Differential Integral Equations 29

Citations From References: 2 From Reviews: 0

MR3513585 45G10 35B38 35J30 35J75 35J91 58J05 Zhu, Meijun [Zhu, Meijun1] (1-OK) Prescribing integral curvature equation. (English summary) Diﬀerential Integral Equations 29 (2016), no. 9-10, 889–904.

n Given any positive, continuous, and antipodally symmetric function R on S , it is proved bαc−n n in this paper that there exists a positive, antipodally symmetric solution u ∈ C (S ) to the following integral equation: Z α−n n+α (1) u(ξ) = R(η)|ξ − η| u(η) n−α dση, n S where α > n. Here, |ξ − η| denotes the chordal distance between ξ and η in Rn+1 and n by antipodally symmetric functions f we mean those f satisfying f(η) = f(−η) on S . Clearly (1) can be thought of as an integral version of the diﬀerential equation

α/2 −1 n+α (2) (−∆) v = (R ◦ π )v n−α n n n on R obtained via the stereographic projection π: S → R . However, (1) and (2) are not equivalent in the case α > n [see Y. S. Choi and X. Xu, J. Differential Equations 246 (2009), no. 1, 216–234; MR2467021; I. A. Guerra, J. Differential Equations 253 (2012), no. 11, 3147–3157; MR2968196; Trinh Viet Duoc and Quôc-AnhNgô,“A´ note on radial 2 −q 3 solutions of ∆ u + u = 0 in R with exactly quadratic growth at infinity”, preprint, arXiv:1511.09171]. To obtain such an existence result for (1), the author seeks critical points of the energy functional ZZ Z n+α α−n 2n n Jα,R(f) = R(ξ)R(η)f(ξ)f(η)|ξ − η| dσξdση f(η) n+α R(η)dση n n n S ×S S n on S . By using a reversed Hardy-Littlewood-Sobolev inequality recently proved by the author and J. B. Dou [see Int. Math. Res. Not. IMRN 2015, no. 19, 9696–9726; MR3431607] (also see [Quôc-AnhNgôand´ Van Hoang Nguyen, “Sharp reversed Hardy- n Littlewood-Sobolev inequality on R ”, preprint, arXiv:1508.02041] for a different proof), it is not hard to see that the functional Jα,R is bounded from below in the set of positive, integrable, and antipodally symmetric functions f. Then by a density argument and variational techniques, the infimum of Jα,R is achieved by a positive and antipodally symmetric function. In the next part of the paper, to pave the way for further research, the author carefully derives several properties, such as the conformal covariant property of the corresponding integral operator of the form Z α−n cn,α |ξ − η| u(η)dσg n S −1 R α−n with c = n |ξ − η| dσg n , for any metric g conformally covariant to the standard n,α S S n n−2 n metric gS on S . Knowing that cn,2|x − y| is the Green function for the Laplacian n ∆ on R with n > 3 and that the Green function Gg of the conformal Laplacian −∆g + n−2 4(n−1) scalg enjoys the conformally covariant property

Ggb(y, x) = φ−1(y)φ−1(x)Gg(y, x) 4/(n−2) if two metrics g and bg are conformal in the sense that bg = φ g, the integral operator Z g α−n [G (y, x)] 2−n u(y)dµg M n deﬁned on any compact Riemannian manifold (M n, g) is also conformally covariant. In the last part of the paper, the author formally proposes the α-curvature Q α,gb problem, which asks whether there exists a function u on a given compact Riemannian n manifold (M , g) of dimension n 6= 2 with positive scalar curvature scalg which solves Z g α−n n+α (3) u(x) = [G (y, x)] 2−n Q (y)u(y) n−α dµ α,gb g M n for a prescribed function Q . It is worth noting that (3) on Rn was studied earlier α,gb by Xu in his pioneering work, where several non-existence results were obtained [see J. Funct. Anal. 247 (2007), no. 1, 95–109; MR2319755]. Qu[U+1ED1]c Anh Ngˆo

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MR3512045 53C21 26D10 58J05 Kombe, Ismail (TR-ITICU); Yener, Abdullah (TR-ITICU) Weighted Hardy and Rellich type inequalities on Riemannian manifolds. (English summary) Math. Nachr. 289 (2016), no. 8-9, 994–1004.

