Convexity Arguments Research Articles

Qualitative properties for a Moore–Gibson–Thompson thermoelastic problem with heat radiation

In this work, we study some qualitative properties arising in the solution of a thermoelastic problem with heat radiation. The so-called Moore–Gibson–Thompson equation is used to model the heat conduction. By using the logarithmic convexity argument, the uniqueness and instability of solutions are proved without imposing any condition on the elasticity tensor. Then, the existence of solutions is obtained assuming that the elastic tensor is positive definite applying the theory of linear semigroups, and the exponential energy decay is shown in the one-dimensional case. Finally, we consider the one-dimensional quasi-static version and we assume that the elastic coefficient is negative. The existence and decay of solutions are proved, and a justification of the quasi-static approach is also provided.

Radial solutions and a local bifurcation result for a singular elliptic problem with Neumann condition

We study the problem − Δ u = λ u − u − 1 with a Neumann boundary condition; the peculiarity being the presence of the singular term − u − 1 . We point out that the minus sign in front of the negative power of u is particularly challenging, since no convexity argument can be invoked. Using bifurcation techniques we are able to prove the existence of solution ( u λ , λ ) with u λ approaching the trivial constant solution u = λ − 1 / 2 and λ close to an eigenvalue of a suitable linearized problem. To achieve this we also need to prove a generalization of a classical two-branch bifurcation result for potential operators. Next we study the radial case and show that in this case one of the bifurcation branches is global and we find the asymptotical behavior of such a branch. This results allows to derive the existence of multiple solutions u with λ fixed.

Rellich inequalities via Riccati pairs on model space forms

We present a simple method for proving Rellich inequalities on Riemannian manifolds with constant, non-positive sectional curvature. The method is built upon simple convexity arguments, integration by parts, and the so-called Riccati pairs, which are based on the solvability of a Riccati-type ordinary differential inequality. These results can be viewed as the higher order counterparts of the recent work by Kajántó, Kristály, Peter, and Zhao, discussing Hardy inequalities using Riccati pairs.

Global minimizers of a large class of anisotropic attractive‐repulsive interaction energies in 2D

AbstractWe study a large family of Riesz‐type singular interaction potentials with anisotropy in two dimensions. Their associated global energy minimizers are given by explicit formulas whose supports are determined by ellipses under certain assumptions. More precisely, by parameterizing the strength of the anisotropic part we characterize the sharp range in which these explicit ellipse‐supported configurations are the global minimizers based on linear convexity arguments. Moreover, for certain anisotropic parts, we prove that for large values of the parameter the global minimizer is only given by vertically concentrated measures corresponding to one dimensional minimizers. We also show that these ellipse‐supported configurations generically do not collapse to a vertically concentrated measure at the critical value for convexity, leading to an interesting gap of the parameters in between. In this intermediate range, we conclude by infinitesimal concavity that any superlevel set of any local minimizer in a suitable sense does not have interior points. Furthermore, for certain anisotropic parts, their support cannot contain any vertical segment for a restricted range of parameters, and moreover the global minimizers are expected to exhibit a zigzag behavior. All these results hold for the limiting case of the logarithmic repulsive potential, extending and generalizing previous results in the literature. Various examples of anisotropic parts leading to even more complex behavior are numerically explored.

Stability of standing waves for a generalized Benney-Roskes system

We analyze the orbital stability of standing waves for a generalized Benney-Roskes system in spatial dimensions N = 2 N=2 , 3 3 . We establish stability of standing waves under certain conditions by reducing the system to a single nonlinear (nonlocal) Schrödinger equation, using the variational characterization of standing waves and a convexity argument.

Information backflow may not indicate quantum memory

We analyze recent approaches to quantum Markovianity and how they relate to the proper definition of quantum memory. We show that the well-known criterion of information backflow does not report a distinction between the quantum or classical character of observed memory effects. Therefore, as a complement to the current description, we propose a definition of a quantum backflow based on the concept of component dynamical maps and convexity arguments. In particular, for a natural choice of component maps, we provide a practical characterization of quantum memory in terms of the new concept of quantum memory witness. In particular, we give a full description of such witnesses in a qubit case. Finally, we discuss a potential generalization of component maps that should capture certain classical features of information backflow.

Theoretical and computational decay results for a Bresse system with one infinite memory in the longitudinal displacement

<p style='text-indent:20px;'>In this paper, we consider a one-dimensional linear Bresse system with only one infinite memory term acting in the third equation (longitudinal displacements). Under a general condition on the memory kernel (relaxation function), we establish a decay estimate of the energy of the system. Our decay result extends and improves some decay rates obtained in the literature such as the one in [<xref ref-type="bibr" rid="b27">27</xref>], [<xref ref-type="bibr" rid="b4">4</xref>], [<xref ref-type="bibr" rid="b33">33</xref>], [<xref ref-type="bibr" rid="b58">58</xref>] and [<xref ref-type="bibr" rid="b34">34</xref>]. The proof is based on the energy method together with convexity arguments. Numerical simulations are given to illustrate the theoretical decay result.</p>

Logarithmic convexity for third order in time partial differential equations

In this short note, we want to describe the logarithmic convexity argument for third order in time partial differential equations. As a consequence, we first prove a uniqueness result whenever certain conditions on the parameters are satisfied. Later, we show the instability of the solutions if the initial energy is less or equal than zero.

On the construction of the elementary trigonometric functions

Abstract The article discusses the construction of the elementary trigonometric functions. It discusses several approaches, but the main message is that to construct the trigonometric functions one needs to follow the same approach as one should use during the construction of the real exponential function, but one should use complex numbers. The key point is that one must construct the complex square root function to define the trigonometric functions on the binary numbers and then one should use some convexity argument to prove their differentiability. This approach, as the power series approach, based on the intimate relation between the trigonometric and the exponential functions.

