Weighted Hardy inequality with higher dimensional ... - CAPDE

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Weighted Hardy inequality with higher dimensional ... - CAPDE

and(1.3) maxΣ kWe put∫(1.4) I k =Σ kqp = 1, η = 0 on Σ k .dσ√1 − (q(σ)/p(σ)), 1 ≤ k ≤ N − 1 and I 0 = ∞.It was shown by Brezis and Marcus in [2] that there exists λ ∗ such that if λ > λ ∗then µ λ (Ω, Σ N−1 ) < 1 4 and it is attained while for λ ≤ λ∗ , µ λ (Ω, Σ N−1 ) = 1 4and itis not achieved for every λ < λ ∗ . The critical case λ = λ ∗ was studied by Brezis,Marcus and Shafrir in [3], where they proved that µ λ ∗(Ω, Σ N−1 ) admits a minimizerif and only if I N−1 < ∞. The case where k = 0 (Σ 0 is reduced to a point on theboundary) was treated by the first author in [8] and the same conclusions hold true.Here we obtain the followingTheorem 1.1 Let Ω be a smooth bounded domain of R N , N ≥ 3 and let Σ k ⊂ ∂Ω bea closed submanifold of dimension k ∈ [1, N − 2]. Assume that the weight functionsp, q and η satisfy (1.2) and (1.3). Then, there exists λ ∗ = λ ∗ (p, q, η, Ω, Σ k ) such that(N − k)2µ λ (Ω, Σ k ) = , ∀λ ≤ λ ∗ ,4(N − k)2µ λ (Ω, Σ k ) < , ∀λ > λ ∗ .4The infinimum µ λ (Ω, Σ k ) is attained if λ > λ ∗ and it is not attained when λ < λ ∗ .Concerning the critical case we getTheorem 1.2 Let λ ∗ be given by Theorem 1.1 and consider I k defined in (1.4).Then µ λ ∗(Ω, Σ k ) is achieved if and only if I k < ∞.By choosing p = q ≡ 1 and η = δ 2 , we obtain the following consequence of the abovetheorems.Corollary 1.3 Let Ω be a smooth bounded domain of R N , N ≥ 3 and Σ k ⊂ ∂Ω bea closed submanifold of dimension k ∈ {1, · · · , N − 2}. For λ ∈ R, put∫∫|∇u| 2 dx − λ |u| 2 dxν λ (Ω, Σ k ) = infu∈H0 1(Ω)Ω ∫Ωδ −2 |u| 2 dx,2Ω


orthonormal frame (X i ) i=2,··· ,N−k in a neighborhood of P . We can therefore definethe parameterization of a neighborhood – in M – of P via the mapping( N−k) ∑(˘y, ȳ) ↦→ h P M(˘y, ȳ) := Exp M f P (ȳ)y i X i ,i=2with ˘y = (y 2 , · · · , y N−k ) and Exp M Q is the exponential map at Q in M endowedwith the metric induced by R N . We now have a parameterization of a neighborhood– in R N – of P defined via the Fermi coordinates by the mapy = (y 1 , ˘y, ȳ) ↦→ F P M(y 1 , ˘y, ȳ) = h P M(˘y, ȳ) + y 1 N M (h P M(˘y, ȳ)).Next we denote by g the metric induced by F P M which has component g αβ(y) =〈∂ α FM P (y), ∂ βFM P (y)〉. Then we have the following expansion(2.1)g 11 (y) = 1g 1β (y) = 0, for β = 2, · · · , Ng αβ (y) = δ αβ + O(|ỹ|), for α, β = 2, · · · , N,where ỹ = (y 1 , ˘y) and O(r m ) is a smooth function in the variable y which is uniformlybounded by a constant (depending only M and Σ k ) times r m .In concordance to the above coordinates, we will consider the “half”-geodesicneighborhood contained in U around Σ k of radius ρ(2.2) U ρ (Σ k ) := {x ∈ U : ˜δ(x) < ρ},with ˜δ is the projection distance function given by√˜δ(x) := |dist M (x, Σ k )| 2 + |x − x| 2 ,where x is the orthogonal projection of x on M and dist M (·, Σ k ) is the geodesicdistance to Σ k on M with the induced metric. Observe that(2.3) ˜δ(FPM (y)) = |ỹ|,where ỹ = (y 1 , ˘y). We also define σ(x) to be the orthogonal projection of x on Σ kwithin M. Lettingˆδ(x) := dist M (x, Σ k ),5


one hasNext we observe thatx = Exp M σ(x) (ˆδ ∇ˆδ) or equivalently σ(x) = Exp M x (−ˆδ ∇ˆδ).(2.4) ˜δ(x) =√ˆδ 2 (¯x) + d 2 (x).In addition it can be easily checked via the implicit function theorem that thereexists a positive constant β 0 = β 0 (Σ k , Ω) such that ˜δ ∈ C ∞ (U β0 (Σ k )).It is clear that for ρ sufficiently small, there exists a finite number of Lipschitzopen sets (T i ) 1≤i≤N0such thatT i ∩ T j = ∅ for i ≠ j and U ρ (Σ k ) =We may assume that each T i is chosen, using the above coordinates, so that⋃N 0i=1T i = F p iM (BN−k + (0, ρ) × D i ) with p i ∈ Σ k ,where the D i ’s are Lipschitz disjoint open sets of R k such thatIn the above setting we have⋃N 0i=1f p i (Di ) = Σ k .Lemma 2.1 As ˜δ → 0, the following expansions hold1. δ 2 = ˜δ 2 (1 + O(˜δ)),2. ∇˜δ · ∇d = d˜δ ,T i .3. |∇˜δ| = 1 + O(˜δ),4. ∆˜δ = N−k−1˜δ+ O(1),where O(r m ) is a function for which there exists a constant C = C(M, Σ k ) suchthat|O(r m )| ≤ Cr m .6