The paper under review is an improvement of the first author’s papers with M. Ozaydın¨ [see Trans. Amer. Math. Soc. 365 (2013), no. 10, 5035–5050; MR3074365; Trans. Amer. Math. Soc. 361 (2009), no. 12, 6191–6203; MR2538592]. In the aforementioned papers, on a complete, non-compact Riemannian manifold (M, g), the Lp-Hardy inequality Z p Z α p C + α + 1 − p α−p p (1) ρ |∇φ| dV > ρ |φ| dV, M p M the Hardy-Poincaré-type inequality Z p Z α+p p C + α + 1 α p (2) ρ |∇ρ · ∇φ| dV > ρ |φ| dV, M p M and the weighted L2-Hardy inequality with two weights Z 2 Z 2 Z 2 α 2 C + α − 1 α φ 1 α |∇δ| 2 (3) ρ |∇φ| dV > ρ 2 dV + ρ 2 φ dV Ω 2 M ρ 4 Ω δ were obtained. In this paper, the authors first provide a new form of (1) involving two weight functions, that is, Z Z Z φ p p p p a(x)|∇φ| dV > b(x)|φ| dV + c(p) a(x)|w| ∇ dV M M M w for suitable functions a, b, and w. Then they improve (2) involving two weight functions with a non-negative remainder term to get Z p Z α+p p C + α + 1 α p aρ |∇ρ · ∇φ| dV > aρ |φ| dV + a remainder term; M p M see Theorem 2.7. Next the authors improve (3) with non-negative remainder terms to obtain Z p Z α 2 C + α + 1 − p α−p p aρ |∇φ| dV > aρ |φ| dV Ω p M Z p c(p) α |∇δ| p + p aρ p |φ| dV + a remainder term; p Ω δ see Theorem 2.8. In the last part of the paper, they obtain improved weighted Rellich- type inequalities with non-negative remainder terms involving two weight functions; see Theorem 3.1. Qu[U+1ED1]c Anh Ngô

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Citations From References: 2 From Reviews: 0

MR3466488 45G10 26A33 26E70 39A13 47H08 Jleli, Mohamed (SAR-RYADS); Mursaleen, Mohammad (6-ALIG); Samet, Bessem (SAR-RYADS) Q-integral equations of fractional orders. (English summary) Electron. J. Diﬀerential Equations 2016, Paper No. 17, 14 pp.

In this paper, the authors provide a sufficient condition for the existence of at least one solution u ∈ C([0, 1]; R) for the following q-integral equation of fractional order: α x(t) = F t, x(a(t)), f(t, x(b(t)))Iq u(·, x(·))(t) α for t ∈ [0, 1], where Iq is the q-fractional integral of order α given by Z t α 1 α−1 Iq h(t) = (t − qs) h(s)dqs. Γq(α) 0 Here, α > 1, q ∈ (0, 1), f, u: [0, 1] × R → R, a, b: [0, 1] → [0, 1], and F : [0, 1] × R × R → R. The approach makes use of a generalized version of Darbo’s fixed-point theorem involving the notion of measures of non-compactness. Qu[U+1ED1]c Anh Ngô

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Citations From References: 1 From Reviews: 0

MR3392897 35J60 35A01 Chen, Tiancong (1-UCSB); Han, Qing [Han, Qing1] (1-NDM) Smooth local solutions to Weingarten equations and σk-equations. (English summary) Discrete Contin. Dyn. Syst. 36 (2016), no. 2, 653–660.

The paper under review concerns the existence of a solution u to the prescribed σk- curvature equation σk(κ1, . . . , κn) = ψ(x), where σk denotes the k-th elementary symmetric polynomial and the κi’s are the principal curvatures of the graph (x, u(x)). ∞ n The main result of the paper states that if ψ is C in a neighborhood of 0 ∈ R , then ∞ the prescribed σk-curvature equation always admits a C -solution in some neighbor- n hood of 0 ∈ R for any 2 6 k 6 n − 1. To prove such a statement, the authors look for a solution u of the form n 1 X u(x) = µ x2 + w(x), 2 i i i=1 where the µi’s are constant and w is a small error. The way to ﬁnd the µi’s follows from the identity σk(µ1, . . . , µn) = ψ(0), while the error term w follows from the clever use of the implicit function theorem. The novelty of the argument is that no sign condition on ψ is assumed; however, for a price, there is less information on how the solution u depends on the prescribed function ψ. Qu[U+1ED1]c Anh Ngˆo

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MR3412397 35J40 35B40 35B44 35J60 35J65 35K35 35K55 58E20 Cooper, Matthew K. (5-NENG-ST) Critical O(d)-equivariant biharmonic maps. (English summary) Calc. Var. Partial Diﬀerential Equations 54 (2015), no. 3, 2895–2919.

