On a class of Rellich inequalities

On a class of Rellich inequalities G. Barbatis A. Tertikas Dedicated to Professor E.B. Davies on the occasion of his 60th birthday Abstract We rove Rellich and imroved Rellich inequalities that involve the distance function from a hyersurface of codimension k, under a certain geometric assumtion. In case the distance is taken from the boundary, that assumtion is the convexity of the domain. We also discuss the best constant of these inequalities. AMS Subject Classification: 35J0 35P0, 35P99, 6D10, 7A75 Keywords: Rellich inequality, best constants, distance function 1 Introduction The classical Rellich inequality states that for > 1 u dt 1 1 0 0 u dt, 1.1 t for all u Cc 0,. A multi-dimensional version of 1.1 for = is also classical and states that for any R N, N 5, there holds u dx N N u dx, 1. 16 x for all u C c. Davies and Hinz [DH] generalized 1. and showed that for any 1, N/ there holds 1N N u u dx x dx, u C c \ {0. 1.3 Inequality 1.1 has also been generalized to higher dimensions in another direction, where the singularity involves the distance dx = distx,. Owen [O] roved among other results that if is bounded and convex then u dx 9 16 u dx dx, u C c. 1. Deartment of Mathematics, University of Ioannina, 5110 Ioannina, Greece Deartment of Mathematics, University of Crete, 7109 Heraklion, Greece and Institute of Alied and Comutational Mathematics, FORTH, 71110 Heraklion, Greece 1

Recently, an imroved version of 1. has been established in [TZ]. Among several other results they showed that for a bounded domain in R N, N 5, there holds u dx N N u 16 x dx NN u 1 8 x X 1X... Xi dx, as well as u dx N u x dx 1 u x X 1X... X i dx, 1.5 for all u Cc \{0. Here X k are iterated logarithmic functions; see 1.6 for recise the definition. Rellich inequalities have various alications in the study of fourth-order ellitic and arabolic PDE s; see e.g. [DH, O, B]. Imroved Rellich inequalities are useful if critical otentials are additionally resent. As a simlest examle, one obtains information on the existence of solution and asymtotic behavior for the equation u t = V for critical otentials V. Corresonding roblems for imroved Hardy s inequalities have recently attracted considerable attention: see [BV, BM, BFT1] and references therein. Our aim in this aer is to obtain shar imroved versions of inequalities 1.3 and 1., where additional non-negative terms are resent in the resective right-hand sides. At the same time we obtain some new imroved Rellich inequalities which are new even at the level of lain Rellich inequalities; these involve the distance to a surface K of intermediate codimension. Statement of results Before stating our theorems let us first introduce some notation. We denote by a domain in R N, N. For the sake of simlicity all functions considered below are assumed to be real-valued; in relation to this we note however that minor modifications of the roofs or a suitable alication of [D, Lemma 7.5] can yield the validity of Theorems 1-3 below for comlex-valued functions u. We let K be a closed, iecewise smooth surface of codimension k, k = 1,..., N. We do not assume that K is connected but only that it has finitely many connected comonents. In the case k = N we assume that K is a finite union of oints while in the case k = 1 we assume that K =. We then set dx = distx, K, and assume that dx is bounded in. We define recursively X 1 t = 1 log t 1, t 0, 1], X i t = X 1 X i 1 t, i =, 3,..., t 0, 1]. 1.6 These are iterated logarithmic functions that vanish at an increasingly slow rate at t = 0 and satisfy X i 1 = 1. Given an integer m 1 we define η m t = X 1 t... X i t, ζ m t = X1t... Xi t. 1.7

