The Nehari manifold for indefinite Kirchhoff problem with Caffarelli-Kohn-Nirenberg type critical growth

In this paper we study the following class of nonlocal problem involving Caffarelli-Kohn-Nirenberg type critical growth $$ L(u)-\lambda h(x)|x|^{-2(1+a)}u=\mu f(x)|u|^{q-2}u+|x|^{-pb}|u|^{p-2}u\quad \text{in } \mathbb R^N, $$% where $h(x)\geq 0$, $f(x)$ is a continuous function which may change sign, $\lambda, \mu$ are positive real parameters and $1< q< 2< 4< p=2N/[N+2(b-a)-2]$, $0\leq a< b< a+1< N/2$, $N\geq 3$. Here $$ L(u)=-M\left(\int_{\mathbb R^N} |x|^{-2a}|\nabla u|^2dx\right)\mathrm {div} \big(|x|^{-2a}\nabla u\big) $$ and the function $M\colon \mathbb R^+_0\to\mathbb R^+_0$ is exactly the Kirchhoff model, given by $M(t)=\alpha+\beta t$, $\alpha, \beta> 0$. The above problem has a double lack of compactness, firstly because of the non-compactness of Caffarelli-Kohn-Nirenberg embedding and secondly due to the non-compactness of the inclusion map $$u\mapsto \int_{\mathbb R^N}h(x)|x|^{-2(a+1)}|u|^2dx,$$ as the domain of the problem in consideration is unbounded. Deriving these crucial compactness results combined with constrained minimization argument based on Nehari manifold technique, we prove the existence of at least two positive solutions for suitable choices of parameters $\lambda$ and $\mu$.

Comparison of the Von Kármán and Kirchhoff models for the post-buckling and vibrations of elastic beams

International audience We compare different models describing the buckling, post-buckling and vibrations of elastic beams in the plane. Focus is put on the first buckled equilibrium solution and the first two vibration modes around it. In the incipient post-buckling regime, the classic Woinowsky-Krieger model is known to grasp the behavior of the system. It is based on the von Kármán approximation, a 2nd order expansion in the strains of the buckled beam. But as the curvature of the beam becomes larger, the Woinowsky-Krieger model starts to show limitations and we introduce a 3rd order model, derived from the geometrically-exact Kirchhoff model. We discuss and quantify the shortcomings of the Woinowsky-Krieger model and the contributions of the 3rd order terms in the new model, and we compare them both to the Kirchhoff model. Different ways to nondi-mensionalize the models are compared and we believe that, although this study is performed for specific boundary conditions, the present results have a general scope and can be used as abacuses to estimate the validity range of the simplified models.

Diffraction by a rigid strip in a plate modelled by Mindlin theory

We consider a plane flexural wave incident on a semi-infinite rigid strip in a Mindlin plate. The boundary conditions on the strip lead to three Wiener–Hopf equations, one of which decouples, leaving a scalar problem and a 2 × 2 matrix problem. The latter is solved using a simple method based on quadrature. The far-field diffraction coefficient is calculated and some numerical results are presented. We also show how the results reduce to the simpler Kirchhoff model in the low-frequency limit.

Degenerate Kirchhoff (p, q)–Fractional systems with critical nonlinearities

This paper deals with the existence of nontrivial solutions for critical possibly degenerate Kirchhoff fractional (p, q) systems. For clarity, the results are first presented in the scalar case, and then extended into the vectorial framework. The main features and novelty of the paper are the (p, q) growth of the fractional operator, the double lack of compactness as well as the fact that the systems can be degenerate. As far as we know the results are new even in the scalar case and when the Kirchhoff model considered is non–degenerate.

Solid circular plate unilaterally supported along two antipodal edge arcs and deflected by a central transverse concentrated force

A solid circular plate unilaterally supported along two antipodal edge arcs and deflected by a static central transverse concentrated force is considered. It is clarified that two distinct mechanical responses are possible, depending on the angular extent of the supports; in the first kind of response, valid for small support angular widths, the plate rotates about the support lateral sides, lifting from the supports along their central zone. The second response is valid for high support angular extents, and according to this response the plate additionally beds over the central portion of the support upper faces. Only the first kind of response is examined in this paper. This plate contact problem is classifiable as receding, and the plate deflection is described in terms of a plate Kirchhoff model. The plate deflection is analytically expressed together with the transitional value of the support angular width that describes the passage from the first to the second mechanical response.

Флексокалорический эффект в тонких пластинах титаната бария и титаната стронция

The flexoelectric effect in a thin plate of a ferroelectric with cubic symmetry is investigated on the basis of the Love-Kirchhoff model. The electric and elastic fields in a ferroelectric are described within the framework of the Landau-Ginzburg thermodynamic potential. The influence of the inhomogeneity of the polarization distribution in the plate is taken into account. Found values for plate bending caused by electric field application make it possible to calculate the dependence of the entropy change on temperature in the barium and strontium titanate plates (flexocaloric effect).

HOMOGENIZATION OF ELASTIC PLATE EQUATION∗

We are interested in general homogenization theory for fourth-order elliptic equation describing the Kirchhoff model for pure bending of a thin solid symmetric plate under a transverse load. Such theory is well-developed for second-order elliptic problems, while some results for general elliptic equations were established by Zhikov, Kozlov, Oleinik and Ngoan (1979). We push forward an approach of Antoni´c and Balenovi´c (1999, 2000) by proving a number of properties of H-convergence for stationary plate equation.