Let Ω ⊂ Rd be a bounded domain and N a closed Riemannian manifold which is iso- metrically embedded in Rκ for some κ ∈ N. A function u:Ω → N is called a biharmonic R 2 2 κ map if it is a critical point of the bi-energy functional Ω |∆u| dx where u ∈ H (Ω; R ) with the property u(x) ∈ N for almost all x ∈ Ω. Denote by O(d) the standard group of orthogonal transformations acting on Rd. A function u:Ω ⊂ Rd → Sd is called O(d)- equivariant if R · u(x) = u(Rx) for all R ∈ O(d) and x ∈ Ω, where Rx denotes the d 0 0 d+1 standard group action of O(d) on R and R · y = (Ry , yd+1) if y = (y , yd+1) ∈ R . In the paper under review, the author restricts himself to the case when d = 4, Ω = B4(0, 1) ⊂ R4, and N = S4, the standard 4-sphere. The primary result of the paper states that for an O(4)-equivariant map from B4(0, 1) into S4, if the normal derivative at the boundary vanishes, then there is a limit on the number of times that an O(4)-equivariant biharmonic map from B4(0, 1) into S4 satisfying the same boundary condition can wind around S4; see Theorem 1. However, in contrast to the harmonic case, there are O(4)-equivariant biharmonic maps from B4(0, 1) into S4 that wind around S4 as many times as we want; see Theorem 4. As a consequence of the primary result, the critical O(4)-equivariant biharmonic map heat flow starting from such initial data must blow up in finite time or at infinity. The final result in the paper shows that non-constant O(4)-equivariant biharmonic maps from R4 into S4 are unique up to dilations, reflections through the origin in the domain, and reflections through the plane {x5 = 0} in the co-domain; see Theorem 3. Qu[U+1ED1]c Anh Ngô

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MR3302613 53C44 31B05 35J20 35R35 47D06 76D27 Onodera, Michiaki (J-KYUS2-INM) Geometric ﬂows for quadrature identities. (English summary) Math. Ann. 361 (2015), no. 1-2, 77–106.

Of importance in potential theory is the question of specifying a closed surface Γ associated with a prescribed electric charge density µ in such a way that the uniform electric charge distribution on the surface coincides with the potential in a neighborhood of the infinity as µ does. Mathematically, this problem can be formulated as follows: For N a prescribed finite positive Radon measure µ with compact support in R with N > 2, find an (N − 1)-dimensional closed surface Γ enclosing a bounded domain Ω such that N for each x ∈ R r Ω, there holds Z Z (1) F (x − y) dµ(y) = F (x − y)dHN−1(y), Γ N N−1 where F denotes the fundamental solution of −∆ in R and H denotes the (N − 1)- dimensional Hausdorff measure. Upon extending each harmonic function to a smooth N function with compact support in R , the above identity is equivalent to the following identity: Z Z (2) h dµ = h dHN−1 Γ for all harmonic functions h defined in a neighborhood of Ω, which is also considered as a generalization of the mean value formula for harmonic functions. From this point of view, for a prescribed measure µ, we can ask whether or not there exists some domain Ω such that Z Z (3) h dµ = h dx. Ω We call a closed surface Γ a quadrature surface of µ for harmonic functions if Γ solves (1). Analogously, a domain Ω is called a quadrature domain of µ for harmonic functions if Ω fulfills (3). The existence of quadrature surfaces as well as quadrature domains of a prescribed measure µ has been studied by a number of authors with different approaches. In one way or another, both quadrature surfaces and domains have a variational characterization; hence, they can be proved to exist by solving some partial differential equation. However, the uniqueness of quadrature surfaces fails to hold in general settings, as Henrot pointed out in 1994 by computing the number of connected quadrature surfaces associated with a family of measures depending on parameter t. This failure of the uniqueness property suggests that one needs to study families of surfaces in order to understand the uniqueness issue. From this, it is natural to ask if there exists some flow of quadrature surfaces. In this interesting paper, the author proposes a geometric flow of vn, the growing speed of some family ∂Ω(t) in the outer normal direction n, in terms of solutions of some partial differential equation, which also includes some well-known flows as particular cases, such as the Hele-Shaw flow in fluid mechanics. More precisely, for prescribed N smooth functions f and g defined in R , the author defines ∂g −1 ∂p (4) v := − Hg + f + n ∂n ∂n for all x ∈ ∂Ω(t) where p solves the following problem: ( −∆p = µ for x ∈ Ω(t), ∂g ∂p Hg + f + ∂n p + g ∂n = 0 for x ∈ ∂Ω(t), with the key condition ∂g (5) Hg + f + > 0 ∂n everywhere on the initial surface ∂Ω(0), where H represents the mean curvature of ∂Ω(t) normalized such that it takes N − 1 for the standard unit sphere and µ denotes a finite positive Radon measure with compact support. The main contribution of the paper under review is to show that for given g > 0 and any f, any C3+α family of surfaces {∂Ω(t)}06t

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Citations From References: 0 From Reviews: 0 MR3220975 45E05 26D15 35A23 45M20 Lei, Yutian (PRC-NJN-MSM) Integrability and boundedness of extremal functions of a Hardy-Sobolev type inequality. (English summary) Math. Inequal. Appl. 17 (2014), no. 1, 75–81.

