Sharp Estimates Research Articles

Eigenvalue estimates for p-Laplace problems on domains expressed in Fermi coordinates

We prove explicit and sharp eigenvalue estimates for Neumann p-Laplace eigenvalues in domains that admit a representation in Fermi coordinates. More precisely, if γ denotes a non-closed curve in R2 symmetric with respect to the y-axis, let D⊂R2 denote the domain of points that lie on one side of γ and within a prescribed distance δ(s) from γ(s) (here s denotes the arc length parameter for γ). Write μ1odd(D) for the lowest nonzero eigenvalue of the Neumann p-Laplacian with an eigenfunction that is odd with respect to the y-axis. For all p>1, we provide a lower bound on μ1odd(D) when the distance function δ and the signed curvature k of γ satisfy certain geometric constraints. In the linear case (p=2), we establish sufficient conditions to guarantee μ1odd(D)=μ1(D). We finally study the asymptotics of μ1(D) as the distance function tends to zero. We show that in the limit, the eigenvalues converge to the lowest nonzero eigenvalue of a weighted one-dimensional Neumann p-Laplace problem.

Asymptotic properties of an optimal principal Dirichlet eigenvalue arising in population dynamics

We consider a shape optimization problem related to the persistence threshold for a biological species, the unknown shape corresponding to the zone of the habitat which is favorable to the population. Analytically, this translates in the minimization of a weighted eigenvalue of the Dirichlet Laplacian, with respect to a bang-bang indefinite weight. For such problem, we provide a full description of the singularly perturbed regime in which the volume of the favorable zone vanishes, with particular attention to the interplay between its location and shape.First, we show that the optimal favorable zone shrinks to a connected, nearly spherical set, in C1,1 sense, which aims at maximizing its distance from the lethal boundary. Secondly, we show that the spherical asymmetry of the optimal favorable zone decays exponentially, with respect to a negative power of its volume, in the C1,α sense, for every α<1. This latter property is based on sharp quantitative asymmetry estimates for the optimization of a weighted eigenvalue problem on the full space, of independent interest.

The nonlinear stability of (n + 1)-dimensional FLRW spacetimes

We prove nonlinear Lyapunov stability of a family of “([Formula: see text])”-dimensional cosmological models of general relativity locally isometric to the Friedman–Lemaître–Robertson–Walker (FLRW) spacetimes including a positive cosmological constant. In particular, we show that the perturbed solutions to the Einstein–Euler field equations around a class of spatially compact FLRW metrics (for which the spatial slices are compact negative Einstein spaces in general and hyperbolic for the physically relevant [Formula: see text] case) arising from regular Cauchy data remain uniformly bounded and decay to a family of metrics with constant negative spatial scalar curvature. To accomplish this result, we employ an energy method for the coupled Einstein–Euler field equations in constant mean extrinsic curvature spatial harmonic (CMCSH) gauge. In order to handle Euler’s equations, we construct energy from a current that is similar to the one derived by Christodoulou [The Formation of Shocks in 3-dimensional Fluids, EMS Monographs in Mathematics, Vol. 2 (European Mathematical Society, 2007)] (and which coincides with Christodoulou’s current on the Minkowski space) and show that this energy controls the desired norm of the fluid degrees of freedom. The use of a fluid energy current together with the CMCSH gauge condition casts the Einstein–Euler field equations into a coupled elliptic–hyperbolic system. Utilizing the estimates derived from the elliptic equations, we first show that the gravity-fluid energy functional remains uniformly bounded in the expanding direction. Using this uniform boundedness property, we later obtain sharp decay estimates if a positive cosmological constant [Formula: see text] is included, which confirms that the accelerated expansion of the physical universe that is induced by the positive cosmological constant is sufficient to control the nonlinearities in the case of small data. A few physical consequences of this stability result are discussed.

Certain sharp estimates of Ozaki close-to-convex functions

The goal of this paper is to establish the sharp estimates on coefficient functionals like Hermitian–Toeplitz determinant of second-order involving logarithmic coefficients, initial logarithmic inverse coefficients as well as sharp bounds on initial order Schwarzian derivatives of the Ozaki close-to-convex functions.

