Bounds of Dirichlet eigenvalues for Hardy-Leray operator
aa r X i v : . [ m a t h . A P ] F e b Bounds of Dirichlet eigenvalues for Hardy-Leray operator
Huyuan Chen Department of Mathematics, Jiangxi Normal University,Nanchang, Jiangxi 330022, PR China
Feng Zhou Center for PDEs, School of Mathematical Sciences, East China Normal University, and NYU-ECNUInstitute of Mathematical Sciences at NYU Shanghai,Shanghai Key Laboratory of PMMP, Shanghai 200062, PR China
Abstract
The purpose of this paper is to study the eigenvalues { λ µ,i } i for the Dirichlet Hardy-Lerayoperator, i.e. − ∆ u + µ | x | − u = λu in Ω , u = 0 on ∂ Ω , where − ∆ + µ | x | is the Hardy-Leray operator with µ ≥ − ( N − and Ω is a smooth boundeddomain with 0 ∈ Ω. We provide lower bounds of { λ µ,i } i together with the Li-Yau’s one for µ > − ( N − and Karachalio’s one for µ ∈ [ − ( N − , λ µ,k . Finally, we get the Weyl’s limit of eigenvalues which is independentof the potential’s parameter µ . This interesting phenomena indicates that the inverse-squarepotential does not play an essential role for the asymptotic behavior of the spectral of theproblem considered. Keywords : Dirichlet eigenvalues; Hardy-Leray operator.
MSC2010 : 35P15; 35J15.
Let Ω be a smooth bounded domain in R N with N ≥
2, 0 ∈ Ω and µ := − ( N − the best constantfor the standard Hardy inequality. The purpose of this paper is to study the bounds of eigenvaluesfor the Dirichlet problem ( L µ u = λu in Ω ,u = 0 on ∂ Ω , (1.1)where L µ := − ∆ + µ | x | for µ ≥ µ is the Hardy-Leray operator.When µ = 0, L µ reduces to the Laplacian and in 1912, Weyl [41] showed that the k -th eigenvalue λ k of the Dirichlet problem for any smooth bounded domain Ω with the Laplacian operator hasthe asymptotic behaviorlim k → + ∞ k − N λ k = c N | Ω | − N , where c N = (2 π ) | B | − N , (1.2) c N is named as Weyl’s constant, | Ω | is the volume of Ω and B is the unit ball centered at theorigin. Later, P´olya [39] (in 1960) proved that λ k ≥ C | Ω | − N k N (1.3)with C = c N , ( N = 2) and for any “plane-covering domain” in R , ( his proof also works indimension N ≥
3) and he also conjectured that (1.3) holds with C = c N for any bounded domain [email protected] [email protected] R N . Later on, Lieb [34] proved (1.3) with some positive constant C in a general bounded domainand then Li-Yau [33] improved this constant in (1.3) to C N := NN + 2 c N = NN + 2 (2 π ) | B | − N . (1.4)The estimate (1.3) with C = C N is also called Berezin-Li-Yau’s inequality because the constant C N is achieved with the help of Legendre transform in an earlier result obtained by Berezin. Here we call C N the Li-Yau’s constant. The Berezin-Li-Yau inequality then is generalized in [14, 17, 29, 34, 36]for degenerate elliptic operators. Upper bounds for the first k -eigenvalues obtained in [30] iscontrolled by c N k N together with lower order terms; later on, Cheng-Yang in [14–16] developed avery interesting upper bound λ k ≤ (cid:16) N (cid:17) k N λ , which is also named Cheng-Yang’s inequality. More results on minimizing the eigenvalues problemscould be referred to [20, 21, 29], subject to the homogeneous Dirichlet boundary condition, to[1, 6] with Neumann boundary condition. Our motivations of the Dirichlet eigenvalues for Hardyoperators are twofold, the one is the Polya’s conjecture associating the zero of the Riemann Zetafunction with the eigenvalue of a Hermitian operator and the latter is the role of the critical potentialin the analysis of PDEs.When µ ≥ µ and µ = 0, the Hardy potential has homogeneity −
2, which is critical fromboth mathematical and physical viewpoint. The Hardy-Leary problems have great applications tovarious fields as molecular physics [31], quantum cosmological models such as the Wheeler-de-Wittequation (see e.g. [2]) and combustion models [25]. The Hardy inequalities play a fundamental rolein the study of Hardy-Leray problems. When the potential’s singularity { } is in Ω, the Hardyinequality [18, (2)] (also see [4, 5]) reads as Z Ω |∇ u ( x ) | dx + µ Z Ω u ( x ) | x | dx ≥ c Z Ω u ( x ) dx, ∀ u ∈ H (Ω) (1.5)for some constant c >
0. The operator L µ is then positive definite for µ ∈ [ µ , + ∞ ). Moreproperties of Hardy-Leray operator could be found in [7, 37]. It is worth noting that the inversesquare potential plays an essential role in isolated singularity of elliptic Hardy equations. Indeed,for a given ‘source’ f defined in Ω, the solutions of the Dirichlet problem L µ u = f in Ω \ { } , u = 0 on ∂ Ωwith isolated singularities at { } are classified fully in [11], thanks to a new formulation of distribu-tional identity associated to some specific weight. Extensive treatments of the associated semilinearproblems are developed in [12, 13] via an introduction of a notion of very weak solution.Due to the inverse square potential, the spectral of the Hardy-Leray operator remains widelyunexplored. In [19] the authors investigated the essential spectral for general Hardy-Leray potential,for problem (1.1), an increasing sequence of eigenvalues { λ µ,k (Ω) } k ∈ N , simply denoted by { λ µ,k } k ∈ N in the sequel, are obtained in [40]; [8] studied the optimal first eigenvalue with respect to the domain,and the lower bounds estimates of eigenvalues for (1.1) in [27,28] are derived by using the estimatesof the related heat kernel and Sobolev inequalities. Precisely, the lower bounds state as following:(i) when N ≥ µ < µ <
0, there holds that for k ∈ N λ µ,k ≥ (cid:16) − µµ (cid:17) N ( N − e ω NN − | Ω | − N k N ; (1.6)(ii) when N ≥ µ = µ , there holds that for k ∈ N , λ µ ,k ≥ e − S N ( N − − N − N k X − N k − N L (Ω) k N , where X ( x ) := (cid:0) − log( | x | D ) (cid:1) − , ∀ x ∈ Ω, D := 2 max x ∈ ∂ Ω | x | , and S N := 2 /N π /N Γ( N +12 )is the best constant of Sobolev inequality in R N and Γ is the well-known Gamma function.2e note that the defect of the lower bound (1.6) is the coefficient N ( N − e (1 − µµ ) → µ → µ +0 .It is natural to ask whether the Hardy-Leray potential plays an essential role in the estimatesof the spectral or how it works on the related eigenvalues. Our main aim in this paper is to answerthis question and establish the lower and upper bounds, and show the limit the eigenvalues as k → + ∞ . Now we state our lower bounds as follows. Theorem 1.1.
