The paper investigates the nonlinear function spaces introduced by Giaquinta, Modica and Souček. It is shown that a function from ${Cart}^p(\Omega ,{𝐑}^m)$ is approximated by ${𝒞}^1$ functions strongly in ${𝒜}^q(\Omega ,{𝐑}^m)$ whenever $q<p$. An example is shown of a function which is in ${cart}^p(\Omega ,{𝐑}^2)$ but not in ${cart}^p(\Omega ,{𝐑}^2)$.

We prove the $L^p$-$L^q$-time decay estimates for the solution of the Cauchy problem for the hyperbolic system of partial differential equations of linear thermoelasticity. In our proof based on the matrix of fundamental solutions to the system we use Strauss-Klainerman’s approach [12], [5] to the $L^p$-$L^q$-time decay estimates.

Some critical remarks are made about the theory of Linear Elasticity, questioning on its validity in general. An alternative linear approximation of the exact theory is proposed.

The Asymptotic Numerical Method (ANM) is a family of algorithms, based on computation of truncated vectorial series, for path following problems [2]. In this paper, we present and discuss some techniques to define local parameterization [4, 6, 7] in the ANM. We give some numerical comparisons of pseudo arc-length parameterization and local parameterization on non-linear elastic shells problems

It is proved that the first eigenfunction of the mixed boundary-value problem for the Laplacian in a thin domain ${\Omega }_h$ is localized either at the whole lateral surface ${\Gamma }_h$ of the domain, or at a point of ${\Gamma }_h$, while the eigenfunction decays exponentially inside ${\Omega }_h$. Other effects, attributed to the high-frequency range of the spectrum, are discussed for eigenfunctions of the mixed boundary-value and Neumann problems, too.

We consider mixed and hybrid variational formulations to the linearized elasticity system in domains with cracks. Inequality type conditions are prescribed at the crack faces which results in unilateral contact problems. The variational formulations are extended to the whole domain including the cracks which yields, for each problem, a smooth domain formulation. Mixed finite element methods such as PEERS or BDM methods are designed to avoid locking for nearly incompressible materials in plane elasticity....