When studying the best constant of the Hardy-Sobolev inequality as well as the more general Caffarelli-Kohn-Nirenberg inequality, one usually considers the following integral equation: Z γ u(y) dy n u(x) = n−α −σ , x ∈ R , n |x − y| |y| R n+α+σ where n > 3, σ 6 0, α + σ > 0, n − α + σ > 0, and γ = n−α−σ . It follows from the author’s joint work with Y. Zhao [see Nonlinear Anal. 75 (2012), no. 4, 1989–1999; s n MR2870892] that if u is a solution of finite energy to the equation, that is, u ∈ L (R ) 2n with s = n−α−σ , then u is radially symmetric and monotone decreasing about the origin t n 1 σ n−α−σ 0. Furthermore, u ∈ L (R ) for all t satisfying t ∈ (− n , n ) and if α + sσ > 0 then u is bounded. In this paper, the author first improves the above result by showing that any solution t n 1 n−α−σ u of finite energy does actually belong to L (R ) for all t ∈ (0, n ). To achieve that goal, the author uses a blend of bootstrap arguments and the weighted Hardy- Littlewood-Sobolev inequality. Using this finding and the fact that any solution u of finite energy is radially symmetric, the author claims that u is in fact bounded without assuming the condition α + sσ > 0. Qu[U+1ED1]c Anh Ngô

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MR3208450 58J05 26D15 31C12 35A23 35J20 35R01 D’Ambrosio, Lorenzo (I-BARI); Dipierro, Serena (I-SISSA-MA) Hardy inequalities on Riemannian manifolds and applications. (English summary) Ann. Inst. H. Poincar´eAnal. Non Lin´eaire 31 (2014), no. 3, 449–475.

The principal result in the paper under review is the derivation of a simple criterion under which the following weighted Hardy inequality holds: Z p Z |u| p p c p |∇ρ| dvg 6 |∇u| dvg, M ρ M ∞ for any u ∈ C0 (M). Here (M, g) is a complete N-dimensional Riemannian manifold and ρ is a weight function on M. More precisely, given an open subset Ω ⊂ M, the authors prove that p Z p Z p − 1 |u| p p p |∇ρ| dvg 6 |∇u| dvg p Ω ρ Ω ∞ 1,p p holds for any u ∈ C0 (Ω), provided the weight 0 6 ρ ∈ Wloc (Ω) satisfies (|∇ρ|/ρ) ∈ 1 Lloc(Ω) and −∆pρ > 0 on Ω in the weak sense. The key point in their proof is to establish a similar Hardy inequality which now involves locally integrable vector fields (see Lemma 2.10). Then a clever choice for such vector fields will give a Hardy inequality. In the last part of the paper and as by-products, various inequalities such as Caccioppoli-type, weighted Gagliardo-Nirenberg, and the uncertain principle are obtained. The question of the best constant is also mentioned. Particular examples, further extensions, and applications are also considered at the end of this paper. Qu[U+1ED1]c Anh Ngô

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MR3160825 53C44 53C21 Ma, Bingqing [Ma, Bing Qing] (PRC-HNO) Some evolution equations under the List’s ﬂow and their applications. (English summary) Comment. Math. Univ. Carolin. 55 (2014), no. 1, 41–52.

Let (M, g) be a closed n-dimensional Riemannian manifold and ϕ a smooth function on M. The author studies the following extended Ricci flow: ∂ g = −2(Ric − α∂ ϕ∂ ϕ) + 2(ar + b)g , (1) t ij g i j ij ∂tϕ = ∆gϕ, where α > 0, a and b are constants and r is the average of the trace of the tensor Ricg − αdϕ ⊗ dϕ commonly known as the generalized Ricci curvature. When a = b = 0, System (1) was studied by B. List in his Ph.D. thesis [Evolution of an extended Ricci flow system, Freie Univ., Berlin, 2006], which somehow indicates that stationary points of the flow correspond to the static Einstein vacuum equation. In the first part of the paper under review, the author derives some evolution equations for the generalized Ricci curvature Ricg − αdϕ ⊗ dϕ as well as for its trace, called 2 the generalized Scalar curvature, that is Scalg − α|dϕ|g. As an application, the author obtains L2-estimates for the generalized scalar curvature and the first variational formula for non-negative eigenvalues of operators involving Laplacian. Finally, some mistakes in [Z. M. Qian, Bull. Sci. Math. 133 (2009), no. 2, 145–168; MR2494463] are pointed out. Correct versions for those mistaken statements are also considered in this paper. Qu[U+1ED1]c Anh Ngô

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MR3117382 35R01 35A01 35A02 35B40 35J61 58J05 Wu, Yen-Lin (RC-NATC); Chen, Zhi-You (RC-NATC); Chern, Jann-Long (RC-NATC); Kabeya, Yoshitsugu (J-OSAKPE) Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space. (English summary) Commun. Pure Appl. Anal. 13 (2014), no. 2, 949–960.