Proof.1. Let P ∈ Σ k . With an abuse of notation, we write x(y) = FM P (y) and we setϑ(y) := 1 2 δ2 (x(y)).The function ϑ is smooth in a small neighborhood of the origin in R NTaylor expansion yieldsandϑ(y) = ϑ(0, ȳ)ỹ + ∇ϑ(0, ȳ)[ỹ] + 1 2 ∇2 ϑ(0, ȳ)[ỹ, ỹ] + O(‖ỹ‖ 3 )(2.5)= 1 2 ∇2 ϑ(0, ȳ)[ỹ, ỹ] + O(‖ỹ‖ 3 ).Here we have used the fact that x(0, ȳ) ∈ Σ k so that δ(x(0, ȳ)) = 0. We writewithΛ il :===Now using the fact thatwe obtain∇ 2 ϑ(0, ȳ)[ỹ, ỹ] =N−k∑i,l=1∂ 2 ϑ∂y i ∂y l / ỹ=0( )∂ ∂∂y l ∂x j (1 2 δ2 (x) ∂xj∂y i ) /ỹ=0Λ il y i y l ,∂ 2∂x i ∂x s (1 2 δ2 )(x) ∂xj ∂x s∂y i ∂y l / ỹ=0 +∂∂x j (δ2 )(x) ∂2 x s∂y i ∂y l / ỹ=0.∂x s∂y l / ỹ=0 = g ls = δ lsand∂∂x j (δ2 )(x)/ỹ=0 = 0,Λ il y i y l = y i y s ∂ 2∂x i ∂x s (1 2 δ2 )(x)/ỹ=0= |ỹ| 2 ,(where we have used the fact that the matrix ∂ 2( 1 ∂x i ∂x s 2 δ2 )(x)/ỹ=0)1≤i,s≤N isthe matrix of the orthogonal projection onto the normal space of T f P (ȳ)Σ k .Hence using (2.5), we getδ 2 (x(y)) = |ỹ| 2 + O(|ỹ| 3 ).This together with (2.3) prove the first expansion.7


2. Thanks to (2.3) and (2.1), we infer thatas desired∇˜δ · ∇d(x(y)) = ∂˜δ(x(y))∂y 1= y1|ỹ| = d(x(y))˜δ(x(y))3. We observe that∂˜δ∂x τ∂˜δ∂x τ (x(y)) = gτα (y)g τβ (y) ∂˜δ(x(y))∂y α∂˜δ(x(y))∂y β ,where (g αβ ) α,β=1,...,N is the inverse of the matrix (g αβ ) α,β=1,...,N . Thereforeusing (2.3) and (2.1), we get the result.4. Finally using the expansion of the Laplace-Beltrami operator ∆ g in [[13],Lemma 3.3] applied to (2.3), we get the last estimate.In the following of – only – this section, q : U → R be such that(2.6) q ∈ C 2 (U), and q ≤ 1 on Σ k .Let M, a ∈ R, we consider the function(2.7) W a,M,q (x) = X a (˜δ(x)) e Md(x) d(x) ˜δ(x) α(x) ,whereX a (t) = (− log(t)) a 0 < t < 1,α(x) = k − N + N − k √1 − q(σ(¯x)) +2 2˜δ(x).In the above setting, we have the following useful result.Lemma 2.2 As δ → 0, we have(N − k)2∆W a,M,q = − q δ −2 W a,M,q − 2 a √˜α X −1 (δ) δ −2 W a,M,q4+ a(a − 1) X −2 (δ) δ −2 W a,M,q + h + 2M W a,M,q + O(| log(δ)| δ − 3 2 ) Wa,M,q ,where ˜α(x) = (N−k)24satisfies()d1 − q(σ(x)) + ˜δ(x) and h = ∆d. Here the lower order term|O(r)| ≤ C|r|,where C is a positive constant only depending on a, M, Σ k , U and ‖q‖ C 2 (U).8


Proof. We put s = (N−k)24. Let w = ˜δ(x) α(x) then the following formula can beeasily verified)(2.8) ∆w = w(∆ log(w) + |∇ log(w)| 2 .Sincelog(w) = α log(˜δ),we get(2.9) ∆ log(w) = ∆α log(˜δ) + 2∇α · ∇(log(˜δ)) + α∆ log(˜δ).We have(2.10) ∆α = ∆ √˜α = √˜α( 12 ∆ log(˜α) + 1 4 |∇ log(˜α)|2 ),∇ log(˜α) = ∇˜α −s∇(q ◦ σ) + s∇˜δ=˜α˜αand using the formula (2.8), we obtain∆ log(˜α) = ∆˜α˜α− |∇˜α|2˜α 2−s∆(q ◦ σ) + s∆˜δ=˜αPutting the above in (2.10), we deduce that(2.11)∆α = 12 √˜α(− s∆(q ◦ σ) + s∆˜δ − 1 2− s2 |∇(q ◦ σ)| 2 + s 2 |∇˜δ| 2˜α 22 ∇(q ◦ σ) · ∇˜δ+ 2s˜α 2 .s 2 |∇(q ◦ σ)| 2 + s 2 |∇˜δ| 2 − 2s 2 )∇(q ◦ σ) · ∇˜δ.˜αUsing Lemma 2.1 and the fact that q is in C 2 (U), together with (2.11) we get(2.12) ∆α = O(˜δ − 3 2 ).On the other handso that∇α = ∇ √˜α = 1 ∇˜α= −2√˜α s2 √˜α ∇(q ◦ σ) + s ∇˜δ2√˜α∇α · ∇˜δ = −s2 √˜α ∇(q ◦ σ) · ∇˜δ + s 2|∇˜δ| 2√˜α= O(˜δ − 1 2 )9


