. Gazzola, T. Weth, Citical gowth bihamonic elliptic poblems unde Steklov-type bounday conditions, Adv. Diff. Eq., 7, [5] D. Cassani, B. Ruf, C. Tasi, Best constants in a bodeline case of second ode Mose type inequalities, to appea in Ann. Inst. H. Poincaé [6] E.B. Davies, A.M. Hinz, Explicit constants fo Rellich inequalities in L p (Ω), Math. Z. 7, 998, 5-53 [7] A. Feeo, F. Gazzola, T. Weth, Positivity, symmety and uniqueness fo minimizes of second-ode Sobolev inequalities, Ann. Mat. Pua Appl. (4) 86, 7, [8] F. Gazzola, H.-Ch. Gunau, E. Mitidiei, Hady inequalities with optimal constants and emainde tems, Tans. Ame. Math. Soc. 356, 4, [9] Y. Ge, Shap Sobolev inequalities in citical dimensions, Michigan Math. J. 5, 3, 7-45 [] G.H. Hady, Notes on some points in the integal calculus, Messenge Math. 48, 99, 7- [] G.H. Hady, J.E. Littlewood, G. Pólya, Inequalities, Cambidge Univesity Pess, 934

11 [] P.L. Lions, The concentation-compactness pinciple in the calculus of vaiations. The limit case. I, Rev. Mat. Ibeoameicana, 985, 45- [3] V.G. Maz ya, Sobolev Spaces, Spinge Seies in Soviet Mathematics, Tanslated fom the Russian by T.O. Shaposhnikova, Spinge-Velag, Belin, 985 [4] E. Mitidiei, A simple appoach to Hady s inequalities, Math. Notes 67,, Russian oiginal: Mat. Zametki 67,, [5] S.A. Nazaov, G. Swees, A hinged plate equation and iteated Diichlet Laplace opeato on domains with concave cones, J. Diffeential Equations 33, 7, 5 8 [6] F. Rellich, Halbbeschänkte Diffeentialopeatoen höhee Odnung, Poceedings of the Intenational Congess of Mathematicians, Amstedam 954, vol. III, Even P. Noodhoff N.V., Goningen, 956, 43-5 [7] E. Rodemich, The Sobolev inequalities with best possible constants, in: Analysis Semina at Califonia Institute of Technology, 966 [8] Ch.A. Swanson, The best Sobolev constant, Appl. Anal. 47, 99, 7-39 [9] G. Talenti, Elliptic equations and eaangements, Ann. Scuola Nom. Sup. Pisa Cl. Sci. (4) 3, 976, [] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pua Appl. (4), 976, [] R.C.A.M. van de Vost, Best constant fo the embedding of the space H H (Ω) into L N/(N 4) (Ω), Diff. Int. Eq. 6, 993, 59-76

A THREE CRITICAL POINTS THEOREM AND ITS APPLICATIONS TO THE ORDINARY DIRICHLET PROBLEM

A THREE CRITICAL POINTS THEOREM AND ITS APPLICATIONS TO THE ORDINARY DIRICHLET PROBLEM DIEGO AVERNA AND GABRIELE BONANNO Abstact. The aim of this pape is twofold. On one hand we establish a thee citical