Let D = max x ∈ ∂ Ω | x | and { λ µ,i } i ∈ N be the increasing sequence of eigenvalues ofproblem (1.1). ( i ) If µ ≤ µ < and N ≥ , then we have that for k ∈ N , k X i =1 λ µ,i ≥ max (cid:26)(cid:16) − µµ (cid:17) C N | Ω | − N k N + µµ λ µ, k, b k e − σ µ k N (cid:27) , where C N is the Li-Yau’s constant defined in (1.4), b k > ( 12 ) N , lim k → + ∞ b k = NN + 2 and σ µ = max n S N ( N − − N − N k X − N k − N L (Ω) , (cid:0) − µµ (cid:1) N ( N − ω NN − | Ω | − N o ; (1.7)( ii ) If µ > and N ≥ , then we have that for k ∈ N , k X i =1 λ µ,i ≥ C N | Ω | − N k N + D − µ k. We remark that the regularity of Ω could be released to ‘bounded’ for the lower bounds andfor µ ≤ µ <
0, the lower bounds in part ( i ) consist of the Berezin-Li-Yau’s type bounds andKarachalio’s type bounds. In fact, the Li-Yau’s method can not be applied for µ = µ thanks tothe factor (cid:0) − µµ (cid:1) , which vanishes as µ → µ +0 . This also occurs for the Karachalio’s estimate(1.6). To overcome this decay, we develop the Karacholio’s method and our lower bound appears e − σ µ k N , where σ µ has the nondecreasing monotonicity for µ ∈ [ µ , σ µ ≥ σ µ = S N ( N − − N − N k X − N k − N L (Ω) > . From the monotonicity of { λ µ,i } i ∈ N , we have the following corollary: Corollary 1.2.
Let { λ µ,i } i ∈ N be the increasing sequence of eigenvalues of problem (1.1). ( i ) If µ ≤ µ < and N ≥ , then for k ∈ N , λ µ,k ≥ max (cid:26)(cid:16) − µµ (cid:17) C N | Ω | − N k N + µµ λ µ, , σ µ k N (cid:27) . ( ii ) If µ > and N ≥ , then for k ∈ N , λ µ,k ≥ C N | Ω | − N k N + D − µ. In order to obtain upper bounds, we extend Cheng-Yang’s type inequality for µ ≥ µ and ourupper bounds state as follows: Theorem 1.3.
Let { λ µ,i } i ∈ N be the increasing sequence of eigenvalues of problem (1.1). ( i ) If µ ≥ and N ≥ , then λ µ,k ≤ (cid:16) N (cid:17) k N λ µ, . (1.8)( ii ) If µ ∈ [ µ , and N ≥ , then λ µ,k ≤ (cid:16) N (cid:17) k N λ , + µD − . (1.9)3t is worth noting that the crucial inequality for obtaining the Cheng-Yang’s type inequality(1.8) is the following k X i =1 ( λ µ,k +1 − λ µ,i ) ≤ a µ k X i =1 ( λ µ,k +1 − λ µ,i ) λ µ,i , (1.10)which holds with a µ = N if µ >
0. However, (1.10) has the constant a µ = N − µ µ − µ if µ ∈ ( µ ,
0) for N ≥
3, then it will give an equality like λ µ,k ≤ (cid:16) a µ (cid:17) k a µ λ µ, , which is too rough since a µ > N and lim µ → µ +0 a µ = + ∞ . So we adopt an approach of comparing with the Laplacian directly.Combining the estimate of the first eigenvalue λ µ, , we have the following corollary: Corollary 1.4.
For N ≥ and µ > , we have that λ µ,k ≤ (cid:0) N (cid:1)(cid:16) λ , + µ k φ , k L ∞ Z Ω | x | − dx (cid:17) k N . (1.11) For N ≥ and µ > , λ µ,k ≤ (cid:0) N (cid:1)(cid:18) λ , + c − τ + ( µ ) k φ , k C Z Ω | x | τ + ( µ ) − dx Z Ω ρ ( x ) | x | τ + ( µ ) dx (cid:19) k N , (1.12) where c > , τ + ( µ ) = − N − + √ µ − µ , ( λ , , φ , ) is the first eigenvalue and associated eigen-functions of (1.1) for µ = 0 and ρ ( x ) = dist( x, ∂ Ω) . It is remarkable that the upper bound (1.11) is much simple, but it can’t be used in the casethat N = 2 and the upper bound (1.12) is available for N ≥ τ + ( µ ) > µ > k -eigenvalues from Corollary 1.2 andTheorem 1.3 in the following table: µ (0 , + ∞ ) [ µ , λ µ,k C N | Ω | − N k N + µD − max n(cid:16) − µµ (cid:17) C N | Ω | − N k N + µµ λ µ, (cid:17) , σ µ k N o UB of λ µ,k (cid:16) N (cid:17) k N λ µ, (cid:16) N (cid:17) k N λ , + µD − where LB and U B stand for Lower bound and Upper bound respectively.Finally, we provide the Weyl’s limit of eigenvalues for Hardy-Leray operators.
Theorem 1.5.