We note that lim t 0 η m t = lim t 0 ζ m t = 0. Now, it has been shown [BFT] that both series in 1.7 converge for any t 0, 1. This allows us to also introduce the functions η and ζ as the infinite series. We fix a arameter s R and we assume that the following inequality holds in the distributional sense: k s, k s d d k 1 0 in \ K. For a detailed discussion of this condition we refer to [BFT1]. Here we simly note that it is satisfied in the following two imortant cases: i it is satisfied as an equality if k = N and K consists of single oint and ii it is also satisfied if K = so k = 1, s 1 < 0 and is convex. Our first theorem involves the functions η m = η m dx/d and ζ m = ζ m dx/d, x, for a large enough arameter D > 0. In any case D will be large enough so that the quantity 1 αη m βη m γζ m is ositive in. We also set H = k s. 1.8 Theorem 1 weighted imroved Hardy inequality Let > 1 and m N {. Let be a domain in R N and K a iecewise smooth surface of codimension k, k = 1,..., N. Suose that k s, that su x dx < and that k s d d k 1 0 in \ K. 1.9 Also, let α, β, γ R be fixed. Then there exists a ositive constant D 0 su x dx such that for any D D 0 and all u Cc \ K there holds d s 1 αη m βηm γζ m u dx H d s u dx H α d s η m u dx H β H Hα d s η m u dx 1 H H Hα H γ d s ζ m u dx, where η m = η m dx/d = m X 1 dx/d... X i dx/d and ζ m = ζ m dx/d = m X 1 dx/d... X i dx/d. We note that the secial case s = α = β = γ = 0 has been roved in [BFT]. To state our next theorem we define the constant Q = 1kk. 1.10 Theorem imroved Rellich inequality I Let > 1. Let be a domain in R N and K a iecewise smooth surface of codimension k, k = 1,..., N. Suose that su x dx <. Suose also that k > and that d d k 1 0, in \ K 3

in the distributional sense. Then there exists a ositive constant D 0 su x dx such that for any D D 0 and all u Cc \ K there holds u dx Q u dx 1.11 d 1 { 3 Q k 1 k u d X 1X... Xi dx, where X j = X j dx/d. It is remarkable that the geometric assumtion of this Theorem only involves d, as in the case of Theorem 1, and not higher-order derivatives of d as one might exect. The above theorem does not cover the imortant case k = 1 which corresonds to dx = distx,. This is done in the following theorem for the case =. Theorem 3 imroved Rellich inequality II Let be convex and such that dx:= distx, is bounded in. Then there exists a ositive constant D 0 su x dx such that for any D D 0 and all u Cc there holds i ii u dx 1 u dx 9 16 where X j = X j dx/d. u d dx 1 u d dx 5 8 u d X 1X... X i dx, 1.1 u d X 1X... X i dx, 1.13 In our last theorem we rove the otimality of the constants aearing in Theorems and 3 above. In a similar manner one can rove the otimality of the constants in Theorem 1; we omit the roof since it follows very closely the roof of [BFT, Proosition 3.1]. Anyway, we note that in some articular cases the otimality of Theorem 1 follows indirectly from the otimality of Theorems and 3, which we do rove. In relation to Theorem see also the remark at the end of the aer. We define and for m N, J m [u] = J 0 [u] = u dx Q 1 { 3 Q Our next theorem reads: u dx Q u dx d u dx d m k 1 k u d X 1X... X i dx. Theorem Let > 1. Let be a domain in R N. i If k N 1 then we take K to be a iecewise smooth surface of codimension k and assume K ; ii if

k = N then we take K = {0 ; iii if k = 1 then we assume K =. For any D su dx we have i inf u dx Cc \K u dx Q ; d J m 1 [u] ii inf Cc \K u X d 1 X... 1 {k X 3 Q 1 k, m 1. m where X j = X j dx/d. It follows in articular that all constants in Theorem and Theorem 3 ii are shar. The sharness of Theorem 3 i follows imlicitly from the sharness of 3 ii. Series exansion for weighted Hardy inequality In this section we give the roof of Theorem 1. We note that in the secial case s = α = β = γ = 0 the theorem has already been roved in [BFT]. In the sequel we shall reeatedly use the differentiation rule which is easily roved by induction. d dt Xβ i t = β t X 1X... X i 1 X 1β i, β 0,.1 Proof of Theorem 1. We set for simlicity ψ = 1 αη m βηm γζ m. If T is a vector field in, then, for any u Cc \ K we first integrate by arts and then use Young s inequality to obtain div T u dx T u u 1 dx d s ψ u dx 1 d s 1 T 1 ψ 1 1 u dx, and thus conclude that d s ψ u dx divt 1d s 1 T 1 ψ 1 1 u dx.. We recall that H = k s / and define T x = H H d s1 x dx 1 α 1 H η mdx/d Bηmdx/D. where D su dx and B R is a free arameter to be chosen later. In any case, once B is chosen, D will be large enough so that the quantity 1 α 1 H η md/d Bηmd/D is ositive on. Note that T is singular on K, but since u Cc \ K all revious calculations are legitimate. In view of., to rove the theorem it is enough to show that there exists D 0 su dx such that for D D 0 divt 1d s 1 T H β H Hα 1 ψ 1 1 d s { H H αη m.3 1 ηm H H Hα H γ ζ m 0 5

for all x. To comute divt we shall need to differentiate η m d/d. For this we note that.1 easily imlies η mt = 1 t X1 X1X X1X X1X... X m X1... Xm, from which follows that tη mt = 1 ζ mt 1 η mt.. We also define θ m on 0, 1 by ζ mt = θ mt, t and, for simlicity, we set A = α 1/H so that T = H Hd s1 d1 Aη m Bη m. We think of η m as an indeendent variable, which we may assume to be small by taking D large enough. Simle comutations together with assumtion 1.9 and the fact that d = 1 give divt.5 = d { H s H Aη m H B H H A η m H H A ζ m H HBηm 3 η m ζ m H Hd s d d s 11 Aη m Bηm d { H s H Aη m H B H H A η m H H A ζ m H HBηm 3 η m ζ m H Hd s d d k 11 Aη m Bηm d { H s H Aη m H B H H A η m H H A ζ m H HBηm 3 η m ζ m.6 Moreover, since d = 1, Taylor s exansion gives T 1 = H d s1 1 { = H d s1 1 1 Aη m Bη m 1 1 A 1 η m AB A3 1 6 1 3 B 1 ηm 3 Oηm A 1 η m.7 and also ψ 1 1 = 1 α 1 η m αβ β 1 α 1 ηm 3 1α3 1 6 1 3 η m γ 1 ζ m αγ 1 η mζ m Oη m..8 Using.6,.7 and.8 we see that the LHS of.3 is greater than or equal to d s times a linear combination of owers of η m, ζ m and θ m lus Oη m. Recalling 6

that A = α 1/H, we easily see that the constant term and the coefficients of η m, η m and ζ m vanish, indeendently of the choice of the arameter B. The remaining two coefficients, that is the coefficients of η 3 m and η m ζ m are, resectively, 1α H β H 1 6 H 3, B γ H. Since ζ m η m mζ m, we conclude that taking B to be large and ositive if H > 0 or large and negative if H < 0, inequality.3 is satisfied rovided η m is small enough, which amounts to D being large enough. This comletes the roof of the theorem. // In the roof of Theorem we are going to use the last theorem in the following secial case which corresonds to taking = and s = q : Secial case. Assume that k q and that k qd d k 1 0 on \ K. Then for D large enough there holds for all u C c d q 1 αη m βηm γζ m u k q dx k q α k q d q η m u β dx 1 k qα \ K. k qα d q u dx d q η mu dx k q γ d q ζ m u dx.9 3 The imroved Rellich inequality In this section we are going to rove Theorems and 3 as well as the corresonding otimality theorem. We begin with the following lemma where, we note, φ is to be understood in the distributional sense. Lemma 5 For any locally bounded function φ with u L loc \ K we have u dx 1 φ u u dx φ 1 φ 1 u dx, 3.10 for all u C c \ K. Proof. Given u Cc \ K we have φ u dx = φ u u udx = φ u u udx 1 φ u u dx 1 φ 1 u dx 1 u dx 1 φ u u dx. which is 3.10. 7

