Periodic solutions with prescribed minimal period to Hamiltonian systems

In this article, we study the existence of periodic solutions to second order Hamiltonian systems. Our goal is twofold. When the nonlinear term satisfies a strictly monotone condition, we show that, for any $T>0$ T > 0 , there exists a T-periodic solution with minimal period T. When the nonlinear term satisfies a non-decreasing condition, using a perturbation technique, we prove a similar result. In the latter case, the periodic solution corresponds to a critical point which minimizes the variational functional on the Nehari manifold which is not homeomorphic to the unit sphere.

Hybrid Finite Element Analysis of Heat Conduction in Orthotropic Media with Variable Thermal Conductivities

A hybrid finite element method is proposed for the heat conduction analysis with variable thermal conductivities. A linear combination of fundamental solutions is employed to approximate the intra-element temperature field while standard one-dimensional shape functions are utilized to independently define the frame temperature field along the element boundary. The influence of variable thermal conductivities embeds in the intra-element temperature field via the fundamental solution. A hybrid variational functional, which involves integrals along the element boundary only, is developed to link the two assumed fields to produce the thermal stiffness equation. The advantage of the proposed method lies that the changes in the thermal conductivity are captured inside the element domain. Numerical examples demonstrate the accuracy and efficiency of the proposed method and also the insensitivity to mesh distortion.

On Nodal Solutions of the Nonlinear Choquard Equation

This paper deals with the general Choquard equation-\Delta u+V(|x|)u=(I_{\alpha}*|u|^{p})|u|^{p-2}u\quad\text{in }\mathbb{R}^{N},where {V\in C([0,\infty),\mathbb{R}^{+})} is bounded below by a positive constant, and {I_{\alpha}} denotes the Riesz potential of order {\alpha\in(0,N)}. In view of the convolution term, the nonlocal property makes the variational functional completely different from the one for local pure power-type nonlinearity. By combining the Brouwer degree and developing some new techniques, a family of radial solutions with a prescribed number of zeros is constructed for {p\in[2,\frac{N+\alpha}{N-2})}, while the degeneracy happens for {p\in(\frac{N+\alpha}{N},2)}. This result complements and improves the ones in the literature in the aspect of the range of p.

Nodal solutions of a nonlocal Choquard equation in a bounded domain

In this paper, we are interested in the least energy nodal solutions to the following nonlocal Choquard equation with a local term: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] is a bounded domain. This problem may be seen as a nonlocal perturbation of the classical Lane–Emden equation [Formula: see text] in [Formula: see text]. The problem has a variational functional with a nonlocal term [Formula: see text]. The appearance of the nonlocal term makes the variational functional very different from the local case [Formula: see text] for which the problem has ground state solutions and least energy nodal solutions if [Formula: see text]. The problem may also be viewed as a nonlocal Choquard equation with a local perturbation term when [Formula: see text]. For [Formula: see text], we show that although ground state solutions always exist, the existence of least energy nodal solution depends on [Formula: see text]: for [Formula: see text], there does not exist a least energy nodal solution while for [Formula: see text], such a solution exists. Note that [Formula: see text] is a critical value. In the case of a linear local perturbation, i.e. [Formula: see text], if [Formula: see text], the problem has a positive ground state and a least energy nodal solution. However, if [Formula: see text], the problem has a ground state which changes sign. Hence, it is also a least energy nodal solution.

General Integral Equations and Bounds of the Effective Moduli of Random Structure Composites

One considers a linear elastic random structure composite material (CM) with a homogeneous matrix. The idea of the effective field hypothesis (EFH, H1) dates back to Faraday, Poisson, Mossotti, Clausius, and Maxwell (1830–1870, see for references and details [1], [2]) who pioneered the introduction of the effective field concept as a local homogeneous field acting on the inclusions and differing from the applied macroscopic one. It is proved that a concept of the EFH (even if this term is not mentioned) is a (first) background of all four groups of analytical methods in physics and mechanics of heterogeneous media (model methods, perturbation methods, self-consistent methods, and variational ones, see for refs. [1]). New GIEs essentially define the new (second) background (which does not use the EFH) of multiscale analysis offering the opportunities for a fundamental jump in multiscale research of random heterogeneous media with drastically improved accuracy of local field estimations (with possible change of sign of predicted local fields). Estimates of the Hashin-Shtrikman (H-S) type are developed by extremizing of the classical variational functional involving either a classical GIE [1] or a new one. In the classical approach by Willis (1977), the H-S functional is extremized in the class of trial functions with a piece-wise constant polarisation tensors while in the current work we consider more general class of trial functions with a piece-wise constant effective fields. One demonstrates a better quality of proposed bounds, that is assessed from the difference between the upper and lower bounds for the concrete numerical examples.

An Improved Fountain Theorem and Its Application

The main aim of the paper is to prove a fountain theorem without assuming the τ-upper semi-continuity condition on the variational functional. Using this improved fountain theorem, we may deal with more general strongly indefinite elliptic problems with various sign-changing nonlinear terms. As an application, we obtain infinitely many solutions for a semilinear Schrödinger equation with strongly indefinite structure and sign-changing nonlinearity.

ENTROPY PRODUCTION OF ENTIRELY DIFFUSIONAL LAPLACIAN TRANSFER AND THE POSSIBLE ROLE OF FRAGMENTATION OF THE BOUNDARIES

The entropy production and the variational functional of a Laplacian diffusional field around the first four fractal iterations of a linear self-similar tree (von Koch curve) is studied analytically and detailed predictions are stated. In a next stage, these predictions are confronted with results from numerical resolution of the Laplace equation by means of Finite Elements computations. After a brief review of the existing results, the range of distances near the geometric irregularity, the so-called "Near Field", a situation never studied in the past, is treated exhaustively. We notice here that in the Near Field, the usual notion of the active zone approximation introduced by Sapoval et al. [M. Filoche and B. Sapoval, Transfer across random versus deterministic fractal interfaces, Phys. Rev. Lett. 84(25) (2000) 5776;1 B. Sapoval, M. Filoche, K. Karamanos and R. Brizzi, Can one hear the shape of an electrode? I. Numerical study of the active zone in Laplacian transfer, Eur. Phys. J. B. Condens. Matter Complex Syst. 9(4) (1999) 739-753.]2 is strictly inapplicable. The basic new result is that the validity of the active-zone approximation based on irreversible thermodynamics is confirmed in this limit, and this implies a new interpretation of this notion for Laplacian diffusional fields.