and from which we deduce that(2.13) ∇α · ∇ log(˜δ) = 1˜δ ∇α · ∇˜δ = O(˜δ − 3 2 ).By Lemma 2.1 we have thatα∆ log(˜δ) = α N − k − 2˜δ 2(1 + O(˜δ)).Taking back the above estimate together with (2.13) and (2.12) in (2.9), we get(2.14) ∆ log(w) = α N − k − 2˜δ 2 (1 + O(˜δ)) + O(| log(˜δ)|˜δ − 3 2 ).We also haveand thus∇(log(w)) = ∇(α log(˜δ)) = α ∇˜δ˜δ+ log(˜δ)∇α|∇(log(w))| 2 = α2 2α log(˜δ)+ ∇˜δ · ∇α + | log(˜δ)|˜δ 2 ˜δ2 |∇α| 2 = α2˜δ + O(| log(˜δ)|˜δ − 3 2 2 ).Putting this together with (2.14) in (2.8), we conclude that(2.15)Now we define the function∆ww = α N − k − 2 + α2˜δ 2 ˜δ + O(| log(˜δ)| ˜δ − 3 2 2 ).v(x) := d(x) w(x),where we recall that d is the distance function to the boundary of U. It is clear that(2.16) ∆v = w∆d + d∆w + 2∇d · ∇w.Notice that∇w = w ∇ log(w) = w(log(˜δ)∇α + α ∇˜δ˜δ)and so(2.17) ∇d · ∇w = wRecall 2. of Lemma 2.1 that we rewrite(2.18) ∇d · ∇˜δ = d˜δ .(log(˜δ)∇d · ∇α + α˜δ )∇d · ∇˜δ .10


Therefore()(2.19) ∇d · ∇α = ∇d · −s2 √˜α ∇(q ◦ σ) + s ∇˜δ2√˜α= s2 √˜αd˜δ −Notice that if x is in a neighborhood of some point P ∈ Σ k one has∇d · ∇(q ◦ σ)(x) =This with (2.19) and (2.18) in (2.17) give(∇d · ∇w = w(2.20)= v∂∂y 1 q(σ(x)) =∂∂y 1 q(f P (y)) = 0.O(˜δ − 3 2 | log(˜δ)|) d +α˜δ 2 d )(O(˜δ − 3 2 | log(˜δ)|) +α˜δ 2 ).s ∇d · ∇(q ◦ σ).2√˜αFrom (2.15), (2.16) and (2.20) (recalling the expression of α above), we get immediately(2.21)∆v ==(α N − k˜δ 2+ α2˜δ 2 )v + O(| log(˜δ)| ˜δ − 3 2 ) v +hd v)(N − k)2 q(x)(− + O(| log(˜δ)|4 ˜δ ˜δ − 3 2 2 ) v + h d v,where h = ∆d. Here we have used the fact that |q(x) − q(σ(¯x))| ≤ C˜δ(x) for x in aneighborhood of Σ k .Recall thatW a,M,q (x) = X a (˜δ(x)) e Md(x) v(x), with X a (˜δ(x)) := (− log(˜δ(x))) a ,where M and a are two real numbers. We have∆W a,M,q = X a (˜δ) ∆(e Md v) + 2∇X a (˜δ) · ∇(e Md v) + e Md v ∆X a (˜δ)and thus(2.22)∆W a,M,q = X a (˜δ)e Md ∆v + X a (˜δ)∆(e Md ) v + 2X a (˜δ)∇v · ∇(e Md )+ 2∇X a (˜δ) · (v∇(e Md ) + e Md ∇v ) + e Md v ∆X a (˜δ).We shall estimate term by term the above expression.First we have form (2.21)(2.23) X a (˜δ)e Md (N − k)2 q∆v = −4 ˜δ W 2 a,M,q + h d W a,M,q + O(| log(˜δ)| ˜δ − 3 2 ) Wa,M,q .11


It is plain that(2.24) X a (˜δ) ∆(e Md ) v = O(1) W a,M,q .It is clear that(2.25) ∇v = w ∇d + d ∇w = w ∇d + d(log(˜δ) ∇α + α ∇˜δ˜δFrom which and (2.18) we get{ (X a (˜δ) ∇v · ∇(e Md ) = M X a (˜δ) e Md w |∇d| 2 + d log(˜δ) ∇d · ∇α + α˜δ )}∇˜δ · ∇d{= M X a (˜δ) e Md w 1 + O(| log(˜δ)| ˜δ}− 12 ) d + O(˜δ−1 ) d{ }M(2.26)= W a,M,qd + O(| log(˜δ)| ˜δ −1 ) .Observe thatThis with (2.25) and (2.18) implies that(2.27)∇X a (˜δ) ·∇(X a (˜δ)) = −a ∇˜δ˜δ X a−1(˜δ).()v ∇(e Md ) + e Md a(α + 1)∇v = − X −1 W a,M,q + O(| log(˜δ)|˜δ˜δ − 3 2 2 ) Wa,M,q .By Lemma 2.1, we have∆(X a (˜δ)) = ã δ 2 X a−1(˜δ){2 + k − N + O(˜δ)} +Therefore we obtain(2.28) e Md v∆(X a (˜δ)) = ã δ 2 {2 + k − N + O(˜δ)} X −1 W a,M,q +)w.a(a − 1)X a−2 (˜δ).˜δ 2a(a − 1)˜δ 2 X −2 W a,M,q .Collecting (2.23), (2.24), (2.26), (2.27) and (2.28) in the expression (2.22), we getas ˜δ → 0(N − k)2∆W a,M,q = − q4˜δ −2 W a,M,q − 2 a √˜α X −1 (˜δ) ˜δ −2 W a,M,q+ a(a − 1) X −2 (˜δ) ˜δ −2 W a,M,q + h + 2M W a,M,q + O(| log(˜δ)|d˜δ − 3 2 ) Wa,M,q .The conclusion of the lemma now follows from 1. of Lemma 2.1.12