In the paper under review, the authors study the existence and uniqueness of “radial solutions” of the following semilinear elliptic equation: p−1 ∆Hn u − λu + |u| u = 0 n n on H r {Q}, where H is the hyperbolic space of dimension n > 10 given by n 2 2 2 H = (x1, . . . , xn, xn+1): x1 + ··· + xn − xn+1 = −1 , n Q = (0,..., 0, 1) ∈ H , p > 1, λ > 0 is a parameter, and ∆Hn is the Laplace-Beltrami operator on Hn. By “radial solutions” we mean those which depend only on xn+1 and which then verify the following ODE:

1 n−1 p−1 (1) (sinh t)ut − λu + |u| u = 0, t > 0. sinhn−1t t

Here, the parameter t and the point (x1, . . . , xn, xn+1) are related through the following condition: 2 2 2 2 2 sinh t = x1 + ··· + xn, cosh t = xn+1. The main result of the paper basically says that if the dimension n > 10 and the power p > p?, where p? is the Joseph-Lundgren exponent given by q (n − 2)2 − 4n + 4 n2 − (n − 2)2 , (n − 2)(n − 10) then for any λ > 0, equation (1) possesses a unique positive solution u∞ which converges to λ1/(p−1) as t → ∞. The analysis of the paper is rather involved. To construct a solution for (1), the authors ﬁrst consider u(t, α), the unique solution of the following initial value problem:

1 n−1 p−1 ((sinh t)ut)t − λu + |u| u = 0, t > 0, sinhn−1t u(0) = α > 0. Then, through several steps regarding the proﬁle of u(t, α) plus the Ascoli-Arzela theorem, the solution u∞ of (1) is found as the locally uniform limit of u(t, αk) as k → ∞ for some sequence (αk)k with αk → +∞. The uniqueness of u∞ is guaranteed since the limit lim(sinh2/(p−1)tu(t)) t→0 exists and depends only on p. Qu[U+1ED1]c Anh Ngˆo

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MR3096597 58J05 35R01 58J60 Barthelm´e,Thomas (CH-NCH) A natural Finsler-Laplace operator. (English summary) Israel J. Math. 196 (2013), no. 1, 375–412.

In the long paper under review, the author gives a new definition for the usual Laplace- Beltrami operator on Riemannian manifolds for Finsler metrics on Finsler manifolds, which are known as nontrivially generalized Riemannian manifolds in the sense that these are not necessarily infinitesimally Euclidean. Given a smooth manifold M of dimension n, unlike the Riemannian context where the Laplace-Beltrami operator ∆ can be defined in terms of all second directional derivatives in orthogonal directions, there is no suitable notion of orthogonality on Finsler manifolds. The key point in the author’s construction is to introduce a suitable angle measure αF associated to the Finsler metric F that allows him to define a Finsler- Laplace operator ∆F as an average with regard to an angle measure of the second directional derivatives. Using this approach, for given f ∈ C2(M), he formally defines ∆F acting on f as follows: Z 2 F d f F ∆ f(x) = cn 2 (cξ(t)) αx (ξ), 1 dt ξ∈Tx M t=0 F where cn is a normalizing constant chosen in such a way that ∆ coincides with the 1 usual Laplace-Beltrami operator ∆ whenever F is Riemannian, Tx M is the unit tangent F bundle over x, cx is the geodesic leaving x in the direction ξ, and α is the conditional on the fibers of the canonical volume form on T 1M. The first and main result in the paper under review is the following theorem. Theorem A. Let F be a Finsler metric on M. Then ∆F is a second-order differential operator. Furthermore, (i) ∆F is elliptic. (ii) ∆F is symmetric, i.e., for any f, g ∈ C∞(M), Z f∆F g − g∆F fΩF = 0. M (iii) Therefore, ∆F is unitarily equivalent to a Schrödingeroperator. (iv) ∆F coincides with the Laplace-Beltrami operator when F is Riemannian. Having ∆F in hand, the author then studies the energy functional associated with ∆F as well as its Rayleigh quotient and the spectrum of ∆F . It turns out that on compact Finsler manifolds, the set of eigenvalues of −∆F and its corresponding set of eigenfunctions behaves exactly the same as that for the compact Riemannian case (see Theorem 4.5). Interestingly, by a direct computation, it holds that the Finsler-Laplace ∆F is conformal invariant in dimension two (see Theorem 4.9). In the last part of the paper, the author gives explicit representations of the Finsler- F 2 Laplace operator ∆ and its spectrum only for Katok-Ziller metrics on the 2-torus T 2 and on the 2-sphere S (see Theorems 5.6 and 5.8). Proofs and calculations necessary for Theorems 5.6 and 5.8 are very clear and precise. In addition to this part, the author F 2 claims that the smallest nonzero eigenvalue of −∆ on S with respect to the Katok- 2 Ziller metrics is 8π/volΩ(S ) (see Theorem B). Qu[U+1ED1]c Anh Ngô

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MR3074365 58J60 26D10 26D15 53C21 Kombe, Ismail (TR-ITICUS-M); Ozaydin,¨ Murad [Ozaydın,¨ Murad] (1-OK) Hardy-Poincar´e,Rellich and uncertainty principle inequalities on Riemannian manifolds. (English summary) Trans. Amer. Math. Soc. 365 (2013), no. 10, 5035–5050.