Sharp estimates for distinguished random walks on affine buildings of type [formula omitted]

We study a distinguished random walk on affine buildings of type A˜r , which was already considered by Cartwright, Saloff-Coste and Woess. In rank r=2, it is the simple random walk and we obtain optimal global bounds for its transition density (same upper and lower bound, up to multiplicative constants). In the higher rank case, we obtain sharp uniform bounds in fairly large space–time regions which are sufficient for most applications.

A Sharp Estimate for the Genus of Embedded Surfaces in the 3-Sphere

By refining the volume estimate of Heintze and Karcher [11], we obtain a sharp pinching estimate for the genus of a surface in S3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb S^{3}$$\\end{document}, which involves an integral of the norm of its traceless second fundamental form. More specifically, we show that if g is the genus of a closed orientable surface Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document} in a 3-dimensional orientable Riemannian manifold M whose sectional curvature is bounded below by 1, then 4π2g(Σ)≤22π2-|M|+∫Σf(|A∘|)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4 \\pi ^{2} g(\\Sigma ) \\le 2\\left( 2 \\pi ^{2}-|M|\\right) +\\int _{\\Sigma } f(|{\\mathop {A}\\limits ^{\\circ }}|)$$\\end{document}, where A∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathop {A}\\limits ^{\\circ }} $$\\end{document} is the traceless second fundamental form and f is an explicit function. As a result, the space of closed orientable embedded minimal surfaces Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document} with uniformly bounded ‖A‖L3(Σ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert A\\Vert _{L^3(\\Sigma )}$$\\end{document} is compact in the Ck\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^k$$\\end{document} topology for any k≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 2$$\\end{document}.

Schwarz-Pick lemma for (α,β)-harmonic functions in the unit disc

We obtain Schwarz–Pick lemma for (α,β)-harmonic functions u in the disc, where α and β are complex parameters satisfying ℜα+ℜβ>−1. We prove sharp estimate of ‖Du(0)‖ for such functions in terms of Lp norm of the boundary function. Also, we give asymptotically sharp estimate of ‖Du(z)‖. This estimate is generalized to higher order derivatives. Our results extend earlier results for α-harmonic and Tα-harmonic functions.

Leveraging a realistic synthetic database to learn Shape-from-Shading for estimating the colon depth in colonoscopy images

Colonoscopy is the choice procedure to diagnose, screening, and treat the colon and rectum cancer, from early detection of small precancerous lesions (polyps), to confirmation of malign masses. However, the high variability of the organ appearance and the complex shape of both the colon wall and structures of interest make this exploration difficult. Learned visuospatial and perceptual abilities mitigate technical limitations in clinical practice by proper estimation of the intestinal depth. This work introduces a novel methodology to estimate colon depth maps in single frames from monocular colonoscopy videos. The generated depth map is inferred from the shading variation of the colon wall with respect to the light source, as learned from a realistic synthetic database. Briefly, a classic convolutional neural network architecture is trained from scratch to estimate the depth map, improving sharp depth estimations in haustral folds and polyps by a custom loss function that minimizes the estimation error in edges and curvatures. The network was trained by a custom synthetic colonoscopy database herein constructed and released, composed of 248400 frames (47 videos), with depth annotations at the level of pixels. This collection comprehends 5 subsets of videos with progressively higher levels of visual complexity. Evaluation of the depth estimation with the synthetic database reached a threshold accuracy of 95.65%, and a mean-RMSE of 0.451cm, while a qualitative assessment with a real database showed consistent depth estimations, visually evaluated by the expert gastroenterologist coauthoring this paper. Finally, the method achieved competitive performance with respect to another state-of-the-art method using a public synthetic database and comparable results in a set of images with other five state-of-the-art methods. Additionally, three-dimensional reconstructions demonstrated useful approximations of the gastrointestinal tract geometry. Code for reproducing the reported results and the dataset are available at https://github.com/Cimalab-unal/ColonDepthEstimation.