2.1 Construction of a subsolutionFor λ ∈ R and η ∈ Lip(U) with η = 0 on Σ k , we define the operator(2.29) L λ := −∆ −(N − k)24where q is as in (2.6). We have the following lemmaq δ −2 + λ η δ −2 ,Lemma 2.3 There exist two positive constants M 0 , β 0 such that for all β ∈ (0, β 0 )the function V ε := W −1,M0 ,q + W 0,M0 ,q−ε (see (2.7)) satisfies(2.30) L λ V ε ≤ 0 in U β , for all ε ∈ [0, 1).Moreover V ε ∈ H 1 (U β ) for any ε ∈ (0, 1) and in addition(2.31)Proof.choose∫U βV 20δ 2 dx ≥ C ∫√1 − q(σ)dσ.Σ k1Let β 1 be a positive small real number so that d is smooth in U β1 . WeM 0 = maxx∈U β1|h(x)| + 1.Using this and Lemma 2.2, for some β ∈ (0, β 1 ), we have()(2.32) L λ W −1,M0 ,q ≤ −2δ −2 X −2 + C| log(δ)| δ − 3 2 + |λ|ηδ−2W −1,M0 ,q in U β .Using the fact that the function η vanishes on Σ k (this implies in particular that|η| ≤ Cδ in U β ), we haveL λ (W −1,M0 ,q) ≤ −δ −2 X −2 W −1,M0 ,q = −δ −2 X −3 W 0,M0 ,q in U β ,for β sufficiently small. Again by Lemma 2.2, and similar arguments as above, wehave(2.33) L λ W 0,M0 ,q−ε ≤ C| log(δ)| δ − 3 2 W0,M0 ,q−ε ≤ C| log(δ)| δ − 3 2 W0,M0 ,q in U β ,for any ε ∈ [0, 1). Therefore we getL λ (W −1,M0 ,q + W 0,M0 ,q−ε) ≤ 0 in U β ,13


if β is small. This proves (2.30).The proof of the fact that W a,M0 ,q ∈ H 1 (U β ), for any a < − 1 2and W 0,M 0 ,q−ε ∈H 1 (U β ), for ε > 0 can be easily checked using polar coordinates (by assumingwithout any loss of generality that M 0 = 0 and q ≡ 1), we therefore skip it.We now prove the last statement of the theorem. Using Lemma 2.1, we have∫U βV 20δ 2 dx ≥ ∫≥≥W0,M 2 0 ,qU βδ 2∫CC= C≥C∑N 0i=1∑N 0U β (Σ k )i=1∑N 0i=1∫dxd 2 (x)˜δ(x) 2α(x)−2 dxd 2 (x)˜δ(x) 2α(x)−2 dxT i∫(y 1 ) 2 |ỹ| 2α(F p iM (y))−2 |Jac(F p iB N−kM+ (0,β)×D i∫)|(y) dyB N−k+ (0,β)×D i(y 1 ) 2 |ỹ| k−N−2+(N−k)√ 1−q(f p i(ȳ)) |ỹ| −√ |ỹ| dy.Here we used the fact that |Jac(F p iM)|(y) ≥ C. Observe that|ỹ| −√ |ỹ| ≥ C > 0 as |ỹ| → 0.Using polar coordinates, the above integral becomesV∫U 2N 00βδ 2 dx ≥ C ∑≥i=1N 0∑∫Ci=1∫D i∫∫ ri1D i 0S N−k−1+We therefore obtainV0∫U 2βδ 2 dx ≥ ∫C≥∫C( ) y1 2 ∫ βdθ r −1+(N−k)√ 1−q(f pi(ȳ)) dȳ|ỹ|0r −1+(N−k)√ 1−q(f p i(ȳ)) |Jac(f p i)|(ȳ) dȳ.∫ βΣ k 0Σ k1r −1+(N−k)√ 1−q(σ) dr dσ√1 − q(σ)dσ.14


2.2 Construction of a supersolutionWe provide a supersolution for the operator L λ defined in (2.29).Lemma 2.4 There exist constants β 0 > 0, M 1 < 0, M 0 > 0 (the constant M 0 is asin Lemma 2.3) such that for all β ∈ (0, β 0 ) the function U := W 0,M1 ,q−W −1,M0 ,q > 0in U β and satisfies(2.34) L λ U a ≥ 0 in U β .Moreover U ∈ H 1 (U β ) provided∫(2.35)√1 − q(σ)dσ < +∞.Σ k1Proof. We consider β 1 as in the beginning of the proof of Lemma 2.3 and we define(2.36) M 1 = − 1 2 maxx∈U β1|h(x)| − 1.Since U(x) = (e M1d(x) − e M0d(x) X −1 (˜δ(x)))d(x)˜δ(x) α(x) , it follows that U > 0 in U βfor β > 0 small. By (2.36) and Lemma 2.2, we get()L λ W 0,M1 ,q ≥ −C| log(δ)| δ − 3 2 − |λ|ηδ−2W 0,M1 ,q.By (2.32) we haveL λ (−W −1,M0 ,q) ≥(2δ −2 X −2 − C| log(δ)| δ − 3 2 − |λ|ηδ−2 ) W −1,M0 ,q.Taking the sum of the two above inequalities, we obtainL λ U ≥ 0 in U β ,which holds true because |η| ≤ Cδ in U β . Hence we get readily (2.34).To prove that U ∈ H 1 (U β ) provided (2.35) holds, it is enough to show that W 0,M1 ,q ∈H 1 (U β ) provided (2.35) holds.15


