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Attainment results for the two-well problem by convex integration. (English) Zbl 0930.35038
Jost, Jürgen (ed.), Geometric analysis and the calculus of variations. Dedicated to Stefan Hildebrandt on the occasion of his 60th birthday. Cambridge, MA: International Press. 239-251 (1996).
Summary: Models for solid-solid phase transitions lead to the problem of finding Lipschitz continuous maps $$u: \Omega\subset \mathbb{R}^n\to \mathbb{R}^m$$ which satisfy $$\nabla u\in K$$ (for a given set $$K$$ of $$m\times n$$ matrices) as well as suitable boundary conditions. We show how Gromov’s method of convex integration can be used to construct such solutions and give an application to the two-well problem where $$K= \text{SO}(2)A\cup \text{SO}(2)B$$.
For the entire collection see [Zbl 0914.00109].

##### MSC:
 35F05 Linear first-order PDEs 35R70 PDEs with multivalued right-hand sides