On the instability for an incremental problem in elastodynamics

In this short note, we consider some issues regarding the instability of some elastodynamical problems when the elasticity tensor is not positive definite. By using the so-called logarithmic convexity argument, we prove the instability of solutions when the time derivative of the elasticity tensor is semi-definite negative or it satisfies another restriction on the coefficients. The uniqueness of the solution is also concluded. Finally, a simple one-dimensional example is provided to demonstrate the numerical behaviour of the instability.

On finite time blow-up for a 3D Davey-Stewartson system

We consider the elliptic-elliptic Davey-Stewartson system in the three-dimensional Euclidean space, and we give sufficient conditions for the existence of finite time blow-up solutions in non-isotropic spaces. The proof is based on some general results on distributions defined via homogeneous symbols, in conjunction with a convexity argument.

Stability result for a class of weakly dissipative second-order systems with infinite memory

In this paper we consider the following abstract class of weakly dissipative second-order systems with infinite memory, \(u''(t)+Au(t)-\displaystyle\int_{0}^{\infty} g(s)A^\alpha u(t-s)ds=0,~t>0,\) and establish a general stability result with a very general assumption on the behavior of \(g\) at infinity; that is \(g'(t) \leq - \xi(t) G \left(g(t)\right),~~t \geq 0.\) where \(\xi\) and \(G\) are two functions satisfying some specific conditions. Our result generalizes and improves many earlier results in the literature. Moreover, we obtain our result with imposing a weaker restrictive assumption on the boundedness of initial data used in many earlier papers in the literature such as the one in [<a href="#1">1</a>,<a href="#2">2</a>,<a href="#3">3</a>,<a href="#4">4</a>,<a href="#5">5</a>]. The proof is based on the energy method together with convexity arguments.

A Stability Result for a Viscoelastic Wave Equation in the Presence of Finite and Infinite Memories

In this study, we consider a viscoelastic wave equation in the presence of finite and infinite memories. Under suitable conditions on the variable coefficients and for a wide class of relaxation functions, we establish an explicit and general decay result. Moreover, we use a weaker boundedness condition, on the initial data than the one used in the literature. The proof of our decay result is based on the multiplier method together with some convexity arguments. This study generalizes and improves previous literature outcomes.

Some operator inequalities via convexity

In this article, we employ a standard convexity argument to obtain new and refined inequalities related to the matrix mean of two accretive matrices, the numerical radius and the Tsallis relative operator entropy.

On a multiple Dirichlet series associated to binary cubic forms

In this paper, we construct and study the analytic properties of a multiple Dirichlet series associated to the space of certain binary cubic forms. Using reduction theory, local Euler factor computation, a functional equation and convexity arguments, we prove that this multiple Dirichlet series can be meromorphically continued to the whole of C2.

A MIXED MULTIFRACTAL ANALYSIS FOR QUASI-AHLFORS VECTOR-VALUED MEASURES

The multifractal formalism for measures in its original formulation is checked for special classes of measures, such as, doubling, self-similar, and Gibbs-like ones. Out of these classes, suitable conditions should be taken into account to prove the validity of the multifractal formalism. In this work, a large class of measures satisfying a weak condition known as quasi-Ahlfors is considered in the framework of mixed multifractal analysis. A joint multifractal analysis of finitely many quasi-Ahlfors probability measures is developed. Mixed variants of multifractal generalizations of Hausdorff, and packing measures, and corresponding dimensions are introduced. By applying convexity arguments, some properties of these measures, and dimensions are established. Finally, an associated multifractal formalism is introduced, and proved to hold for the class of quasi-Ahlfors measures. Besides, some eventual applications, and motivations, especially, in AI are discussed.

Sharp conditions for scattering and blow-up for a system of NLS arising in optical materials with χ 3 nonlinear response

We study the asymptotic dynamics for solutions to a system of nonlinear Schrödinger equations with cubic interactions, arising in nonlinear optics. We provide sharp threshold criteria leading to global well-posedness and scattering of solutions, as well as formation of singularities in finite time for (anisotropic) symmetric initial data. The free asymptotic results are proved by means of Morawetz and interaction Morawetz estimates. The blow-up results are shown by combining variational analysis and an ODE argument, which overcomes the unavailability of the convexity argument based on virial-type identities.

The One‐Dimensional Log‐Gas Free Energy Has a Unique Minimizer

We prove that, at every positive temperature, the infinite‐volume free energy of the one‐dimensional log‐gas, or beta‐ensemble, has a unique minimizer, which is the Sine‐beta process arising from random matrix theory. We rely on a quantitative displacement convexity argument at the level of point processes, and on the screening procedure introduced by Sandier‐Serfaty. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Uniqueness and exponential instability in a new two-temperature thermoelastic theory

<abstract> In this work we consider the temperature-rate dependent two temperatures thermoelastic theory. It has been proposed very recently. We study the case in which the elasticity tensor may not be positive definite. Thus, the problem can be ill posed in the sense of Hadamard. We adapt the logarithmic convexity argument to the specific situation proposed by this theory. That is, we define a suitable function on the solutions satisfying that the logarithm is convex. Uniqueness and instability of the solutions under suitable conditions on the constitutive tensors are proved. </abstract>

Strongly Convex Divergences.

We consider a sub-class of the f-divergences satisfying a stronger convexity property, which we refer to as strongly convex, or -convex divergences. We derive new and old relationships, based on convexity arguments, between popular f-divergences.