The limit behavior of a periodic assembly of a finite number of elasto-plastic phases is investigated as the period becomes vanishingly small. A limit quasi-static evolution is derived through two-scale convergence techniques. It can be thermodynamically viewed as an elasto-plastic model, albeit with an infinite number of internal variables.

Sufficient conditions for the problem $$\frac{{\partial }^{2}u}{\partial x\partial y}=P_0(x,y)u+P_1(x,y)\frac{\partial u}{\partial x}+P_2(x,y)\frac{\partial u}{\partial y}+q(x,y),u(x+{\omega }_1,y)=u(x,y),u(x,y+{\omega }_2)=u(x,y)$$ to have the Fredholm property and to be uniquely solvable are established, where ${\omega }_1$ and ${\omega }_2$ are positive constants and $P_j:R^2\to R^{n\times n}$$(j=0,1...$

the existence of an $\omega$-periodic solution of the equation $\frac{{\partial }^{2}u}{\partial t^2}+\alpha \frac{{\partial }^{4}u}{\partial x^4}+\gamma \frac{{\partial }^{5}u}{\partial x^4\partial t}-\tilde{\gamma }\frac{{\partial }^{3}u}{\partial x^2\partial t}+\delta \frac{\partial u}{\partial t}-\left[\beta +\aleph {\int }_0^n{\left(\frac{\partial u}{\partial x}\right)}^2(·,\xi )d\xi +\sigma {\int }_0^n\frac{{\partial }^{2}u}{\partial x\partial t}(·,\xi )\frac{\partial u}{\partial x}(·,\xi )d\xi \right]\frac{{\partial }^{2}u}{\partial x^2}=f$ sarisfying the boundary conditions $u(t,0)=u(t,\pi )=\frac{{\partial }^{2}u}{\partial x^2}\left(t,0\right)=\frac{{\partial }^{2}u}{\partial x^2}\left(t,\pi \right)=0$ is proved for every $\omega$-periodic function $f\in C\left(\left[0,\omega \right],L_2\right)$.

The paper deals with the existence of time-periodic solutions to the beam equation, in which terms expressing torsion and damping are also considered. The existence of periodic solutions is proved in the cas of time-periodic outer forces by means of an apriori estimate and the Fourier method.

Let G be a homogeneous Lie group with a left Haar measure dg and L the action of G as left translations on $L_p(G;dg)$. Further, let H = dL(C) denote a homogeneous operator associated with L. If H is positive and hypoelliptic on $L_2$ we prove that it is closed on each of the $L_p$-spaces, p ∈ 〈 1,∞〉, and that it generates a semigroup S with a smooth kernel K which, with its derivatives, satisfies Gaussian bounds. The semigroup is holomorphic in the open right half-plane on all the $L_p$-spaces, p ∈ [1,∞]. Further extensions...

In this paper, we consider the existence and nonexistence of positive solutions of degenerate elliptic systems $$\left\}\begin{array}{cccc}-{\Delta }_{p}u=f(x,u,v),\hfill & \text{in}\Omega ,-{\Delta }_{p}v=g(x,u,v),\hfill & \text{in}\Omega ,u=v=0,& \text{on}\partial \Omega ,\end{array}\right.$$ where $-{\Delta }_p$ is the $p$-Laplace operator, $p>1$ and $\Omega$ is a $C^{1,\alpha }$-domain in ${ℝ}^n$. We prove an analogue of [7, 16] for the eigenvalue problem with $f(x,u,v)={\lambda }_1{v}^{p-1}$, $g(x,u,v)={\lambda }_2{u}^{p-1}$ and obtain a non-existence result of positive solutions for the general systems.

We consider the Euler equation for an incompressible fluid on a three dimensional torus, and the construction of its solution as a power series in time. We point out some general facts on this subject, from convergence issues for the power series to the role of symmetries of the initial datum. We then turn the attention to a paper by Behr, Nečas and Wu, ESAIM: M2AN 35 (2001) 229–238; here, the authors chose a very simple Fourier polynomial as an initial datum for the Euler equation and analyzed...

A description of all «power-logarithmic» solutions to the homogeneous Dirichlet problem for strongly elliptic systems in a $n$-dimensional cone $K=K_d\times {\mathbb{R}}^{n-d}$ is given, where $K_d$ is an arbitrary open cone in ${\mathbb{R}}^d$ and $n>d>1$.