In this work we consider a solid body $\Omega \subset {ℝ}^3$ constituted by a nonhomogeneous elastoplastic material, submitted to a density of body forces $\lambda f$ and a density of forces $\lambda g$ acting on the boundary where the real $\lambda$ is the loading parameter. The problem is to determine, in the case of an unbounded convex of elasticity, the Limit load denoted by $\overline{\lambda }$ beyond which there is a break of the structure. The case of a bounded convex of elasticity is done in [El-Fekih and Hadhri,   (1995) 391–419]. Then assuming...

We study the asymptotic behavior of $\lambda v_{\lambda }$ as $\lambda \to 0^{+}$, where $v_{\lambda }$ is the viscosity solution of the following Hamilton-Jacobi-Isaacs equation (infinite horizon case) $$\lambda v_{\lambda }+H(x,D{v}_{\lambda })=0,$$ with $$H(x,p):=\underset{b\in B}{min}\underset{a\in A}{max}\{-f(x,a,b)·p-l(x,a,b)\}.$$ We discuss the cases in which the state of the system is required to stay in an -dimensional torus, called periodic boundary conditions, or in the closure of a bounded connected domain $\Omega \subset {}^n$ with sufficiently smooth boundary. As far as the latter is concerned, we treat both the case of the Neumann boundary conditions (reflection...

Let be one solution to $${\partial }_{t}y(t,x)-\sum _{i,j=1}^n{\partial }_j\left(a_{ij}\left(x\right){\partial }_{i}y(t,x)\right)=h(t,x),0<t<T,x\in \Omega$$ with a non-homogeneous term , and ${y|}_{(0,T)\times \partial \Omega }=0$, where $\Omega \subset {ℝ}^n$ is a bounded domain. We discuss an inverse problem of determining unknown functions by $\{{\partial }_{\nu }y\left(h_{\ell }\right){|}_{(0,T)\times {\Gamma }_0}$, $y\left(h_{\ell }\right)(\theta ,·){\}}_{1\le \ell \le {\ell }_0}$ after selecting input sources $h_1,...,h_{{\ell }_0}$ suitably, where ${\Gamma }_0$ is an arbitrary subboundary, ${\partial }_{\nu }$ denotes the normal derivative, $0<\theta <T$ and ${\ell }_0\in \mathbb{N}$. In the case of ${\ell }_0={(n+1)}^{2}n/2$, we prove the Lipschitz stability in the inverse problem if we choose $(h_1,...,h_{{\ell }_0})$ from a set $\mathscr{H}\subset {\left\{C_0^{\infty }((0,T)\times \omega )\right\}}^{{\ell }_0}$ with an arbitrarily fixed subdomain $\omega \subset \Omega$. Moreover we can take ${\ell }_0=(n+3)n/2$ by making special choices...

We consider a class of semilinear elliptic equations of the form 15.7cm -${\epsilon }^2\Delta u(x,y)+a\left(x\right)W^{\text{'}}\left(u(x,y)\right)=0,(x,y)\in {ℝ}^2$ where $\epsilon >0$, $a:ℝ\to ℝ$ is a periodic, positive function and $W:ℝ\to ℝ$ is modeled on the classical two well Ginzburg-Landau potential $W\left(s\right)={(s^2-1)}^2$. We look for solutions to ([see full textsee full text]) which verify the asymptotic conditions $u(x,y)\to \pm 1$ as $x\to \pm \infty$ uniformly with respect to $y\in ℝ$. We show variational methods that if is sufficiently small and is not constant, then ([see full textsee full text]) admits infinitely many of such solutions,...