Assume that N ≥ , µ ≥ µ and { λ µ,i } i ∈ N is the increasing sequence of eigenvaluesof problem (1.1). Then there holds lim k → + ∞ λ µ,k k − N = c N | Ω | − N , (1.13) where c N is the Weyl’s constant given in (1.2). We notice that the limit of { λ µ,k k − N } k ∈ N is independent of µ . Actually, this limit coincidesthe one for Laplacian, see (1.2). This answers our question that the inverse square potential termis a second role for the asymptotic eigenvalues.The rest of this paper is organized as follows. In Section 2, we build the Berezin-Li-Yau’s typelower bounds and the Karachalio’s type lower bounds in Theorem 1.1. Section 3 is devoted to theYang’s inequality and proof of Theorem 1.3. Finally, we prove the Weyl’s limit of eigenvalues inTheorem 1.5 in Section 4. 4 Lower bounds
Proposition 2.1.
Assume that N ≥ , µ < µ < and { λ µ,i } i ∈ N is the increasing sequence ofeigenvalues of problem (1.1). Then we have ( i ) for µ < µ < and k ∈ N there holds k X i =1 λ µ,i ≥ (cid:16) − µµ (cid:17) C N | Ω | − N k N + µµ λ µ, k ;( ii ) for µ > and k ∈ N there holds k X i =1 λ µ,i ≥ C N | Ω | − N k N + D − µ k, where we recall D = max x ∈ ∂ Ω | x | . For N ≥ µ ≥ µ , we denote H µ (Ω) as the completion of C ∞ (Ω) with the norm k u k µ = sZ Ω |∇ u | dx + µ Z Ω u | x | dx and it is a Hilbert space with the inner product h u, v i µ = Z Ω ∇ u · ∇ vdx + µ Z Ω uv | x | dx. We remark that H µ (Ω) = H (Ω) for µ > µ and if µ = 0, H µ (Ω) ' H (Ω) . The following lemmais crucial to get the Berezin-Li-Yau’s lower bound whch is appeared in [33].
Lemma 2.2. [33, Lemma 1] If f is a real-valued function defined on R N with ≤ f ≤ M and Z R N f ( z ) | z | dz ≤ M , then we have Z R N f ( z ) dz ≤ ( M | B | ) N +2 M NN +2 (cid:16) N + 2 N (cid:17) NN +2 . Proof of Proposition 2.1.
Let ( λ µ,k , φ k ) be the eigenvalue and eigenfunction pair of (1.1) suchthat k φ k k L (Ω) = 1 . Then λ µ,k = min n Z Ω (cid:16) |∇ u | + µ | x | u (cid:17) dx : u ∈ H k (Ω) with k u k L (Ω) = 1 o , where H (Ω) = H µ (Ω) and H k (Ω) = { u ∈ H µ (Ω) : Z Ω uφ j dx = 0 for j = 1 , . . . , k − } for l > { φ k ∈ H µ (Ω) : k ∈ N } is an orthonormal basis of L (Ω).Denote Φ k ( x, y ) := k X j =1 φ j ( x ) φ j ( y )5nd its Fourier transform is thenˆΦ k ( z, y ) = (2 π ) − N Z R N Φ k ( x, y ) e i x · z dx. Note that Z R N Z R N | ˆΦ k ( z, y ) | dzdy = Z Ω Z R N | Φ k ( x, y ) | dxdy = k X j =1 Z Ω φ j ( x ) dx = k and Z R N | ˆΦ k ( z, y ) | dy = (2 π ) − N Z Ω (cid:12)(cid:12) Z Ω Φ k ( x, y ) e i x · z dx (cid:12)(cid:12) dy = (2 π ) − N (cid:12)(cid:12) Z Ω k X j =1 φ j ( x ) e i x · z dx (cid:12)(cid:12) , which implies by Bessel’s inequality (see [35, (1.2)]) that Z R N | ˆΦ k ( z, y ) | dy ≤ (2 π ) − N Z Ω | e i x · z | dx = (2 π ) − N | Ω | . Meanwhile, the Hardy inequality (1.5) implies that for µ < µ < Z Ω |∇ u | dx + µ Z Ω u | x | dx = µµ (cid:16) Z Ω |∇ u | dx + µ Z Ω u | x | dx (cid:17) + (1 − µµ ) Z Ω |∇ u | dx ≥ (1 − µµ ) Z Ω |∇ u | dx + µµ λ µ, Z Ω u dx, thus, we have that Z R N Z R N | ˆΦ k ( z, y ) | | z | dydz = Z R N Z Ω |∇ Φ k ( x, y ) | dydx = Z Ω (cid:12)(cid:12)(cid:12) k X j =1 ∇ φ j ( x ) (cid:12)(cid:12)(cid:12) dx ≤ (1 − µµ ) − (cid:16) Z Ω (cid:12)(cid:12)(cid:12) ∇ k X j =1 φ j ( x ) (cid:12)(cid:12)(cid:12) dx + µ Z Ω P kj =1 φ j ( x ) | x | dx − µµ λ µ, Z Ω k X j =1 φ j ( x ) dx (cid:17) = (1 − µµ ) − (cid:16) k X j =1 λ µ,j − µµ λ µ, k (cid:17) . For µ >
0, we have that Z Ω |∇ u | dx + µ Z Ω u | x | dx ≥ Z Ω |∇ u | dx + µD − Z Ω u dx, and then Z R N Z R N | ˆΦ k ( z, y ) | | z | dydz = Z R N Z Ω |∇ Φ k ( x, y ) | dydx ≤ Z Ω (cid:12)(cid:12)(cid:12) k X j =1 ∇ φ j (cid:12)(cid:12)(cid:12) dx + µ Z Ω P kj =1 φ j | x | dx − D − µ Z Ω (cid:16) k X j =1 φ j (cid:17) dx = k X j =1 λ µ,j − µD − k. Case: N ≥ , µ < µ < . We apply Lemma 2.2 to the function f ( z ) = Z Ω | ˆΦ k ( z, y ) | dy with M = (2 π ) − N | Ω | and M = (1 − µµ ) − (cid:16) k X j =1 λ µ,j − µµ λ µ, k (cid:17) , then we conclude that k = Z R N f ( z ) dz ≤ ( M | B | ) N +2 M NN +2 (cid:16) N + 2 N (cid:17) NN +2 ≤ (cid:16) (2 π ) − N | Ω | | B | (cid:17) N +2 h (1 − µµ ) − (cid:16) k X j =1 λ µ,j − µµ λ µ, k (cid:17)i NN +2 (cid:16) N + 2 N (cid:17) NN +2 and k X j =1 λ µ,j ≥ (cid:16) − µµ (cid:17) C N | Ω | − N k N + µµ λ µ, k. For
Case: µ > . We again apply Lemma 2.2 to the same function f ( z ) and M , but M = k X j =1 λ µ,j − µD − k, then we conclude that k = Z R N f ( z ) dz ≤ ( M | B | ) N +2 M NN +2 (cid:16) N + 2 N (cid:17) NN +2 ≤ (cid:16) (2 π ) − N | Ω | | B | (cid:17) N +2 (cid:16) k X j =1 λ µ,j − µD − k (cid:17) NN +2 (cid:16) N + 2 N (cid:17) NN +2 and k X j =1 λ µ,j ≥ C N | Ω | − N k N + µD − k. We complete the proof. (cid:3)
Corollary 2.3.