Non-symmetric stable processes: Dirichlet heat kernel, Martin kernel and Yaglom limit

We study a d-dimensional non-symmetric strictly α-stable Lévy process X, whose spherical density is bounded and bounded away from the origin. First, we give sharp two-sided estimates on the transition density of X killed when leaving an arbitrary κ-fat set. We apply these results to get the existence of the Yaglom limit for arbitrary κ-fat cone. In the meantime we also obtain the spacial asymptotics of the survival probability at the vertex of the cone expressed by means of the Martin kernel for Γ and its homogeneity exponent. Our results hold for the dual process X̂, too.

FIRST EIGENVALUE CHARACTERISATION OF CLIFFORD HYPERSURFACES AND VERONESE SURFACES

Abstract We give a sharp estimate for the first eigenvalue of the Schrödinger operator $L:=-\Delta -\sigma $ which is defined on the closed minimal submanifold $M^{n}$ in the unit sphere $\mathbb {S}^{n+m}$ , where $\sigma $ is the square norm of the second fundamental form.

Estimating potential functions with applications to elliptic problems in half space

We first derive sharp estimates on some potential functions on the half space 0 \\} $ ]]> R + n := { x = ( x 1 , … , x n ) ∈ R n ; x n > 0 } , n ≥ 3 . Then, these estimates are applied to demonstrate the existence of a positive solution with an explicit asymptotic behavior for a semi-linear elliptic problem in the half space. Our approach is founded on Karamata's theory for regularly varying functions, complemented by the use of monotonicity methods.

Asymptotically sharp estimates for the area of multiplexers in the cellular circuit model

Abstract A general cellular circuit of functional and switching elements (CCFSE) is a mathematical model of integral circuits (ICs), which takes into account peculiarities of their physical synthesis. A principal feature of this model distinguishing it from the well-known classes of circuits of gates (CGs) is the presence of additional requirements on the geometry of the circuit which ensure the accounting of the necessary routing resources for IC creation. The complexity of implementation of a multiplexer function of Boolean algebra (FBA) in different classes of circuits has been extensively studied. In the present paper, we give asymptotically sharp upper and lower estimates for the area of a CCFSE implementing a multiplexer FBA of order n. We construct a family of circuit multiplexers of order n of area equal to the halved upper estimate, and provide a method of delivering the corresponding lower estimate.

Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces

This paper deals with local criteria for the convergence to a global minimiser for gradient flow trajectories and their discretisations. To obtain quantitative estimates on the speed of convergence, we consider variations on the classical Kurdyka–Łojasiewicz inequality for a large class of parameter functions. Our assumptions are given in terms of the initial data, without any reference to an equilibrium point. The main results are convergence statements for gradient flow curves and proximal point sequences to a global minimiser, together with sharp quantitative estimates on the speed of convergence. These convergence results apply in the general setting of lower semicontinuous functionals on complete metric spaces, generalising recent results for smooth functionals on R n \mathbb {R}^n . While the non-smooth setting covers very general spaces, it is also useful for (non)-smooth functionals on R n \mathbb {R}^n .

Sharp spectral gap estimates for higher-order operators on Cartan–Hadamard manifolds

The goal of this paper is to provide sharp spectral gap estimates for problems involving higher-order operators (including both the clamped and buckling plate problems) on Cartan–Hadamard manifolds. The proofs are symmetrization-free — thus no sharp isoperimetric inequality is needed — based on two general, yet elementary functional inequalities. The spectral gap estimate for clamped plates solves a sharp asymptotic problem from [Q.-M. Cheng and H. Yang, Universal inequalities for eigenvalues of a clamped plate problem on a hyperbolic space, Proc. Amer. Math. Soc. 139(2) (2011) 461–471] concerning the behavior of higher-order eigenvalues on hyperbolic spaces, and answers a question raised in [A. Kristály, Fundamental tones of clamped plates in nonpositively curved spaces, Adv. Math. 367(39) (2020) 107113] on the validity of such sharp estimates in high-dimensional Cartan–Hadamard manifolds. As a byproduct of the general functional inequalities, various Rellich inequalities are established in the same geometric setting.