We argue as in the proof of Lemma 2.3. We have∫|∇W 0,M1 ,q| 2U β≤ C∫d 2 (x)˜δ(x) 2α(x)−2 dxU β≤≤≤∑N 0 ∫Cd 2 (F p ii=1B N−kM (y))˜δ(F p ipM (y))2α(F iM (y))−2 |Jac(F p iM )|(y)dy+ (0,β)×D i∑N 0 ∫C(y 1 ) 2 |ỹ| 2α(F p iM (y))−2 |Jac(F p ii=1B N−kM)|(y) dy+ (0,β)×D i∑N 0 ∫C(y 1 ) 2 |ỹ| k−N−2+(N−k)√ 1−q(f pi(ȳ)) |ỹ| −√ |ỹ| dy.B N−k+ (0,β)×D ii=1Here we used the fact that |Jac(F p iM)|(y) ≤ C. Note thatUsing polar coordinates, it follows that∫∑N 0|∇W 0,M1 ,q| 2 ≤ CU β≤Ci=1∑N 0i=1|ỹ| −√ |ỹ| ≤ C as |ỹ| → 0.∫D i∫∫S N−k−1+D i1( ) y1 2 ∫ βdθ r −1+(N−k)√ 1−q(f pi(ȳ)) dr dȳ|ỹ|√1 − q(f p i(ȳ))dȳ.Racalling that |Jac(f p i)|(ȳ) = 1 + O(|ȳ|), we deduce that∑N 0i=1∫√1 − q(f p i(ȳ))dȳ ≤ CD i1∑N 0i=1∫= C∫0√1 − q(f p i(ȳ))|Jac(f)|(ȳ) dȳD i1√1 − q(σ)dσ.Σ k1Therefore∫∫|∇W 0,M1 ,q| 2 dx ≤ CU β√1 − q(σ)dσΣ k1and the lemma follows at once.16


3 Existence of λ ∗We start with the following local improved Hardy inequality.Lemma 3.1 Let Ω be a smooth domain and assume that ∂Ω contains a smoothclosed submanifold Σ k of dimension 1 ≤ k ≤ N − 2. Assume that p, q and η satisfy(1.2) and (1.3). Then there exist constants β 0 > 0 and c > 0 depending only onΩ, Σ k , q, η and p such that for all β ∈ (0, β 0 ) the inequality∫Ω βp|∇u| 2 dx −holds for all u ∈ H 1 0 (Ω β).(N − k)24∫Ω βq |u|2δ 2dx ≥ c ∫|u| 2Ω βδ 2 | log(δ)| 2 dxProof. We use the notations in Section 2 with U = Ω and M = ∂Ω.Fix β 1 > 0 small and(3.1) M 2 = − 1 2 maxx∈Ω β1(|h(x)| + |∇p · ∇d|) − 1.Since p q ∈ C1 (Ω), there exists C > 0 such that(3.2)p(x)∣q(x) − p(σ(¯x))q(σ(¯x)) ∣ ≤ Cδ(x) ∀x ∈ Ω β,for small β > 0. Hence by (1.3) there exits a constant C ′ > 0 such that(3.3) p(x) ≥ q(x) − C ′ δ(x) ∀x ∈ Ω β .Consider W 12 ,M (in Lemma 2.2 with q ≡ 1). For all β > 0 small, we set2,1(3.4) ˜w(x) = W 12 ,M 2,1 (x), ∀x ∈ Ω β.Notice that div(p∇ ˜w) = p∆ ˜w + ∇p · ∇ ˜w. By Lemma 2.2, we havediv(p∇ ˜w) (N − k)2− ≥ pδ −2 + p ˜w44 δ−2 X −2 (δ) + O(| log(δ)|δ − 3 2 ) in Ωβ .This together with (3.3) yieldsdiv(p∇ ˜w) (N − k)2− ≥ qδ −2 + c 0˜w44 δ−2 X −2 (δ) + O(| log(δ)|δ − 3 2 ) in Ωβ ,17