This short paper is a continuation of the authors’ previous study on basic inequalities on a complete non-compact Riemannian manifold M of dimension n > 1 such as the Hardy, Rellich, Hardy-Poincaré,and Heisenberg-Pauli-Weyl inequalities [see Trans. Amer. Math. Soc. 361 (2009), no. 12, 6191–6203; MR2538592]. In the previous study, improved Hardy, Rellich, and Lp-uncertainty principle inequalities were obtained on M; see Theorems 2.3–2.5 and Corollary 2.1. Besides, sharp improved Hardy and Rellich inequalities were also derived when working on the hyper- n bolic space H instead of M; see Theorems 3.1 and 3.3. In the paper under review, the authors prove new weighted Hardy-Poincaréand Rellich type inequalities as well as an improved version of the L2-uncertainty principle inequality on M. In the first part of the paper, the authors derive a Hardy-Poincaré type inequality on general complete non-compact Riemannian manifolds with a weight function ρ modeling some distance from a fixed point. Their first result is the following: Theorem 2.1 (Hardy-Poincarétype inequality). Let M be a complete non-compact Riemannian manifold of dimension n > 1. Let ρ be a nonnegative function on M such C that |∇ρ| = 1 and ∆ρ > ρ in the sense of distribution, where C > 0. Then the following inequality holds: Z p Z α+p p C + α + 1 α p ρ |∇ρ · ∇φ| dV > ρ |φ| dV M p M ∞ −1 for all compactly supported smooth functions φ ∈ C0 (M r ρ {0}), 1 < p < ∞, and C + α > −1. n In particular, when M = H , the following sharp version of Theorem 2.1 is obtained: n ∞ n Theorem 2.2 (Sharp Hardy-Poincarétype inequality on H ). Let φ ∈ C0 (H ), d := 1+|x| d(0, x) = log 1−|x| , n > 2, 1 < p < ∞ and α > −n. Then we have Z p Z α+p p n + α α p n n d |∇H d · ∇H φ| dV > d |φ| dV, n p n H H n+α p where the constant p is sharp. Next, the authors prove an L2-weighted Hardy inequality involving two weight functions ρ and δ modeling distance functions from a fixed point and from the boundary of a domain with smooth boundary: Theorem 2.3 (Weighted L2-Hardy inequality with two weights). Let M be a complete non-compact Riemannian manifold of dimension n > 1. Let ρ and δ be nonnegative C 1−C functions on M such that |∇ρ| = 1, ∆ρ > ρ and −div(ρ ∇δ) > 0 in the sense of distribution, where C > 1. Then we have Z 2 Z 2 Z 2 α 2 C + α − 1 α φ 1 α |∇δ| 2 ρ |∇φ| dV > ρ 2 dV + ρ 2 φ dV Ω 2 M ρ 4 Ω δ ∞ −1 for all φ ∈ C0 (Ω r ρ {0}), α ∈ R, and C + α − 1 > 0. n Then the authors obtain a sharp weighted Hardy inequality on H with a weighted Sobolev term: n Theorem 2.4 (Sharp weighted Hardy inequality on H with a weighted Sobolev term). ∞ n 1+|x| Let φ ∈ C0 (H ), d = log 1−|x| , α ∈ R, n > 2 and n + α − 2 > 0. Then we have Z 2 Z 2 Z 2/q α 2 n + α − 2 α φ (2−n)(2−q)+αq q n 2 d |∇H φ| dV > d 2 dV + c d φ dx , n 2 n d e n H H H n−2 n 2n 2 |S | 1−2/q n+α−2 2 where 2 6 q 6 n−1 + 2ε, ε > 0, ec = c2 2 and the constant 2 is sharp. In the next part of the paper, an improved Rellich inequality is obtained. First, the authors prove the following: Theorem 3.1 (Weighted Rellich inequality). Let M be a complete Riemannian manifold of dimension n > 1. Let ρ be a nonnegative function on M such that |∇ρ| = 1 and C ∆ρ > ρ in the sense of distribution, where C > 0. Then the following inequality holds: Z 2 Z 2 α 2 C + 1 − α α |∇φ| ρ |∆φ| dV > ρ 2 dV M 2 M ρ ∞ −1 7−C for all compactly supported smooth functions φ ∈ C0 (M r ρ {0}), 3 < α < 2. n Then, in the case M = H , the authors are also able to derive a sharp version of Theorem 3.1 as follows: n ∞ n Theorem 3.2 (Sharp weighted Rellich inequality on H ). Let φ ∈ C0 (H ), d = 1+|x| 8−n log 1−|x| , n > 2, and 3 < α < 2. Then we have Z 2 Z 2 α 2 n − α α |∇ n φ| n H d |∆H φ| dV > d 2 dV. n 2 n d H H Replacing M by a bounded domain with smooth boundary, the authors also prove the following result: Theorem 3.3 (Weighted Rellich inequality on bounded domains). Let Ω be a bounded domain with smooth boundary ∂Ω in a complete Riemannian manifold of dimen- C sion n > 1. Let ρ be a nonnegative function on M such that |∇ρ| = 1, ∆ρ > ρ and 1−C −div(ρ ∇δ) > 0 in the sense of distribution, where C > 1. Then the following inequality is valid: Z 2 Z 2 Z 2 α 2 C + 1 − α α |∇φ| α−2 |∇δ| 2 ρ |∆φ| dV > ρ 2 dV + K(C, α) ρ 2 φ dV Ω 2 Ω ρ Ω δ ∞ −1 7−C for all compactly supported smooth functions φ ∈ C0 (M r ρ {0}), 3 < α < 2 and 1 K(C, α) = 16 (C + 1 − α)(C + 3α − 7). Finally, an improved Heisenberg-Pauli-Weyl inequality, known as the uncertainty principle inequality, is considered. For example, in the context of general Riemannian manifolds, the authors prove the following result, which has a better constant than that of Corollary 2.1 in their previous work [op. cit.]: Theorem 4.1 (Heisenberg-Pauli-Weyl inequality). Let M be a complete Riemannian manifold of dimension n > 2. Let ρ be a nonnegative function on M such that |∇ρ| = C 1 and ∆ρ > ρ in the sense of distribution, where C > 0. Then the following inequality holds: Z Z 2 Z 2 α 2 2 C + 1 2 ρ φ dV |∇φ| dV > φ dV M M 2 M ∞ for all compactly supported smooth functions φ ∈ C0 (M). n While in the context of the hyperbolic space H , the following sharp version of Theorem 4.1 is obtained: n ∞ n Theorem 4.2 (Sharp Heisenberg-Pauli-Weyl inequality on H ). Let φ ∈ C0 (H ), d = 1+|x| log 1−|x| and n > 2. Then