Let { λ µ,i } i ∈ N be the increasing sequence of eigenvalues of problem (1.1). ( i ) For N ≥ , µ < µ < and k ∈ N , we have that λ µ,k ≥ (cid:16) − µµ (cid:17)(cid:16) C N | Ω | − N k N + µµ λ µ, (cid:17) . ( ii ) For N ≥ , µ > and k ∈ N , we have that λ µ,k ≥ C N | Ω | − N k N + µD − . Proof.
From the increasing monotonicity of k λ µ,k , we have that λ µ,k ≥ k k X j =1 λ µ,j , which implies the lower bounds for λ µ,k by Proposition 2.1. (cid:3) .2 Karachalio’s type lower bound Recall that S N = 2 /N π /N Γ( N +12 ) is the best constant of Sobolev inequality in R N and Γ is theGamma function. Now we set X ( x ) = (cid:0) − log( | x | D ) (cid:1) − and X ( x ) ≡ , ∀ x ∈ Ω , where we recall D = max x ∈ ∂ Ω | x | . Note that k X − N − k L (Ω) ≤ Z B D X − N ( y ) dy ≤ ω N − Z D r N − ( − log rD ) N − dr < + ∞ and k X − N − k L (Ω) = | Ω | . Proposition 2.4.
Assume that N ≥ , µ ≤ µ < and { λ µ,k } k ∈ N is the increasing sequence ofeigenvalues of problem (1.1). Then for k ∈ N we have that λ µ,k ≥ e − σ µ k N , where σ µ is defined in (1.7). Proof.
When µ = µ , the proof is refereed for [28] and we focus on the case µ < µ < L µ be the Hardy-Leray operator with its domain defined as D ( L µ ) = C ∞ (Ω) and itsFriedrich’s extension, still denoted by L µ , with its domain defined as D ( L µ ) := (cid:8) u ∈ H µ (Ω) : L µ u ∈ L (Ω) (cid:9) , which is a nonnegative self-adjoint operator on L (Ω) and the operator gives rise to the semigroupof operators e −L µ t for every t >
0, possessing an integral kernel K ( x, y, t ) > x, y, t ) ∈ Ω × Ω × (0 , + ∞ ). Then L µ has compact resolvent and K can be represented as K ( x, y, t ) = ∞ X i =1 e − λ µ,i t φ i ( x ) φ i ( y ) , which solves the problem ∂ t K + L µ K = 0 in Ω × Ω × (0 , + ∞ ) ,K ( x, y, t ) > × Ω × (0 , + ∞ ) ,K ( x, y, t ) = 0 on ∂ Ω × ∂ Ω × (0 , + ∞ ) . (2.1)Since { φ i } i ≥ is an orthonormal basis of L (Ω), it follows that h ( t ) := ∞ X i =1 e − λ µ,i t = Z Ω Z Ω K ( x, y, t ) dxdy. By the improved Hardy-Sobolev inequalities, [23, Theorem A] for µ ≥ µ , (the following in-equality is sharp for µ = µ ) Z Ω |∇ u | dx + µ Z Ω u | x | dx ≥ σ (cid:16) Z Ω | u | ∗ X ( x ) − N − N − dx (cid:17) ∗ , ∀ u ∈ C ∞ (Ω)and for µ > µ Z Ω |∇ u | dx + µ Z Ω u | x | dx ≥ σ (cid:16) Z Ω | u | ∗ dx (cid:17) ∗ , ∀ u ∈ C ∞ (Ω) , ∗ = NN − , σ = S N ( N − − N − N and σ = N ( N − − µµ ) ω N − . By H¨older inequality, we get that for j = 1 , h ( t ) ≤ Z Ω (cid:16)(cid:0) Z Ω X j ( y ) N − N − K ∗ ( x, y, t ) dy (cid:1) ∗− (cid:0) Z Ω X j ( y ) − N − N − ∗− K ( x, y, t ) dy (cid:1) ∗− ∗− (cid:17) dx ≤ h Z Ω (cid:16) Z Ω X j ( y ) N − N − K ∗ ( x, y, t ) dy (cid:17) ∗ dx i ∗ ∗− (cid:16) Z Ω Q j ( x, t ) dx (cid:17) ∗− ∗− , (2.2)where N − N − ∗ − = N − and Q j ( x, t ) = Z Ω X j ( y ) − N − K ( x, y, t ) dy. We observe that Q j ( x, t ) is the solution of the Cauchy-Dirichlet problem: ∂ t Q j + L µ Q j = 0 in Ω × (0 , + ∞ ) ,Q j ( x,
0) = X − N j in Ω ,Q j ( x, t ) = 0 on ∂ Ω × (0 , + ∞ ) . (2.3)Multiplying the above equation (2.3) by Q j , we get the energy equation12 ddt k Q j ( t ) k L (Ω) + k Q j ( t ) k µ = 0and then k Q j ( t ) k L (Ω) ≤ k Q j (0) k L (Ω) = k X − N j k L (Ω) . With the help of above inequalities, letting C j := k X − N j k ∗− ∗ L (Ω) for j = 1 , , for µ > µ , from (2.2) we obtain that h ∗− ∗ ( t ) ≤ C j Z Ω (cid:16) Z Ω X j ( y ) N − N − K ∗ ( x, y, t ) dy (cid:17) ∗ dx ≤ C j σ j Z Ω Z Ω (cid:16) |∇ y K ( x, y, t ) | dx + µ K ( x, y, t ) | y | (cid:17) dxdy ≤ C j σ j k Q j ( t ) k µ = − C j σ j ddt k Q j ( t ) k L (Ω) ≤ − C j σ j dh ( t ) dt , which, using h (0) = + ∞ , implies that ∞ X i =1 e − λ µ,i t = h ( t ) ≤ (cid:16) ∗ σ j (2 ∗ − (cid:17) ∗ ∗− k X − N j k L (Ω) t − ∗ ∗− for j = 1 , . Now we choose t = 2 ∗ ∗ −
2) 1 λ µ,k ke − ∗ ∗− ≤ ∞ X i =1 e − ∗ λµ,iλµ,k (2 ∗− ≤ σ − ∗ ∗− j λ ∗ ∗− µ,k k X − N j k L (Ω) for j = 1 , . Note that ∗ ∗ − = N and we conclude that λ µ,k ≥ e − σ j k X − N j k − N L (Ω) k N for j = 1 , . As a consequence, we obtain that λ µ,k ≥ e − σ µ k N , where σ µ = max n σ k X − N k − N L (Ω) , σ | Ω | − N o . We complete the proof. (cid:3)
Proof of Theorem 1.1.
