Existence and non-existence of minimizers for Poincaré–Sobolev inequalities

Existence and non existence results for minimizers of the Ginzburg-Landau energy with prescribed degrees

Let D = Ω\ω ⊂ R be a smooth annular type domain. We consider the simplified Ginzburg-Landau energy Eε(u) = 12 ∫

Existence of Minimizers on Drops

Existence of Energy Minimizers for Magnetostrictive Materials

The existence of a deformation and magnetization minimizing the magnetostrictive free energy is given. Mathematical challenges are presented by a free energy that includes elastic contributions defined in the reference configuration and magnetic contributions defined in the spatial frame. The one-to-one a.e. and orientation-preserving property of the deformation is demonstrated, and the satisfa...

Existence of Minimizers of Nonlocal Interaction Energies

We investigate nonlocal-interaction energies on the space of probability measures. We establish sharp conditions for the existence of minimizers for a broad class of nonlocalinteraction energies. The condition is closely related to the notion of H-stability of pairwise interaction potentials in statistical mechanics. Our approach uses the direct method of calculus of variations.

The Existence of Finite Element Minimizers

We present a general theorem on the existence of nite element minimizers for the approximation of variational problems of multiple inte-grals. Our theorem applies to variational problems for which a minimum does not exist in innnite-dimensional spaces of functions. Such problems occur in models for microstructure in martensitic and ferromagnetic crystals .