Proof of Theorem. Let m N be fixed and let η m and ζ m be as in 1.7. We aly 3.10 with φx = λdx 1 αη m βη m, λ > 0, where, as always, η m = η m dx/d and D is yet to be determined. We thus obtain where u dx T 1 T T 3 3.11 T 1 = 1 φ u u dx, T = φ u dx, T 3 = 1 φ 1 u dx. To estimate T 1 we set v = u / and aly.9 for q =, T 1 = 1λ 1λ 1 k α d 1 αη m βη m v dx { k d k α ζ m k α η m k β ηm u dx 3.1 To estimate T we first note that φ = λd { 1 11 αη m βηm α η m ζm βηm 3 η m ζ m d and hence comute { φ = λd 1 d d 11 αη m βηm α η m ζ m βηm 3 η m ζ m λd { 1αηm ζ m βηm 3 η m ζ m α η3 m η m ζ m θ m Oηm Using the geometric assumtion d d k 1 0 and collecting similar terms we conclude that T λ d { 1k 1k αη m 1k β k α k αζ m k 1β α ηm 3.13 ηm 3 η m ζ m α θ m Oηm dx. 8

From Taylor s theorem we have 1 αη m βη m 1 = 1 α from which follows that 1 η m αβ { T 3 = 1 λ 1 d 1 α β 1 α3 1 6 1 3 1 η m αβ α3 1 6 1 3 β 1 η m α 1 ηm 3 Oηm α 1 ηm 3 Oηm η m dx. 3.1 Using the above estimates on T 1, T and T 3 and going back to 3.11 we obtain the inequality u dx d V u dx 3.15 where the otential V has the form V x = r 0 r 1 η m r η m r ζ m r 3 η 3 m r 3η m ζ m r 3θ m. We comute the coefficients r i, r i by adding the corresonding coefficients from 3.1, 3.13 and 3.1. We ignore for now the coefficients of the third-order terms. For the others we find kk r 0 = 1 λ λ 1 1kk r 1 = αλ α λ 1 k k 1kk β r = αλ βλ 1 1 α 1 λ 1 1 r k k = α λ We now make a secific choice for α and λ. We recall that Q = 1kk /, and choose λ = Q 1 1k k, α =. Q We then have r 0 = Q, r 1 = r = 0, irresective of the value of β. We also have r = 1 Q Q Substituting these values in 3.15 we thus obtain u dx Q d u dx 1 Q Q k k. d r 3 ηm 3 r 3η m ζ m r 3θ m Oηm u dx 9 k k d ζ m u dx

We still have not imosed any restriction on β. We now observe that r 3 and r 3 are indeendent of β, while r 3 = c 1 β c with c 1 = Q 1 k/ k 3 < 0. Hence, since the functions ηm, 3 η m ζ m and θ m are comarable in size to each other, the integral is made ositive by choosing β to be large and negative and η m small enough, which amounts to D being large enough. Hence we have roved that for D D 0 there holds u dx Q 1 u dx d Q 1 1 Q R m u d X 1X... X i dx, where X j = X j dx/d. This concludes the roof of the theorem. // Remark. Let us mention here that in the roofs of Theorems 1 and we did not use at any oint the assumtion that k is the codimension of the set K. Indeed, a careful look at the two roofs shows that K can be any closed set such that distx, K is bounded in and for which the condition d d k 1 0 or 0 is satisfied; the roof does not even require k to be an integer. Of course, the natural realizations of these conditions are that K is smooth and k = codimk. However, the argument also alies in the case where K is a union of sets of different codimensions; see [BFT1]. We next rove Theorem 3. Proof of Theorem 3. We note that the convexity of imlies that d 0 on in the distributional sense [EG, Theorem 6.3.]. Now, let u Cc be given. Alying Theorem 1 with k = 1, =, s = 0 and α = β = γ = 0 to the artial derivatives u xi we have u dx = = 1 n { n 1 u xi dx u x i d dx 1 u 1 ζ m dx d u x i d ζ mdx, 3.16 for D large enough, where ζ m = ζ m dx/d. Alying Theorem 1 once more this time with k = 1, =, s =, α = β = 0 and γ = 1 we obtain u d 1 ζ m dx 9 Combining 3.16 and 3.17 we obtain u dx 9 16 u d dx 5 u d ζ mdx. 3.17 u d dx 5 u 8 d ζ mdx, which is the stated inequality. // We next give the roof of Theorem. We recall that is a domain in R N and that K is a iecewise smooth surface of codimension k such that K, unless k = 1 in which case K =. All the calculations below are local, in a small ball of radius 10