When N ≥ µ ≤ µ <
0, the lower bounds k X i =1 λ µ,i ≥ (cid:16) − µµ (cid:17)(cid:16) C N | Ω | − N k N + µµ λ µ, k (cid:17) follows from Proposition 2.1 part ( i ). Note that for µ = µ , the above bounds is true obviouslysince 1 − µµ = 0. Moreover, from Proposition 2.4, we have that λ µ,k ≥ e − σ µ k N , which implies that k X i =1 λ µ,i (Ω) ≥ e − σ µ k N k X i =1 (cid:0) ik (cid:1) N ≥ b k e − σ µ k N +1 , where b k = 1 k k X i =1 (cid:0) ik (cid:1) N → NN + 2 as k → + ∞ and for any k ≥ b k = 1 k k X i =1 (cid:0) ik (cid:1) N ≥ k k X i =[ k ] (cid:0) ik (cid:1) N ≥ ( 12 ) N +1 . When µ >
0, it comes from Proposition 2.1 part ( ii ) directly and we complete the proof. (cid:3) Proof of Corollary 1.2.
It follows form Corollary 2.3 and Proposition 2.4 directly. (cid:3)
Now we establish the upper bounds, more precisely we will use the following Cheng-Yang’s typeinequality and extend it for µ ≥ µ . Proposition 3.1. [9, Proposition 3.1] Let < ν ≤ ν ≤ · · · be a sequence of numbers satisfyingthe following inequality k X i =1 ( ν k +1 − ν i ) ≤ ̺ k X i =1 ( ν k +1 − ν i ) ν i , (3.1) where ̺ is a positive constant. Then we have ν k +1 ≤ (1 + 2 ̺ ) k ̺ ν . ̺ . Tothis end, we have the following inequalities. Proposition 3.2.
Assume that µ > µ and let { λ µ,i } i ∈ N be the increasing sequence of eigenvaluesof problem (1.1).Then ( i ) for N ≥ and µ > , one has k X i =1 ( λ µ,k +1 − λ µ,i ) ≤ N k X i =1 ( λ µ,k +1 − λ µ,i ) λ µ,i ; (3.2)( ii ) for N ≥ and µ < µ < , one has k X i =1 ( λ µ,k +1 − λ µ,i ) ≤ N − µ µ − µ k X i =1 ( λ µ,k +1 − λ µ,i ) λ µ,i . (3.3) Proof.
The proof for µ = 0 can be found in [14, Theorem 2.1]. Here the difference is to deal withthe Hardy term and we give the proof for reader’s convenience.Let ( λ µ,i , φ i ) be the i -th eigenvalue and eigenfunctions of (1.1) with R Ω φ i φ j dx = δ ij , where δ ij = 0 if i = j and δ ii = 1. Let x = ( x , · · · , x N ) and g = x m with m = 1 , · · · , N define a trialfunction ϕ i by ϕ i := gφ i − k X j =1 a ij φ j , a ij := Z Ω gφ i φ j dx = a ji . By the orthonormality of φ i and the definition of a ij , ϕ i is perpendicular to φ j , that is, for i, j =1 , · · · , k , there holds Z Ω ϕ i φ j dx = 0. Let b ij = Z Ω φ j ∇ g · ∇ φ i dx and then λ µ,j a ij = Z Ω g ( L µ φ j ) φ i dx = Z Ω (cid:16) − φ j ∇ g · ∇ φ i + g ( L µ φ i ) φ j dx (cid:17) = − b ij + λ µ,i a ij , i.e. 2 b ij = ( λ µ,i − λ µ,j ) a ij . (3.4)Note that L µ ϕ i = λ µ,i gφ i − ∇ g · ∇ φ i − k X j =1 a ij λ µ,j φ j . Hence, we infer that k ϕ i k µ = λ µ,i Z Ω ϕ i dx − Z Ω ϕ i ∇ g · ∇ φ i dx = λ µ,i Z Ω ϕ i dx − Z Ω ( g ∇ g ) · ( φ i ∇ φ i ) dx + 2 k X j =1 a ij Z Ω φ j ∇ g · ∇ φ i dx = λ µ,i Z Ω ϕ i dx + 1 + k X j =1 ( λ µ,i − λ µ,j ) a ij , − Z Ω ϕ i ∇ g · ∇ φ i dx = 1 + k X j =1 ( λ µ,i − λ µ,j ) a ij . (3.5)From the Rayleigh-Ritz inequality, we have that λ µ,k +1 Z Ω ϕ i dx ≤ k ϕ i k µ = λ µ,i Z Ω ϕ i dx + 1 + k X j =1 ( λ µ,i − λ µ,j ) a ij , that is ( λ µ,k +1 − λ µ,i ) Z Ω ϕ i dx ≤ k X j =1 ( λ µ,i − λ µ,j ) a ij . (3.6)Multiplying (3.5) by ( λ µ,k +1 − λ µ,i ) and then taking sum on i from 1 through k , we obtain that − k X i =1 ( λ µ,k +1 − λ µ,i ) Z Ω ϕ i ∇ g · ∇ φ i dx = k X i =1 ( λ µ,k +1 − λ µ,i ) + k X i,j =1 ( λ µ,i − λ µ,j )( λ µ,k +1 − λ µ,i ) a ij = k X i =1 ( λ µ,k +1 − λ µ,i ) − k X i,j =1 ( λ µ,k +1 − λ µ,i ) b ij := θ by the symmetry of a ij and the anti-symmetry of b ij . Multiplying (3.6) by ( λ µ,k +1 − λ µ,i ) andthen taking sum on i from 1 through k , we obtain that k X i =1 ( λ µ,k +1 − λ µ,i ) Z Ω ϕ i dx ≤ k X i =1 ( λ µ,k +1 − λ µ,i ) − k X i,j =1 ( λ µ,k +1 − λ µ,i ) b ij = θ. Note that for arbitrary constant d ij (to be determined later) θ = (cid:18) − k X i =1 ( λ µ,k +1 − λ µ,i ) Z Ω ϕ i ∇ g · ∇ φ i dx (cid:19) = 4 (cid:18) k X i =1 Z Ω ( λ µ,k +1 − λ µ,i ) ϕ i ∇ g · ∇ φ i dx − ( λ µ,k +1 − λ µ,i ) Z Ω k X j =1 d ij ϕ i φ j dx (cid:19) ≤ (cid:18) k X i =1 ( λ µ,k +1 − λ µ,i ) Z Ω ϕ i (cid:19)(cid:18) k X i =1 Z Ω (cid:16) ( λ µ,k +1 − λ µ,i ) ∇ g · ∇ φ i − k X j =1 d ij φ j (cid:17) dx (cid:19) ≤ θ Z Ω (cid:16) k X i =1 ( λ µ,k +1 − λ µ,i ) | ∂ m φ i | − k X i,j =1 d ij ( λ µ,k +1 − λ µ,i ) φ j ∇ g · ∇ φ i + ( k X i,j =1 d ij φ j ) (cid:17) dx, then we have that θ ≤ k X i =1 ( λ µ,k +1 − λ µ,i ) Z Ω | ∂ m φ i | dx + 4 (cid:16) − k X i,j =1 d ij ( λ µ,k +1 − λ µ,i ) b ij + k X i,j =1 d ij (cid:17) . d ij = ( λ µ,k +1 − λ µ,i ) b ij , we have that θ = k X i =1 ( λ µ,k +1 − λ µ,i ) − k X i,j =1 ( λ µ,k +1 − λ µ,i ) b ij ≤ k X i =1 ( λ µ,k +1 − λ µ,i ) Z Ω | ∂ m φ i | dx − k X i,j =1 ( λ µ,k +1 − λ µ,i ) b ij . Thus, k X i =1 ( λ µ,k +1 − λ µ,i ) ≤ k X i =1 ( λ µ,k +1 − λ µ,i ) Z Ω | ∂ m φ i | dx. Finally, we take sum on m from 1 through N and obtain that N k X i =1 ( λ µ,k +1 − λ µ,i ) ≤ k X j =1 ( λ µ,k +1 − λ µ,i ) Z Ω |∇ φ i | dx. Therefore we conclude that for µ > N k X i =1 ( λ µ,k +1 − λ µ,i ) < k X i =1 ( λ µ,k +1 − λ µ,i ) (cid:16) Z Ω |∇ φ i | dx + µ Z Ω φ i | x | dx (cid:17) = 4 k X i =1 ( λ µ,k +1 − λ µ,i ) λ µ,i . While for µ < µ <
0, since we have that Z Ω |∇ φ i | dx ≤ − µ µ − µ (cid:16) Z Ω ( |∇ φ i | + µ φ i | x | ) dx (cid:17) = − µ µ − µ λ µ,i and then N k X i =1 ( λ µ,k +1 − λ µ,i ) ≤ = 4 − µ µ − µ k X i =1 ( λ µ,k +1 − λ µ,i ) λ µ,i . We complete the proof. (cid:3)
Proof of Theorem 1.3.
Part ( i ) : Case of µ > . Theorem 1.3 follows Proposition 3.1 and(3.2) with ̺ = N . To be convenient, we sketch the proof here. DenoteΛ k = 1 k k X i =1 λ µ,i , T k = 1 k k X i =1 λ µ,i , F k = (1 + 2 N )Λ k − T k , then we have F k +1 ≤ C ( N, k )( k + 1 k ) N F k , (3.7)where 0 < C ( N, k ) := 1 −
16 ( kk + 1 ) N (1 + N )(1 + N )( k + 1) < , and then F k ≤ C ( N, k − kk − N F k − < ( kk − N F k − ≤ · · · < ( kk − N ( k − k − N · · · ( 21 ) N F = 2 N k N λ µ, .