with c 0 = min Ωβ1p > 0. Thereforediv(p∇ ˜w) (N − k)2(3.5) − ≥ qδ −2 + c δ −2 X −2 (δ) in Ω β ,˜w4for some positive constant c depending only on Ω, Σ k , q, η and p.Let u ∈ C ∞ c (Ω β ) and put ψ = ũ w . Then one has |∇u|2 = | ˜w∇ψ| 2 + |ψ∇ ˜w| 2 +∇(ψ 2 ) · ˜w∇ ˜w. Therefore |∇u| 2 p = | ˜w∇ψ| 2 p + p∇ ˜w · ∇( ˜wψ 2 ). Integrating by parts,we get∫∫|∇u| 2 p dx = | ˜w∇ψ| 2 p dx +Ω β Ω β∫Ω β(−Putting (3.5) in the above equality, we get the result.)div(p∇ ˜w)u 2 dx.˜wLemma 3.2 Let Ω be a smooth bounded domain and assume that ∂Ω contains asmooth closed submanifold Σ k of dimension 1 ≤ k ≤ N − 2. Assume that (1.2) and(1.3) hold. Then there exists λ ∗ = λ ∗ (Ω, Σ k , p, q, η) ∈ R such thatµ λ (Ω, Σ k ) =µ λ (Ω, Σ k ) λ ∗ .(3.6) sup µ λ (Ω, Σ k ) ≤λ∈R(N − k)2.4Indeed, we know that ν 0 (R N + , R k ) = (N−k)24, see e.g. [12]. Given τ > 0, we letu τ ∈ Cc∞ (R N + ) be such that(3.7)∫( ) (N − k)|∇u τ | 2 2∫dy ≤+ τ |ỹ| −2 u 2 τ dy.4R N +R N +By (1.3), we can let σ 0 ∈ Σ k be such thatq(σ 0 ) = p(σ 0 ).Now, given r > 0, we let ρ r > 0 such that for all x ∈ B(σ 0 , ρ r ) ∩ Ω(3.8) p(x) ≤ (1 + r)q(σ 0 ), q(x) ≥ (1 − r)q(σ 0 ) and η(x) ≤ r.18


We choose Fermi coordinates near σ 0 ∈ Σ k given by the map F σ 0∂Ω(as in Section 2)and we choose ε 0 > 0 small such that, for all ε ∈ (0, ε 0 ),Λ ε,ρ,r,τ := F σ 0∂Ω (ε Supp(u τ )) ⊂ B(σ 0 , ρ r ) ∩ Ωand we define the following test functionv(x) = ε 2−N2 u τ(ε −1 (F σ 0∂Ω )−1 (x) ) , x ∈ Λ ε,ρ,r,τ .Clearly, for every ε ∈ (0, ε 0 ), we have that v ∈ Cc∞ (Ω) and thus by a change ofvariable, (3.8) and Lemma 2.1, we have∫∫p|∇v| 2 dx + λ δ −2 ηv 2 dxµ λ (Ω, Σ k ) ≤Ω ∫Ωq(x) δ −2 v 2 dx∫Ω(1 + r) |∇v| 2 dxΛ≤ε,ρ,r,τ r|λ|∫+(1 − r) δ −2 v 2 (1 − r)q(σdx0 )Λ∫ε,ρ,r,τ(1 + r) |∇v| 2 dxΛ≤ε,ρ,r,τ r|λ|∫+(1 − cr) v 2 (1 − r)q(σdx0 )Λ ε,ρ,r,τ∫(1 + r)ε 2−N ε −2 (g ε ) ij ∂ i u τ ∂ j u τ | √ g ε (y) dyR≤+∫(1 − cr) ε 2−N |εỹ| −2 u 2 √+ crτ g ε 1 − r ,(ỹ) dyR N +where g ε is the scaled metric with components g ε αβ (y) = ε−2 〈∂ α F σ 0∂Ω (εy), ∂ βF σ 0∂Ω (εy)〉for α, β = 1, ·, N and where we have used the fact that ˜δ(F σ 0∂Ω (εy)) = |εỹ|2 for everyỹ in the support of u τ . Since the scaled metric g ε expands a g ε = I + O(ε) on thesupport of u τ , we deduce thatµ λ (Ω, Σ k ) ≤ 1 + r1 − cr1 + cε1 − cε19∫∫R N +R N +|∇u τ | 2 dy|ỹ| −2 u 2 τ dy+ cr1 − r ,


where c is a positive constant depending only on Ω, p, q, η and Σ k . Hence by (3.7)we concludeµ λ (Ω, Σ k ) ≤ 1 + r1 − cr1 + cε1 − cε( (N − k)24)+ τ + cr1 − r .Taking the limit in ε, then in r and then in τ, the claim follows.Step 2: We claim that there exists ˜λ ∈ R such that µ˜λ(Ω, Σ k ) ≥ (N−k)24.Thanks to Lemma 3.1, the proof uses a standard argument of cut-off function andintegration by parts (see [2]) and we can obtain∫∫∫δ −2 u 2 q dx ≤ |∇u| 2 p dx + C δ −2 u 2 η dx ∀u ∈ Cc ∞ (Ω),ΩΩfor some constant C > 0. We skip the details. The claim now follows by choosing˜λ = −CΩFinally, noticing that µ λ (Ω, Σ k ) is decreasing in λ, we can set}(3.9) λ ∗ (N − k)2:= sup{λ ∈ R : µ λ (Ω, Σ k ) =4so that µ λ (Ω, Σ k ) < (N−k)24for all λ > λ ∗ .4 Non-existence resultLemma 4.1 Let Ω be a smooth bounded domain of R N , N ≥ 3, and let Σ k be asmooth closed submanifold of ∂Ω of dimension k with 1 ≤ k ≤ N − 2. Then, thereexist bounded smooth domains Ω ± such that Ω + ⊂ Ω ⊂ Ω − and∂Ω + ∩ Ω = ∂Ω − ∩ Ω = Σ k .Proof. Consider the mapsx ↦→ g ± (x) := d ∂Ω (x) ± 1 2 δ2 (x),where d ∂Ω is the distance function to ∂Ω. For some β 1 > 0 small, g ± are smooth inΩ β1and since |∇g ± | ≥ C > 0 on Σ k , by the implicit function theorem, the sets{x ∈ Ω β : g ± = 0}20