Voting protocols on the star graph

AbstractLet $$G=(V,E)$$ G = ( V , E ) be a finite graph together with an initial assignment $$V\rightarrow \{0,1\}$$ V → { 0 , 1 } that represents the opinion of each vertex. Then discordant push voting is a discrete, non-deterministic protocol that alters the opinion of one vertex at a time until a consensus is reached. More precisely, at each round a discordant vertex u (i.e., one that has a neighbor with a different opinion) is chosen uniformly at random, and then we choose a neighbor v with different vote uniformly at random, and force v to change its opinion to that of u. In case of the discordant pull protocol we simply choose a discordant vertex uniformly at random and change its opinion. In this paper, we give asymptotically sharp estimations for the worst expected runtime of the discordant push and pull protocols on the star graph.

Speed and duration of drawdown under general Markov models

We propose an efficient computational method based on continuous-time Markov chain (CTMC) approximation to compute the distributions of the speed and duration of drawdown for general one-dimensional (1D) time-homogeneous Markov processes. We derive linear systems for the Laplace transforms of drawdown quantities and show how to solve them efficiently by recursion. In addition, we prove the convergence of our method and obtain a sharp estimate of the convergence rate under some assumptions. As applications, we consider pricing two financial options that provide protection against drawdown risk. Finally, we demonstrate the accuracy and efficiency of our method through extensive numerical experiments.

Sharp Lower Estimations for Invariants Associated with the Ideal of Antiderivatives of Singularities

Sharp Lower Estimations for Invariants Associated with the Ideal of Antiderivatives of Singularities

The ℓp$\ell ^p$ norm of the Riesz–Titchmarsh transform for even integer p$p$

AbstractThe long‐standing conjecture that for the norm of the Riesz–Titchmarsh discrete Hilbert transform is the same as the norm of the classical Hilbert transform, is verified when or , for . The proof, which is algebraic in nature, depends in a crucial way on the sharp estimate for the norm of a different variant of this operator for the full range of . The latter result was recently proved by the authors (Duke Math. J. 168 (2019), no. 3, 471–504).

Towards optimal sensor placement for inverse problems in spaces of measures

The objective of this work is to quantify the reconstruction error in sparse inverse problems with measures and stochastic noise, motivated by optimal sensor placement. To be useful in this context, the error quantities must be explicit in the sensor configuration and robust with respect to the source, yet relatively easy to compute in practice, compared to a direct evaluation of the error by a large number of samples. In particular, we consider the identification of a measure consisting of an unknown linear combination of point sources from a finite number of measurements contaminated by Gaussian noise. The statistical framework for recovery relies on two main ingredients: first, a convex but non-smooth variational Tikhonov point estimator over the space of Radon measures and, second, a suitable mean-squared error based on its Hellinger–Kantorovich distance to the ground truth. To quantify the error, we employ a non-degenerate source condition as well as careful linearization arguments to derive a computable upper bound. This leads to asymptotically sharp error estimates in expectation that are explicit in the sensor configuration. Thus they can be used to estimate the expected reconstruction error for a given sensor configuration and guide the placement of sensors in sparse inverse problems.

Stable determination of an impedance obstacle by a single far-field measurement

We establish sharp stability estimates of logarithmic type in determining an impedance obstacle in R2 . The obstacle is the polygonal shape and the surface impedance parameter is non-zero constant. We establish the stability results using a single far-field pattern, constituting a longstanding problem in the inverse scattering theory. This is the first stability result in the literature in determining an impedance obstacle by a single far-field measurement. The stability in simultaneously determining the obstacle and the boundary impedance is established in terms of the classical Hausdorff distance. Several technical novelties and developments in the mathematical strategy developed for establishing the aforementioned stability results exist. First, the stability analysis is conducted around a corner point in a micro-local manner. Second, our stability estimates establish explicit relationships between the obstacle’s geometric configurations and the wave field’s vanishing order at the corner point. Third, we develop novel error propagation techniques to tackle singularities of the wave field at a corner with the impedance boundary condition.