13n the other hand, (3.7) implies that (cid:16) λ µ,k +1 − NN Λ k (cid:17) ≤ NN F k − NN NN Λ k and then 24 + N λ µ,k +1 + 4 + N N (cid:16) λ µ,k +1 − NN Λ k (cid:17) ≤ NN F k , that means λ µ,k +1 ≤ NN + 2 ( N + 4 N ) F k ≤ (1 + 4 N ) k N λ µ, . Thus, we have that λ µ,k ≤ (cid:0) N (cid:1) k N λ µ, . (3.8) Part ( ii ) : Case of µ ≤ µ < . In this case, the constant N − µ µ − µ > N , which will producea higher order for parameter k from Proposition 3.1. So we shall derive the upper bounds bycomparing with the eigenvalues of Laplacian Dirichlet problem.Let ˜ L µ = L µ − µD − and λ ˜ L µ ,k be the k -th Dirichlet eigenvalue of ˜ L µ . Clearly we have λ ˜ L µ ,k = λ µ,k − µD − . Then for µ ∈ [ µ , h ˜ L µ u, u i < h− ∆ u, u i , ∀ u ∈ C ∞ (Ω)It follows by [3, Theorem 10.2.2] that λ µ,k − µD − ≤ λ ,k , k = 1 , , · · · together with the upper bound in [14, (2.10)] λ ,k ≤ (cid:0) N (cid:1) λ , k N , we imply that λ µ,k ≤ (cid:0) N (cid:1) λ , k N + µD − , k = 1 , , · · · . The proof is complete. (cid:3)
Proof of Corollary 1.4:
Recall that ( λ , , φ , ) is the first Dirichlet eigenvalue and associatedeigenfunctions of Laplacian, then for µ > N ≥ λ µ, ≤ Z Ω |∇ φ , | dx + µ Z Ω φ , | x | dx ≤ λ , + µ k φ , k L ∞ Z Ω | x | − dx, which, together with (3.8), implies that λ µ,k ≤ (cid:0) N (cid:1)(cid:16) λ , + µ k φ , k L ∞ Z Ω | x | − dx (cid:17) k N . For µ > N ≥
2, take test function u = φ , Γ µ where Γ µ ( x ) = | x | τ + ( µ ) with τ + ( µ ) = − N − + √ µ − µ , which satisfying L µ Γ µ = 0 in R N \ { } . Then λ µ, ≤ Z Ω |∇ u | dx + µ Z Ω u | x | dx k u k L (Ω) λ , + Z Ω ∇ φ , · ∇ Γ µ dx Z Ω φ , Γ µ dx ≤ λ , + c − τ + ( µ ) k φ , k C Z Ω | x | τ + ( µ ) − dx Z Ω ρ ( x ) | x | τ + ( µ ) dx , where c > φ , ( x ) ≥ c ρ ( x ) , ∀ x ∈ Ω . Together with (3.8), we obtain that λ µ,k ≤ (cid:0) N (cid:1)(cid:16) λ , + c − τ + ( µ ) k φ , k C Z Ω | x | τ + ( µ ) − dx Z Ω ρ ( x ) | x | τ + ( µ ) dx (cid:17) k N . The proof is complete. (cid:3)
Our proof of Weyl’s limit of eigenvalues relies on estimates of the partition function Z µ ( t ) := ∞ X i =1 e − λ µ,i t , t > , which can be written as Z µ ( t ) = Z ∞ e − βt d N µ ( β ) , (4.1)where N µ ( β ) = P λ µ,i ≤ β Z µ ( t ) = Z Ω p µ, Ω ( x, x, t ) dx, (4.2)where p µ, Ω is the heat kernel of Hardy operators L µ in domain Ω × Ω × (0 , + ∞ ). More propertiesof heat kernel with Hardy potentials could be found in [22, 37]. In the whole space Ω = R N , wedenote p µ = p µ, R N . Particularly, p ( x, y, t ) = p ( x − y, t ) = 1(4 πt ) N e − | x − y | t for ( x, y, t ) ∈ R N × R N × (0 , + ∞ )is the usual heat kernel of the Laplacian in R N .The following lemma concerns the estimate of heat kernel, which is the essential part for theWeyl’s limit. Lemma 4.1.
Let µ ≥ µ and γ µ ( x, y, t ) = p ( x − y, t ) − p µ, Ω ( x, y, t ) for ( x, y, t ) ∈ R N × R N × (0 , + ∞ ) . Then ( i ) For µ > , there exists c > independent of µ such that ≤ γ µ ( x, x, t ) ≤ c µ | x | − t − N +1 , ∀ x ∈ Ω \ { } , ∀ t ∈ (0 , . ii ) For µ ∈ ( µ , , there exists c > such that | γ µ ( x, x, t ) | ≤ c | x | τ + ( µ ) − t − N +1 , ∀ x ∈ Ω \ { } , ∀ t ∈ (0 , . ( iii ) For µ = µ , there exists c > such that | γ µ ( x, x, t ) | ≤ c (cid:16) | x | − N + N +28( N − t − N +1 + | x | − N + t − N + (cid:17) , ∀ x ∈ Ω \ { } , ∀ t ∈ (0 , . Proof.
Case of µ > . Note that elementary properties infer directly that p µ, Ω ( x, y, t ) ≤ p , Ω ( x, y, t ) ≤ p ( x − y, t ) in Ω × Ω × (0 , + ∞ )and for fixed y ∈ Ω, we have that ∂ t γ µ − ∆ γ µ = µ | x | p µ in Ω × (0 , + ∞ ) ,γ µ = 0 on ∂ Ω × (0 , + ∞ ) ,γ µ = 0 in Ω × { } . (4.3)Note that γ µ could be expressed by heat kernel, i.e. for y ∈ Ω \ { } , γ µ ( x, y, t ) = Z t Z Ω p , Ω ( x, z, t − s ) µ | z | p µ, Ω ( z, y, s ) dzds ≤ µ Z t Z R N p ( x − z, t − s ) 1 | z | p ( z − y, s ) dzds ≤ µ (4 π ) N Z t t − s ) N s N Z R N | z | e − | x − z | t − s ) e − | z − y | s dzds. For N ≥ x ∈ Ω \ { } , we have that γ µ ( x, x, t ) ≤ µ (4 π ) N (cid:16) Z t t − s ) N s N Z R N \ B | x | ( x ) | z | e − | x − z | t − s ) e − | z − y | s dzds + Z t t − s ) N s N Z B | x | ( x ) | z | e − | x − z | t − s ) e − | x − z | s dzds (cid:17) < µ (4 π ) N (cid:16) | x | − Z t t − s ) N s N Z R N e − ( t − s + s ) | z | dzds + Z t t − s ) N s N e − | x | t ( t − s ) s Z B | x | (0) | z | dzds (cid:17) ≤ µ (4 π ) N (cid:16) N +2 | x | t − N +1 Z R N e −| ˜ z | d ˜ z + c ω N − N − | x | N − Z t t − s ) N s N ( | x | − t ( t − s ) s ) − N ds (cid:17) = µ (4 π ) N | x | − t − N +1 (cid:16) N +2 Z R N e −| z | dz + c ω N − N − N +2 (cid:17) , where c > e − a ≤ c a − N for all a ≥ . Case of µ ∈ ( µ , . It is known that for y ∈ Ω \ { } , γ µ ( x, y, t ) = Z t Z Ω p , Ω ( x, z, t − s ) µ | z | p µ, Ω ( z, y, s ) dzds.
