Sharp Estimates Research Articles

Wave equation on general noncompact symmetric spaces

abstract: We establish sharp pointwise kernel estimates and dispersive properties for the wave equation on noncompact symmetric spaces of general rank. This is achieved by combining the stationary phase method and the Hadamard parametrix, and in particular, by introducing a subtle spectral decomposition, which allows us to overcome a well-known difficulty in higher rank analysis, namely the fact that the Plancherel density is not a differential symbol in general. Consequently, we deduce the Strichartz inequality for a large family of admissible pairs and prove global well-posedness results for the corresponding semi-linear equation with low regularity data as on hyperbolic spaces.

Free boundary regularity and support propagation in mean field games and optimal transport

We study the behavior of solutions to the first-order mean field games system with a local coupling, when the initial density is a compactly supported function on the real line. Our results show that the solution is smooth in regions where the density is strictly positive, and that the density itself is globally continuous. Additionally, the speed of propagation is determined by the behavior of the cost function near small values of the density. When the coupling is entropic, we demonstrate that the support of the density propagates with infinite speed. On the other hand, for a power-type coupling, we establish finite speed of propagation, leading to the formation of a free boundary. We prove that under a natural non-degeneracy assumption, the free boundary is strictly convex and enjoys C1,1 regularity. We also establish sharp estimates on the speed of support propagation and the rate of long time decay for the density. Moreover, the density and the gradient of the value function are both shown to be Hölder continuous up to the free boundary. Our methods are based on the analysis of a new elliptic equation satisfied by the flow of optimal trajectories. The results also apply to mean field planning problems, characterizing the structure of minimizers of a class of optimal transport problems with congestion.

Sharp Coefficient Estimates for Analytic Functions Associated with Lemniscate of Bernoulli

The main purpose of this work is to study the third Hankel determinant for classes of Bernoulli lemniscate-related functions by introducing new subclasses of star-like functions represented by SLλ* and RLλ. In many geometric and physical applications of complex analysis, estimating sharp bounds for problems involving the coefficients of univalent functions is very important because these coefficients describe the fundamental properties of conformal maps. In the present study, we defined sharp bounds for function-coefficient problems belonging to the family of SLλ* and RLλ. Most of the computed bounds are sharp. This study will encourage further research on the sharp bounds of analytical functions related to new image domains.

L2$L^{2}$‐growth property for the wave equation with a higher derivative term

AbstractWe consider the Cauchy problem in for the wave equation with a higher derivative term. We derive sharp growth estimates of the ‐norm of the solution itself for the case of and . By imposing the weighted ‐initial velocity, we can get the lower and upper bound estimates of the solution itself. For the case of , we observe that the ‐growth behavior of the solution never occurs in the ‐framework of the initial data.

The asynchronous DeGroot dynamics

AbstractWe analyze the asynchronous version of the DeGroot dynamics: In a connected graph with nodes, each node has an initial opinions in and an independent Poisson clock. When a clock at a node rings, the opinion at is replaced by the average opinion of its neighbors. It is well known that the opinions converge to a consensus. We show that the expected time to reach ‐consensus is poly in undirected graphs and in Eulerian digraphs, but for some digraphs of bounded degree it is exponential. Our main result is that in undirected graphs and Eulerian digraphs, if the degrees are uniformly bounded and the initial opinions are i.i.d., then for every fixed . We give sharp estimates for the variance of the limiting consensus opinion, which measures the ability to aggregate information (“wisdom of the crowd”). We also prove generalizations to non‐reversible Markov chains and infinite graphs. New results of independent interest on fragmentation processes and coupled random walks are crucial to our analysis.

Projective tensor products of approximation spaces associated with positive operators

In this paper the projective tensor products of approximation spaces associated with positive operators in Banach spaces are characterized. We show that the tensor products of approximation spaces can be considered as the interpolation spaces generated by $K$-method of real interpolation. The inequalities that provide a sharp estimates of best approximations by analytic vectors of positive operators on projective tensor products are established. Application to spectral approximations of the regular elliptic operators on projective tensor products of Lebesgue spaces is shown.

Robust mixed finite element methods for a quad-curl singular perturbation problem

Robust mixed finite element methods are developed for a quad-curl singular perturbation problem. Lower order H(gradcurl)-nonconforming but H(curl)-conforming finite elements are constructed, which are extended to nonconforming finite element Stokes complexes and the associated commutative diagrams. Then H(gradcurl)-nonconforming finite elements are employed to discretize the quad-curl singular perturbation problem, which possess the sharp and uniform error estimates with respect to the perturbation parameter. The Nitsche’s technique is exploited to achieve the optimal convergence rate in the case of the boundary layers. Numerical results are provided to verify the theoretical convergence rates. In addition, the regularity of the quad-curl singular perturbation problem is established.

Bernoulli numbers in embedding constants for Sobolev spaces with diverse boundary conditions

The Sobolev spaces W 2 n [ 0 ; 1 ] W^n_2[0;1] are treated with five different boundary conditions (periodic, antiperiodic, of even and odd orders, and of even-odd order). Sharp estimates for the derivatives of order k = 0 , 1 , … , n − 1 k=0,1,\dots , n-1 are obtained, sharp constants for the embedding of W 2 n [ 0 ; 1 ] W^n_2[0;1] into W ∞ k [ 0 ; 1 ] W^k_\infty [0;1] are found, it is shown that they are rationally expressed in terms of Bernoulli numbers and, therefore, are rational. The exact embedding constants can also be expressed in terms of the Riemann ζ \zeta -function. The reproducing kernels of these spaces are calculated.

Positive links with arrangements of pseudocircles as shadows

An arrangement of pseudocircles A is a collection of Jordan curves in the plane that pairwise intersect (transversally) at exactly two points. How many non-equivalent links have A as their shadow? Motivated by this question, we study the number of non-equivalent positive oriented links that have an arrangement of pseudocircles as their shadow. We give sharp estimates on this number when A is one of the three unavoidable arrangements of pseudocircles.

Spectral stability under removal of small segments

In the present paper, we deepen the works of L. Abatangelo, V. Felli, L. Hillairet and C. Léna on the asymptotic estimates of the eigenvalue variation under removal of segments from the domain in R 2 \mathbb {R}^2 . We get a sharp asymptotic estimate when the eigenvalue is simple and the removed segment is tangent to a nodal line of the associated eigenfunction. Moreover, we extend their results to the case when the eigenvalue is not simple.

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.