Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians

We consider the classical functional of the Calculus of Variations of the form $$\begin{array}{} \displaystyle I(u)=\int\limits_{{\it\Omega}}F(x, u(x), \nabla u(x))\,dx, \end{array}$$ where Ω is a bounded open subset of ℝn and F : Ω × ℝ × ℝn → ℝ is a Carathéodory convex function; the admissible functions u coincide with a prescribed Lipschitz function ϕ on ∂Ω. We formulate some conditions under which a given function in ϕ + $\begin{array}{} \displaystyle W^{1,p}_0 \end{array}$(Ω) with I(u) < +∞ can be approximated in the W1,p norm and in energy by a sequence of smooth functions that coincide with ϕ on ∂Ω. As a particular case we obtain that the Lavrentiev phenomenon does not occur when F(x, u, ξ) = f(x, u) + h(x, ξ) is convex and x ↦ F(x, 0, 0) is sufficiently smooth.

Multivalued elliptic operators with nonstandard growth

The paper is devoted to the Dirichlet problem for monotone, in general multivalued, elliptic equations with nonstandard growth condition. The growth conditions are more general than the well-known {p(x)} growth. Moreover, we allow the presence of the so-called Lavrentiev phenomenon. As consequence, at least two types of variational settings of Dirichlet problem are available. We prove results on the existence of solutions in both of these settings. Then we obtain several results on the convergence of certain types of approximate solutions to an exact solution.

Rate-independent elastoplasticity at finite strains and its numerical approximation

Gradient plasticity at large strains with kinematic hardening is analyzed as quasistatic rate-independent evolution. The energy functional with a frame-indifferent polyconvex energy density and the dissipation is approximated numerically by finite elements and implicit time discretization, such that a computationally implementable scheme is obtained. The nonself-penetration as well as a possible frictionless unilateral contact is considered and approximated numerically by a suitable penalization method which keeps polyconvexity and simultaneously bypasses the Lavrentiev phenomenon. The main result concerns the convergence of the numerical scheme toward energetic solutions. In the case of incompressible plasticity and of nonsimple materials, where the energy depends on the second derivative of the deformation, we derive an explicit stability criterion for convergence relating the spatial discretization and the penalizations.

Singular sets and the Lavrentiev phenomenon

We show that non-occurrence of the Lavrentiev phenomenon does not imply that the singular set is small. Precisely, given a compact Lebesgue null subsetE⊆ ℝ and an arbitrary superlinearity, there exists a smooth strictly convex Lagrangian with this superlinear growth such that all minimizers of the associated variational problem have singular set exactlyEbut still admit approximation in energy by smooth functions.