16t follows from [37, Theorem 3.10] that the corresponding heat kernel verifies that for t ∈ (0 , ≤ p µ, Ω ( x, y, t ) ≤ p µ, R N ( x, y, t ) ≤ c ( | x || y | ) τ + ( µ ) t − N e − c | x − y | t , where c , c > τ + ( µ ) <
0. We see that there is no order for p µ, Ω and p . Thus, for x ∈ Ω \ { } and t ∈ (0 , | γ µ ( x, x, t ) | ≤ | µ | Z t Z Ω p , Ω ( x, z, t − s ) 1 | z | p µ, Ω ( z, x, s ) dzds ≤ c | µ | (4 π ) N | x | τ + ( µ ) Z t t − s ) N s N Z R N | z | τ + ( µ ) − e − | x − z | t − s ) − c | x − z | s dzds ≤ c | µ | (4 π ) N | x | τ + ( µ ) (cid:16) Z t t − s ) N s N Z R N \ B | x | ( x ) | z | τ + ( µ ) − e − | x − z | t − s ) − c | x − z | s dzds + Z t t − s ) N s N Z B | x | ( x ) | z | τ + ( µ ) − e − | x − z | t − s ) − c | x − z | s dzds (cid:17) < c | µ | (4 π ) N | x | τ + ( µ ) (cid:16) ( | x | τ + ( µ ) − Z t t − s ) N s N Z R N e − | x − z | t − s ) − c | x − z | s dzds + Z t t − s ) N s N e − | x | t − s ) − c | x | s Z B | x | | z | τ + ( µ ) − dzds (cid:17) ≤ c | µ | (4 π ) N | x | τ + ( µ ) (cid:16) ( | x | τ + ( µ ) − Z R N e −| z | dz Z t ( c t + (4 − c ) s ) − N ds + ω N − N − | x | N + τ + ( µ ) − c Z t t − s ) N s N (cid:16) | x | t − s ) + c | x | s (cid:17) − N ds (cid:17) ≤ c | µ | (4 π ) N (cid:16) − τ + ( µ ) Z R N e −| z | dz + c ω N − N − N − τ + ( µ )+2 c − N (cid:17) | x | τ + ( µ ) − t − N +1 , where 2 τ + ( µ ) − > − N for µ ∈ ( µ ,
0) and c , c > Case of µ = µ . When µ = µ , the above inequality holds true, but the factor | x | τ + ( µ ) − = | x | − N is non-integrable in Ω. So we have to modify the above estimates.From [22, Theorem 1.1] we have that t ∈ (0 , ≤ p µ, Ω ( x, y, t ) ≤ c ( | x || y | ) − N t − N e − c | x − y | t . We choose r = 14 | x | − N − and θ = 14 , then we have that | γ µ ( x, x, t ) | ≤ c | µ | (4 π ) N | x | − N (cid:16) Z t t − s ) N s N Z R N \ B r ( x ) | z | − N +22 e − | x − z | t − s ) − c | x − z | s dzds + Z t t − s ) N s N Z B r ( x ) | z | − N +22 e − | x − z | t − s ) − c | x − z | s dzds (cid:17) < c | µ | (4 π ) N | x | − N (cid:16) r − N +22 Z t t − s ) N s N Z R N e − | x − z | t − s ) − c | x − z | s dzds + Z t t − s ) N s N e − | x | t − s ) − c | x | s Z B r (0) | z | − N +22 dzds (cid:17) c | µ | (4 π ) N | x | − N (cid:16) r − N +22 Z R N e −| z | dz Z t ( c t + (4 − c ) s ) − N ds + ω N − N − r N − c Z t t − s ) N s N (cid:16) | x | t − s ) + c | x | s (cid:17) − N − θ ds (cid:17) ≤ c | x | − N r − N +22 t − N +1 + c | x | θ − N r N − t − N − θ Z t ( t − s ) − θ s − θ ds = c | x | − N + N +28( N − t − N +1 + c | x | − N + t − N + , where c , c > c > e − a ≤ c a − N − for all a ≥ . This completes the proof. (cid:3)
Proof of Theorem 1.5.
From Lemma 4.1, we see that for t ∈ (0 ,
1) and µ ≥ µ , t N (cid:12)(cid:12)(cid:12) Z Ω p µ, Ω ( x, x, t ) dx − Z Ω p (0 , t ) dx (cid:12)(cid:12)(cid:12) ≤ t N Z Ω | r µ ( x, x, t ) | dx ≤ c t Z Ω max {| x | − , | x | τ + ( µ ) − } dx → t → + , then we obtain that lim t → + t N | Z µ ( t ) − Z ( t ) | = 0 , where Z is the partition function for Laplacian and direct computation shows thatlim t → + t N Z ( t ) = | Ω | (4 π ) N . So there holds lim t → + t N Z µ ( t ) = | Ω | (4 π ) N . (4.4)From (4.4) and Karamata’s Tauberian theorem [38, Theorem 10.3], we get that N µ ( β ) β − N −→ | Ω | Γ( N + 1)(4 π ) N as β → + ∞ , which, taking β = λ µ,k and N ( λ µ,k ) = k , implies thatlim k → + ∞ λ − N µ,k k = | Ω | Γ( N ) N (4 π ) N = | Ω || B | (2 π ) N , (4.5)where | B | = 1 N ω N − = 2 π N N Γ( N ) . Note that (4.5) is equivalent to (1.13) and the proof is completed. (cid:3)
Acknowledgements:
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