δ, and indeed, it would be enough to assume that K has a smooth art. We also note that for k = N distance from a oint the subsequent calculations are substantially simlified, whereas for k = 1 distance from the boundary one should relace B δ by B δ. This last change entails some minor modifications, the arguments otherwise being the same. Proof. We shall only give the roof of ii since the roof of i is much simler. For the roof we shall use some of the ideas and tools develoed in [BFT]. All our analysis will be local, say, in a fixed ball of Bx 0, δ where x 0 K and δ is small, but fixed throughout th roof. We therefore fix a smooth, non-negative function φ such that φx = 1 on { x x 0 < δ/ and φx = 1 on { x x 0 > δ. For given ɛ 0, ɛ 1..., ɛ m > 0 we then define the function u = φd kɛ 0 =: φv. 1ɛ 1 X1 X 1ɛ... X 1ɛm m A standard argument using cut-off functions shows that u belongs in W, 0 \ K and therefore is a legitimate test-function for the infimum above. We intend to see how J m 1 [u] behaves as the ɛ i s tend to zero. We shall not be interested in terms that remain bounded for small values of the ɛ i s. To distinguish such terms we shall need the following fact, cf. [BFT, 3.8]: we have φ d kβ 0 X 1β 1 1 d/d... Xm 1βm d/ddx < β 0 > 0 or β 0 = 0 and β 1 > 0 or β 0 = β 1 = 0 and β > 0 or β 0 = β 1 =... = β m 1 = 0 and β m > 0. Now, we have u = φ v φ v v φ and hence, using the inequality we have u dx φ v dx c φ v dx 3.18 a b a c a 1 b b, 3.19 {φ v 1 φ v v φ φ v v φ v 1 v v v v. The first integral involves d to the ower k ɛ 0 / see below and is therefore imortant. On the other hand, all terms in the second integral involve d to the ower that is larger than k and in fact bounded away from k, indeendently of ɛ 0 ; hence u dx = φ v dx O1, 3.0 where the O1 is uniform in all the ɛ i s. 11

We next define the function gt = ɛ 0 1 ɛ 1 X 1 1 ɛ X 1 X 1 ɛ m X 1 X... X m, t > 0, where X i = X i t/d. We shall always think of gdx as a small quantity. Recalling.1 one easily sees that dη m dt = 1 t =: Also, for any β there holds d dt βɛ0 t 1ɛ 1 X1 X 1ɛ 1 i j m 1 ɛ j X 1... X i X i1... X j ht. 3.1 t 1ɛm... Xm = t β ɛ 0 1ɛ 1 X1 X 1ɛ... X 1ɛm m [ β gt ]. 3. Alying 3. first for β = k, then for β = k and using 3.1 we obtain { v = d kɛ 1ɛ 0 1 1ɛ 1ɛm k X1 X... Xm d d g k g h, where, here and below, we use g, h and X i to denote gdx, hdx and X i dx/d. Now, by [AS, Theorem 3.] we have d d = k 1 Od as dx 0. Hence, the exression in the braces equals Q R g 1 g 1 h Od as dx 0, where R = k k /. The Od gives a bounded contribution by an alication of 3.19 as was done earlier. Hence 3.0 gives u dx = φ d kɛ 0 X 1ɛ 1 1... Xm 1ɛm Q R g 1 g 1 dx h O1. 3.3 To estimate this we take the Taylor s exansion of Q t about t = 0. We obtain { u dx = φ d kɛ 0 X 1ɛ 1 1... Xm 1ɛm Q Q QRg 1 Q Q 1 Q R g Q Qζ m Og 3 Ogh Oh dx O1. 3. Using 3.19 once again, it is not difficult to see that the terms Og 3, Ogh and Oh give a contribution that is bounded uniformly in the ɛ i s and can therefore be droed. At this oint, and in order to simlify the notation, we introduce some auxiliary quantities. For 0 i j m we define A 0 = φ d kɛ 0 X 1ɛ 1 1... Xm 1ɛm dx 1

