Asymptotic analysis of contact problems between an elastic material and thin-rigid plates

We consider an elastic material in contact with a three-dimensional rigid plate of varying thickness. We suppose that a perfect adhesion occurs along thin zones disposed in a self-similar way on the interface between the two materials. We suppose that the elasticity coefficients in the plate depend on its thickness and tend to infinity as this thickness tends to zero. We derive the effective material properties using Γ-convergence methods.     

No-gap optimality conditions for an optimal control problem with pointwise control-state constraints

An optimal control problem with pointwise mixed constraints of the instationary three-dimensional Navier–Stokes–Voigt equations is considered. We derive second-order optimality conditions and show that there is no gap between second-order necessary optimality conditions and second-order sufficient optimality conditions. In addition, the second-order sufficient optimality conditions for the problem where the objective functional does not contain a Tikhonov regularization term are also discussed.     

Stabilization of the uniform Timoshenko beam by one locally distributed feedback

In this article, we study the energy decay rate for an elastic Timoshenko system. This system consists of two coupled wave equations. Only the equation about the rotation angle is damped by one locally distributed feedback at the neighbourhood of the boundary. The equation for the transverse displacement of the beam is only indirectly damped through the coupling. First, we establish an exponential energy decay rate in the case of the same speed of propagation. Next, when the wave speeds are different, a polynomial-type decay rate is obtained. These results are proved by verifying the frequency domain conditions.     

Mass,momentum and energy conservation laws in zero-pressure gas dynamics and δ-shocks: II

We study δ-shocks in a one-dimensional system of zero-pressure gas dynamics. In contrast to well-known papers (see References) this system is considered in the form of mass, momentum and energy conservation laws. In order to define such singular solutions, special integral identities are introduced which extend the concept of classical weak solutions. Using these integral identities, the Rankine–Hugoniot conditions for δ-shocks are obtained. It is proved that the mass, momentum and energy transport processes between the area outside the of one-dimensional δ-shock wave front and this front are going on such that the total mass, momentum and energy are independent of time, while the mass and energy concentration processes onto the moving δ-shock wave front are going on. At the same time the total kinetic energy transforms into total internal energy.     

Contact problems with friction for hemitropic solids: boundary variational inequality approach

We study the interior and exterior contact problems for hemitropic elastic solids. We treat the cases when the friction effects, described by Tresca friction (given friction model), are taken into consideration either on some part of the boundary of the body or on the whole boundary. We equivalently reduce these problems to a boundary variational inequality with the help of the Steklov–Poincaré type operator. Based on our boundary variational inequality approach we prove existence and uniqueness theorems for weak solutions. We prove that the solutions continuously depend on the data of the original problem and on the friction coefficient. For the interior problem, necessary and sufficient conditions of solvability are established when friction is taken into consideration on the whole boundary.     

Stochastic homogenization of heat transfer in polycrystals with nonlinear contact conductivities

The heat transfer problem in a polycrystal with nonlinear jump conditions on the grain boundaries will be homogenized using the method of stochastic two-scale convergence developed by Zhikov and Pyatnitskii [V.V. Zhikov and A.L. Pyatnitskii, Homogenization of random singular structures and random measures, Izv. Math. 70(1) (2006), pp. 19–67] and recently extended by the author [M. Heida, An extension of stochastic two-scale convergence and application, Asympt. Anal. (2010) (in press)]. It will be shown that for monotone Lipschitz jump conditions differentiable in 0, the nonlinearity vanishes in the limit. Additionally, existing Poincaré inequalities will be extended to more general geometric settings with the only restriction of local C ¹-interfaces with finite intensity. In particular, the result can now be applied to the Poisson–Voronoi tessellation.     

An electro-viscoelastic frictional contact problem with damage

We consider a quasistatic frictional contact problem between a piezoelectric body and a foundation. The contact is modeled with normal compliance and friction is modeled with a general version of Coulomb's law of dry friction; the process is quasistatic and the material's behavior is described by an electro-viscoelastic constitutive law with damage. We derive a variational formulation for the model which is in the form of a system involving the displacement field, the electric potential field, and the damage field. Then we provide the existence of a unique weak solution to the model. The proof is based on arguments of evolutionary variational inequalities and fixed point.     

Symmetry in the geometry of metric contact pairs

We prove that the universal covering of a complete locally symmetric normal metric contact pair manifold with decomposable ? is a Calabi‐Eckmann manifold or the Riemannian product of a sphere and . We show that a complete, simply connected, normal metric contact pair manifold with decomposable ?, such that the foliation induced by the vertical subbundle is regular and reflections in the integral submanifolds of the vertical subbundle are isometries, is the product of globally ?‐symmetric spaces or the product of a globally ?‐symmetric space and . Moreover in the first case the manifold fibers over a locally symmetric space endowed with a symplectic pair.