Abstract for the talk on 18.12.2019 (11:00 h)
Arbeitsgemeinschaft ANGEWANDTE ANALYSISLauro Morales (Universidad Nacional Autónoma de México)
Some Results on quasiconvex hull for a n-well problem in 2D under Geometrically Linear Elastic regime
In shape-memory alloys, it is common to analyze pattern formation induced by the existence of different zero free energy phases in the material. These microstructure provokes the shape-memory effect. The classical model used to study the problem is
| <center class="math-display"> <img src="/fileadmin/lecture_img/tex_31209c0x.png" alt=" ∫ inf ϕ (∇y)dx, kerϕ = {u,u , ...,u }⊕ ℝ2×2 , y∈W 1,∞ (Ω) Ω 1 2 n skew y|∂Ω=Fx " class="math-display"></center> | (1) |
where {u1,u2,…,un}⊂ ℝsym2×2 and the function ϕ : ℝ2×2 → ℝ satisfies mild growth conditions and it is invariant under addition of skew-symmetric matrices to its argument.
In this talk, I will present some recent results about the relaxed problem via quasiconvexification. More precisely, it will be proved that the quasiconvex hull of kerϕ equals its convex hull if the n wells are pairsewise symmetrized-rank-one connected. Particularly, in the three-well problem if one of this connection is lost, then the contention of the quasiconvex hull of kerϕ in its convex hull is strict.