A i = Γ 0j = Γ ij = φ d kɛ 0 X 1ɛ 1 1... X 1ɛ i i X 1ɛ i1 i1... Xm 1ɛm φ d kɛ 0 X ɛ 1 1... Xɛ i i X 1ɛ i1 i1... Xm 1ɛm dx φ d kɛ 0 X 1ɛ 1 1... X 1ɛ i i X ɛ i1 i1... Xɛ j j X 1ɛ j1 j1... Xm 1ɛm dx, with the convention that Γ ii = A i. It is then easily seen that φ d kɛ 0 X 1ɛ 1 1... Xm 1ɛm g dx = ɛ 0 A 0 1 ɛ i Γ 0i φ d kɛ 0 X 1ɛ 1 1... Xm 1ɛm g dx = ɛ 0A 0 1 ɛ i m A i ɛ 0 1 ɛ i Γ 0i dx i<j1 ɛ i 1 ɛ j Γ ij φ d kɛ 0 X 1ɛ 1 1... Xm 1ɛm h dx = ɛ i A i 1 1 ɛ j Γ ij. i<j Here and below i<j means 1 i<j m. Let us also define the constant Going back to 3. and noting that we obtain J m 1 [u] P = 1 Q Q 1 Q R. m 1 = Q QR ɛ 0 A 0 P ɛ 0A 0 Now, by [BFT, 18], u d dx = A 0 u d X 1... X i dx = m 1 A i. 1 ɛ i Γ 0i 1 ɛ i m A i ɛ 0 ɛ i Γ 0i 1 1 ɛ i 1 ɛ j Γ ij i<j Q m Q ɛ i A i 1 1 ɛ j Γ ij G i<j m 1 A i O1. 3.5 ɛ m 0 ɛ 0 1 ɛ i Γ 0i = i ɛ ɛ i A i ɛ i 11 ɛ j Γ ij O1. i<j For the sake of simlicity, we set G = 1 {k 3 Q 1 k. 13

One then easily sees that P Q Q = G. Hence, collecting similar terms, J m 1 [u] = Q m QRɛ 0 A 0 1 ɛ j Γ 0j G ɛ i A i 1 ɛ j Γ ij j=1 i<j GA m O1, where the O1 is uniform for small ɛ i s. U to this oint the arameters ɛ 0, ɛ 1,..., ɛ m where ositive. We intend to take limits as they tend to zero in that order. Due to 3.18, as ɛ 0 0 all terms have finite limits excet those involving A 0 and Γ 0j which, when viewed searately, diverge. However a simle argument involving an integration by arts see [BFT, 3.9] shows that ɛ 0 A 0 1 ɛ j Γ 0j = O1 3.6 j=1 uniformly in ɛ 0,..., ɛ m. Hence, letting ɛ 0 0 we conclude that m J m 1 [u] = G ɛ i A i 1 ɛ j Γ ij GA m O1 ɛ 0 = 0 i<j Now as was the case with 3.6 an integration by arts shows that see [BFT, 3.9] if ɛ 0 = ɛ 1 =... = ɛ i 1 = 0, then ɛ i A i 1 ɛ j Γ ij = O1. 3.7 j=1 We now let ɛ 1 0. Again, all terms have finite limits excet those involving A 1 and Γ 1j which diverge. Using 3.7 we see that when combined these terms stay bounded in the limit α 1 0. We roceed in this way and after letting ɛ m 1 0 we are left with J m 1 [u] = G1 ɛ m A m O1 ɛ 0 =... = ɛ m 1 = 0. Let us denote by G the infimum in the left-hand side of art ii of Theorem. We have thus roved that G G1 ɛ ma m O1 A m ɛ 0 =... = ɛ m 1 = 0. Letting now ɛ m 0 we have A m by 3.18, and thus conclude that G G, as required. // Remark. Slightly modifying the above argument one can also rove the otimality of the ower Xm of the imroved Rellich inequalities 1.11 and 1.13. Namely, for any ɛ > 0 there holds inf u Cc \K J m 1 [u] u X d 1 X... X ɛ m dx = 0. Acknowledgment We acknowledge artial suort by the RTN Euroean network Fronts Singularities, HPRN-CT-00-007. 1

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