Z Z 2 Z 2 2 2 2 n 2 n d φ dV |∇H φ| dV > φ dV . n n 2 n H H H 2 Moreover, equality holds if φ = Ae−αd , where A ∈ , α = n−1 n − 1 + 2π Cn−2 and R n−2 Cn R −αd2 Cn = n e dV . H Together with Theorem 3.1, the authors are able to derive a Heisenberg-Pauli-Weyl type inequality involving ∆ as follows: Theorem 4.3 (Second-order Heisenberg-Pauli-Weyl inequality). Let M be a complete Riemannian manifold of dimension n > 2. Let ρ be a nonnegative function on M such C that |∇ρ| = 1 and ∆ρ > ρ in the sense of distribution, where C > 7. Then the following inequality holds: Z Z 4 Z 2 4 2 2 C + 1 2 ρ φ dV |∆φ| dV > φ dV M M 2 M ∞ −1 for all compactly supported smooth functions φ ∈ C0 (M r ρ {0}). Qu[U+1ED1]c Anh Ngˆo

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Citations From References: 2 From Reviews: 0

MR3057347 53C44 58J50 Zhao, Liang [Zhao, Liang7] (PRC-NAA) The ﬁrst eigenvalue of p-Laplace operator under powers of the mth mean curvature ﬂow. (English summary) Results Math. 63 (2013), no. 3-4, 937–948.

In the paper under review, the author studies the first eigenvalue, denoted by λ1,p(t), g(t) g(t) p−2 of the p-Laplace operator −∆p given by −∆p = divg(t)(|d · |g(t) d·), on compact Riemannian manifolds (M, g(t)) without boundary of dimension n > 3. The author proves that if g(t) satisfies some unnormalized powers of the mth mean curvature flow and if at the initial time t = 0 the mean curvature vector and the metric satisfy the following inequality: 1 1 h εHg, ε ∈ , , > p n then λ1,p(t) is non-decreasing and differentiable almost everywhere along the flow. Qu[U+1ED1]c Anh Ngô

