Positive Solutions for a Second-Order p

The author considers an impulsive boundary value problem involving the one-dimensional p-Laplacian -(φp (u′))′=λωtft,u,  00 and μ>0 are two parameters. Using fixed point theories, several new and more general existence and multiplicity results are derived in terms of different values of λ>0 and μ>0. The exact upper and lower bounds for these positive solutions are also given. Moreover, the approach to deal with the impulsive term is different from earlier approaches. In this paper, our results cover equations without impulsive effects and are compared with some recent results by Ding and Wang

Asymptotic expansions for harmonic functions at conical boundary points

We prove three theorems about the asymptotic behavior of solutions $u$ to the homogeneous Dirichlet problem for the Laplace equation at boundary points with tangent cones. First, under very mild hypotheses, we show that the doubling index of $u$ either has a unique finite limit, or goes to infinity; in other words, there is a well-defined order of vanishing. Second, under more quantitative hypotheses, we prove that if the order of vanishing of $u$ is finite at a boundary point $0$, then locally $u(x) = |x|^m \psi(x/|x|) + o(|x|^m)$, where $|x|^m \psi(x/|x|)$ is a homogeneous harmonic function on the tangent cone. Finally, we construct a convex domain in three dimensions where such an expansion fails at a boundary point, showing that some quantitative hypotheses are necessary in general. The assumptions in all of the results only involve regularity at a single point, and in particular are much weaker than what is necessary for unique continuation, monotonicity of Almgren's frequency, Carleman estimates, or other related techniques

The cubic moment of central values of automorphic L-functions

The authors study the central values of L-functions in certain families; in particular they bound the sum of the cubes of these values.Contents:Comment: 42 pages, published versio

Some recent developments on the Steklov eigenvalue problem

The Steklov eigenvalue problem, first introduced over 125 years ago, has seen a surge of interest in the past few decades. This article is a tour of some of the recent developments linking the Steklov eigenvalues and eigenfunctions of compact Riemannian manifolds to the geometry of the manifolds. Topics include isoperimetric-type upper and lower bounds on Steklov eigenvalues (first in the case of surfaces and then in higher dimensions), stability and instability of eigenvalues under deformations of the Riemannian metric, optimisation of eigenvalues and connections to free boundary minimal surfaces in balls, inverse problems and isospectrality, discretisation, and the geometry of eigenfunctions. We begin with background material and motivating examples for readers that are new to the subject. Throughout the tour, we frequently compare and contrast the behavior of the Steklov spectrum with that of the Laplace spectrum. We include many open problems in this rapidly expanding area.Comment: 157 pages, 7 figures. To appear in Revista Matem\'atica Complutens

Around stability for functional inequalities

Les inégalités fonctionnelles sont des inégalités qui encodent beaucoup d'information, tant de nature probabiliste (concentration de la mesure), qu'analytique (théorie spectrale des opérateurs) ou encore géométrique (profil isopérimétrique). L'inégalité de Poincaré en est un exemple fondamental. Dans cette thèse, nous obtenons des résultats de stabilité dans le cadre d'hypothèses de normalisation de moments, ainsi que dans le cadre de conditions de courbure-dimension. Un résultat de stabilité est une façon de quantifier la différence entre deux situations dans lesquelles les mêmes inégalités fonctionnelles sont presque vérifiées. Les résultats de stabilité obtenus dans cette thèse sont en particulier basés sur la méthode de Stein, qui est une méthode en plein développement ces dernières années, provenant du domaine des statistiques et permettant d'établir des estimations quantitatives sur des résultats de convergence. Par ailleurs, une partie de cette thèse est consacrée à l'étude des constantes optimales des inégalités de Bobkov, qui sont des inégalités fonctionnelles à caractère isopérimétrique.Functional inequalities are inequalities that encode a lot of information, both of a probabilistic (the concentration of measure phenomenon), analytical (the spectral theory of operators) and geometric (isoperimetric profile) nature. The Poincaré inequality is a fundamental example. In this thesis, we obtain stability results under moment normalisation assumptions, as well as under curvature-dimension conditions. A stability result is a way to quantify the difference between two situations where almost the same functional inequalities are verified. The stability results obtained in this thesis are in particular based on the Stein method, which is a method in full development in recent years, coming from the field of statistics and allowing to establish quantitative estimates on convergence results. In addition, a part of this thesis is devoted to the study of the optimal constants of Bobkov inequalities, which are functional inequalities of isoperimetric character

A Multireference Density Functional Approach to the Calculation of the Excited States of Uranium Ions

An accurate and efficient hybrid Density Functional Theory (DFT)/Multireference Configuration Interaction (MRCI) model for computing electronic excitation energies in heavy element atoms and molecules was developed. This model incorporated relativistic effects essential for accurate qualitative and quantitative spectroscopic predictions on heavy elements, while simultaneously removing spin-multiplicity limitations inherent in the original model on which it is based. This model was used to successfully compute ground and low-lying electronic states for atoms in the first two rows of the period table, which were used for calibration. Once calibrated, calculations on carbon monoxide, bromine fluoride, the bromine atom, uranium +4 and +5 ions and the uranyl (UO22+) ion showed the model achieved reductions in relative error with respect to Time Dependent Density Functional Theory (TDDFT) of 11-42%, with a corresponding reduction in computational effort in terms of MRCI expansion sizes of a factor of 25-64