68 research outputs found
Diffeomorphic approximation of Sobolev homeomorphisms
Every homeomorphism h : X -> Y between planar open sets that belongs to the
Sobolev class W^{1,p}(X,Y), 1<p<\infty, can be approximated in the Sobolev norm
by diffeomorphisms.Comment: 21 pages, 5 figure
Quasiconvexity at the boundary and the nucleation of austenite
Motivated by experimental observations of H. Seiner et al., we study the nucleation of austenite in a single crystal of a CuAlNi shape-memory alloy stabilized as a single variant of martensite. In the experiments the nucleation process was induced by localized heating and it was observed that, regardless of where the localized heating was applied, the nucleation points were always located at one of the corners of the sample - a rectangular parallelepiped in the austenite. Using a simplified nonlinear elasticity model, we propose an explanation for the location of the nucleation points by showing that the martensite is a local minimizer of the energy with respect to localized variations in the interior, on faces and edges of the sample, but not at some corners, where a localized microstructure, involving austenite and a simple laminate of martensite, can lower the energy. The result for the interior, faces and edges is established by showing that the free-energy function satisfies a set of quasiconvexity conditions at the stabilized variant in the interior, faces and edges, respectively, provided the specimen is suitably cut
Approximation of W1,p Sobolev homeomorphism by diffeomorphisms and the signs of the Jacobian
Let ⊠â R n, n â„ 4, be a domain and 1 †p 0 on a set of positive measure and Jf < 0 on a set of positive measure. It follows that there are no diffeomorphisms (or piecewise affine homeomorphisms) fk such that fk â f in W1,p .peerReviewe
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