References

1. Cabezas-Rivas, E., Sinestrai, C.: Volume-preserving flow by powers of the mth mean curvature. Calc. Var. 38, 441–469 (2010) MR2647128 2. Li, J.-F.: Eigenvalues and energy functionals with monotonicity formulae under Ricci flow, Math. Ann. 388, 927–946 (2007) MR2317755 3. Ma, L.: Eigenvalue monotonicity for the Ricci flow. Ann. Glob. Anal. Geom. 337(2), 435–441 (2006) MR2248073 4. Ling, J.: A class of monotonic quantities along the Ricci flow, Arxiv: math.DG/0710.4291v2 5. Ni, L.: The entropy formula for linear heat equation. J. Geom. Anal. 14, 369–374 (2004) MR2051693 6. Cao, X.-D.: First eigenvalues of geometric operators under the Ricci flow. Proc. AMS 136, 4075–4078 (2008) MR2425749 R 7. Cao, X.-D.: Eigenvalues of (−∆ + 2 ) on manifolds with nonnegative curvature operator. Math. Ann. 337(2), 435–441 (2007) MR2262792 8. Wu, J.-Y.: First eigenvalue monotonicity for the p-Laplace operator under the Ricci flow. Acta Math. Sin.(Engl. Ser.) 27(8), 1591–1598 (2011) MR2822831 9. Wu, J.-Y., Wang, E.-M., Zheng, Y.: First eigenvalue of the p-Laplace operator along the Ricci flow. Ann. Glob. Anal. Geom. 38, 27–55 (2010) MR2657841 Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors.

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MR2958348 58C40 53C21 Wang, Lin Feng (PRC-NTU-SSC) Eigenvalue estimate for the weighted p-Laplacian. (English summary) Ann. Mat. Pura Appl. (4) 191 (2012), no. 3, 539–550.

On an n-dimensional closed Riemannian manifold (M, g), by a theorem due to Lich- nerowicz, it is well known that if the Ricci curvature of M is bounded from below by Ric > (n − 1)K for some constant K > 0, then the first nonzero eigenvalue of the Laplace-Beltrami operator ∆ on M given by divg(∇) must satisfy λ > nK. In the presence of some smooth (but fixed) function h on M, instead of using the √ p−2 volume form dV (x) = det g dx and the p-Laplacian ∆p given by ∆p = divg(|∇·| ∇·), one can consider the weighted volume form dµ(x) = eh(x)dV (x) and the weighted p- p−2 Laplacian ∆µ,p given by ∆µ,p = ∆p + |∇ · | ∇ · ∇h. When considering the volume form dµ, the m-dimensional Bakry-Emery´ curvature ∇h ⊗ ∇h Ric = Ric − Hess h − m m − n is often used to replace the Ricci curvature with m > n, and m = n if and only if h is constant. As in the case of ∆p, the first nonzero eigenvalue of ∆µ,p can be characterized by the following minimizing problem: Z Z Z p p p−2 λµ,p = inf |∇u| dµ ; |u| dµ = 1, |u| udµ = 0 . 1,p u∈W M M M

In the paper under review, the author obtains some lower bound estimates of λµ,p for ∆µ,p when the m-dimensional Bakry-Emery´ curvature has a positive lower bound. To be precise, the author ﬁrst proves the following result: Theorem 1.1. Let (M, g) be an n-dimensional closed manifold and dµ = ehdV be the weighted measure, where h is a smooth function. If the m-dimensional Bakry-Emery´ curvature on M is bounded by

Ricm > (m − 1)K for some constant K > 0, then for p > 2, the ﬁrst nonzero eigenvalue of ∆µ,p is bounded by λµ,p > C(m, p, K) with p−1 1− p p 2−p − p p C(m, p, K) = 2 m 2 (m − 1) 2 (p − 1) p 2 K 2 p q 2 × (p − 1)m + 2(p − 2) + m2(p − 1)2 − 4m(p − 2)

p−2 q 2 × m(p2 − 3p + 2) + 2(p − 2)2 + p m2(p − 1)2 − 4m(p − 2)

q 1−p × m(p − 1) + 2(p − 3) + m2(p − 1)2 − 4m(p − 2) .

It turns out that Theorem 1.1 also holds for p = 2 as the author proves the following result: Theorem 1.2. Let (M, g) be an n-dimensional closed manifold and dµ = ehdV be the weighted measure, where h is a smooth function. If the m-dimensional Bakry-Emery´ curvature on M is bounded by

Ricm > (m − 1)K for some constant K > 0, then the ﬁrst nonzero eigenvalue of ∆µ is bounded by

λµ > mK. In particular, if h is constant, we recover Lichnerowicz’s theorem. The proofs of Theorems 1.1 and 1.2 make use of Theorem 3.1 in the paper under review which provides a more reﬁned estimate of λµ,p from below. Qu[U+1ED1]c Anh Ngˆo

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