are smooth (N − 1)-dimensional submanifolds of R N , for some β > 0 small. Inaddition, by construction, they can be taken to be part of the boundaries of smoothbounded domains Ω ± with Ω + ⊂ Ω ⊂ Ω − and such that∂Ω + ∩ Ω = ∂Ω − ∩ Ω = Σ k .Now, we prove the following non-existence result.Theorem 4.2 Let Ω be a smooth bounded domain of R N and let Σ k be a smoothclosed submanifold of ∂Ω of dimension k with 1 ≤ k ≤ N − 2 and let λ ≥ 0.Assume that p, q and η satisfy (1.2) and (1.3). Suppose that u ∈ H0 1 (Ω) ∩ C(Ω) is anon-negative function satisfying(4.1) −div(p∇u) −If ∫ √ 1Σ kdσ = +∞ then u ≡ 0.1−p(σ)/q(σ)(N − k)2qδ −2 u ≥ −ληδ −2 u in Ω.4Proof. We first assume that p ≡ 1. Let Ω + be the set given by Lemma 4.1. Wewill use the notations in Section 2 with U = Ω + and M = ∂Ω + . For β > 0 small wedefineΩ + β := {x ∈ Ω+ :δ(x) < β}.We suppose by contradiction that u does not vanish identically near Σ k and satisfies(4.1) so that u > 0 in Ω β by the maximum principle, for some β > 0 small.Consider the subsolution V ε defined in Lemma 2.3 which satisfies(4.2) L λ V ε ≤ 0 in Ω + β, ∀ε ∈ (0, 1).Notice that ∂Ω + β ∩ Ω+ ⊂ Ω thus, for β > 0 small, we can choose R > 0 (independenton ε) so thatR V ε ≤ R V 0 ≤ u on ∂Ω + β ∩ Ω+ ∀ε ∈ (0, 1).Again by Lemma 2.3, setting v ε = R V ε − u, it turns out that v ε+ = max(v ε , 0) ∈H0 1(Ω+ β ) because V ε = 0 on ∂Ω + β \ ∂Ω+ β ∩ Ω+ . Moreover by (4.1) and (4.2),L λ v ε ≤ 0 in Ω + β, ∀ε ∈ (0, 1).21


Multiplying the above inequality by v ε+ and integrating by parts yields∫∫∫|∇v +Ω + ε | 2 (N − k)2dx − δ −2 q|v +4βΩ + ε | 2 dx + λ ηδ −2 |vβΩ + ε + | 2 dx ≤ 0.βBut then Lemma 3.1 implies that v ε + = 0 in Ω + βprovided β small enough because|η| ≤ Cδ near Σ k . Therefore u ≥ R V ε for every ε ∈ (0, 1). In particular u ≥ R V 0 .Hence we obtain from Lemma 2.3 that∫u 2 ∫∞ >δ 2 ≥ R2Ω + βΩ + βV 20δ 2≥ ∫√1 − q(σ)dσΣ k1which leads to a contradiction. We deduce that u ≡ 0 in Ω + β. Thus by the maximumprinciple u ≡ 0 in Ω.For the general case p ≠ 1, we argue as in [3] by setting(4.3) ũ = √ pu.This function satisfies−∆ũ −(N − k)24qp δ−2 ũ ≥ −λ η (p δ−2 ũ + − ∆p )2p + |∇p|24p 2 ũ in Ω.Hence since p ∈ C 2 (Ω) and p > 0 in Ω, we get the same conclusions as in the casep ≡ 1 and q replaced by q/p.5 Existence of minimizers for µ λ (Ω, Σ k )Theorem 5.1 Let Ω be a smooth bounded domain of R N and let Σ k be a smoothclosed submanifold of ∂Ω of dimension k with 1 ≤ k ≤ N − 2. Assume that p, q andη satisfy (1.2) and (1.3). Then µ λ (Ω, Σ k ) is achieved for every λ < λ ∗ .Proof. The proof follows the same argument of [2] by taking into account the factthat η = 0 on Σ k so we skip it.22


Next, we prove the existence of minimizers in the critical case λ = λ ∗ .Theorem 5.2 Let Ω be a smooth bounded domain of R N and let Σ k be a smoothclosed submanifold of ∂Ω of dimension ∫ k with 1 ≤ k ≤ N − 2. Assume that p, q andη satisfy (1.2) and (1.3). If √ dσ < ∞ then µ λ ∗ = µ λ ∗(Ω, Σ k ) is1 − p(σ)/q(σ)achieved.Σ k1Proof. We first consider the case p ≡ 1.Let λ n be a sequence of real numbers decreasing to λ ∗ . By Theorem 5.1, there exitsu n minimizers for µ λn = µ λn (Ω, Σ k ) so that(5.1) −∆u n − µ λn δ −2 qu n = −λ n δ −2 ηu n in Ω.We may assume that u n ≥ 0 in Ω. We may also assume that ‖∇u n ‖ L 2 (Ω) = 1. Henceu n ⇀ u in H 1 0 (Ω) and u n → u in L 2 (Ω) and pointwise. Let Ω − ⊃ Ω be the set givenby Lemma 4.1. We will use the notations in Section 2 with U = Ω − and M = ∂Ω − .It will be understood that q is extended to a function in C 2 (Ω − ). For β > 0 smallwe defineWe have thatwith |b n | ≤ C in Ω \ Ω − β2Ω − β := {x ∈ Ω− :δ(x) < β}.∆u n + b n (x) u n = 0 in Ω,for all integer n. Thus by standard elliptic regularity theory,(5.2) u n ≤ C in Ω \ Ω − β .2We consider the supersolution U in Lemma 2.4. We shall show that there exits aconstant C > 0 such that for all n ∈ N(5.3) u n ≤ CU in Ω − β .Notice that Ω ∩ ∂Ω − β ⊂ Ω− thus by (5.2), we can choose C > 0 so that for any nu n ≤ C U on Ω ∩ ∂Ω − β .Again by Lemma 2.4, setting v n = u n − C U, it turns out that v n+H0 1(Ω− β ) because u n = 0 on ∂Ω ∩ Ω − β. Hence we have= max(v n , 0) ∈L λn v n ≤ −C(µ λ ∗ − µ n )qU − C(λ ∗ − λ n )ηU ≤ 0 in Ω − β ∩ Ω.23


Multiplying the above inequality by v n + and integrating by parts yields∫∫∫|∇v n + | 2 dx − µ λn δ −2 q|v n + | 2 dx + λ n ηδ −2 |v n + | 2 dx ≤ 0.Ω − βΩ − βΩ − βHence Lemma 3.1 implies that∫∫C δ −2 X −2 |v n + | 2 dx + λ nΩ − βΩ − βηδ −2 |v + n | 2 dx ≤ 0.Since λ n is bounded, we can choose β > 0 small (independent of n) such that v + n ≡ 0on Ω − β(recall that |η| ≤ Cδ). Thus we obtain (5.3).Now since u n → u in L 2 (Ω), we get by the dominated convergence theorem and(5.3), thatSince u n satisfies∫1 =Ωδ −1 u n → δ −1 u|∇u n | 2 = µ λn∫Ωin L 2 (Ω).δ −2 qu 2 n + λ n∫Ωδ −2 ηu 2 n,∫taking the limit, we have 1 = µ λ ∗Ω δ−2 qu 2 + λ ∗ ∫ Ω δ−2 ηu 2 . Hence u ≠ 0 and it is aminimizer for µ λ ∗ = (N−k)24.For the general case p ≠ 1, we can use the same transformation as in (4.3). So (5.3)holds and the same argument as a above carries over.6 Proof of Theorem 1.1 and Theorem 1.2Proof of Theorem 1.1: Combining Lemma 3.2 and Theorem 5.1, it remains only tocheck the case λ < λ ∗ . But this is an easy consequence of the definition of λ ∗ andof µ λ (Ω, Σ k ), see [[2], Section 3].Proof of Theorem 1.2: Existence is proved in Theorem 5.2 for I k < ∞. Since theabsolute value of any minimizer for µ λ (Ω, Σ k ) is also a minimizer, we can applyTheorem 4.2 to infer that µ λ ∗(Ω, Σ k ) is never achieved as soon as I k = ∞.24


AcknowledgmentsThis work started when the first author was visiting CMM, Universidad de Chile.He is grateful for their kind hospitality. M. M. Fall is supported by the Alexandervon-HumboldtFoundation. F. Mahmoudi is supported by the Fondecyt proyect n:1100164 and Fondo Basal CMM.References[1] Adimurthi and Sandeep K., Existence and non-existence of the first eigenvalueof the perturbed Hardy-Sobolev operator. Proc. Roy. Soc. Edinburgh Sect. A 132(2002), no. 5, 1021-1043.[2] Brezis H. and Marcus M., Hardy’s inequalities revisited. Dedicated to Ennio DeGiorgi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 217-237.[3] Brezis H., Marcus M. and Shafrir I., Extermal functions for Hardy’s inequalitywith weight, J. Funct. Anal. 171 (2000), 177-191.[4] Caldiroli P., Musina R., On a class of 2-dimensional singular elliptic problems.Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), 479-497.[5] Caldiroli P., Musina R., Stationary states for a two-dimensional singularSchrödinger equation. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 4-B(2001), 609-633.[6] Cazacu C., On Hardy inequalities with singularities on the boundary. C. R.Math. Acad. Sci. Paris 349 (2011), no. 5-6, 273-277.[7] Fall M. M., Nonexistence of distributional supersolutions of a semilinear ellipticequation with Hardy potential. Preprint http://arxiv.org/abs/1105.5886.[8] Fall M. M., On the Hardy-Poincaré inequality with boundary singularities. Commun.Contemp. Math., to appear. http://arxiv.org/abs/1008.4785.[9] Fall M. M., A note on Hardy’s inequalities with boundary singularities. NonlinearAnal. 75 (2012) no. 2, 951-963.25


[10] Fall M. M., Musina R., Hardy-Poincaré inequalities with boundary singularities.Proc. Roy. Soc. Edinburgh. A 142, 1-18, 2012.[11] Fall M. M., Musina R., Sharp nonexistence results for a linear elliptic inequalityinvolving Hardy and Leray potentials. Journal of Inequalities and Applications,vol. 2011, Article ID 917201, 21 pages, 2011. doi:10.1155/2011/917201.[12] Filippas S.; Tertikas A. and Tidblom J., On the structure of Hardy-Sobolev-Maz’ya inequalities . J. Eur. Math. Soc., 11(6), (2009), 1165-1185.[13] Mahmoudi, F., Malchiodi, A., Concentration on minimal submanifolds for asingularly perturbed Neumann problem, Adv. in Math. 209 (2007) 460-525.[14] Nazarov A. I., Hardy-Sobolev Inequalities in a cone, J. Math. Sciences, 132,(2006), (4), 419-427.[15] Nazarov A. I., Dirichlet and Neumann problems to critical Emden-Fowler typeequations. J. Glob. Optim. (2008) 40, 289-303.[16] Pinchover Y., Tintarev K., Existence of minimizers for Schrödinger operatorsunder domain perturbations with application to Hardy’s inequality. Indiana Univ.Math. J. 54 (2005), 1061-1074.26

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