Existence of Lipschitz minimizers for the three-well problem in solid-solid phase transitions

Sergio Conti; Georg Dolzmann; Bernd Kirchheim

Displaying similar documents to “Existence of Lipschitz minimizers for the three-well problem in solid-solid phase transitions”

On lower-semicontinuity of variational integrals

Šverák, Vladimír

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New convexity conditions in the calculus of variations and compensated compactness theory

Krzysztof Chełmiński, Agnieszka Kałamajska (2006)

ESAIM: Control, Optimisation and Calculus of Variations

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We consider the lower semicontinuous functional of the form $I_{f} (u) = \int_{Ω} f (u) d x$ where $u$ satisfies a given conservation law defined by differential operator of degree one with constant coefficients. We show that under certain constraints the well known Murat and Tartar’s $Λ$ -convexity condition for the integrand $f$ extends to the new geometric conditions satisfied on four dimensional symplexes. Similar conditions on three dimensional symplexes were recently obtained by the second author. New conditions apply...

A complete characterization of invariant jointly rank- convex quadratic forms and applications to composite materials

Vincenzo Nesi, Enrico Rogora (2007)

ESAIM: Control, Optimisation and Calculus of Variations

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The theory of compensated compactness of Murat and Tartar links the algebraic condition of rank- convexity with the analytic condition of weak lower semicontinuity. The former is an algebraic condition and therefore it is, in principle, very easy to use. However, in applications of this theory, the need for an efficient classification of rank- convex forms arises. In the present paper, we define the concept of extremal -forms and characterize them in the rotationally invariant...

A note on equality of functional envelopes

Martin Kružík (2003)

Mathematica Bohemica

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We characterize generalized extreme points of compact convex sets. In particular, we show that if the polyconvex hull is convex in $ℝ^{m \times n}$ , $min (m, n) \leq 2$ , then it is constructed from polyconvex extreme points via sequential lamination. Further, we give theorems ensuring equality of the quasiconvex (polyconvex) and the rank-1 convex envelopes of a lower semicontinuous function without explicit convexity assumptions on the quasiconvex (polyconvex) envelope.

Remarks on the quasiconvex envelope of some functions depending on quadratic forms

M. Bousselsal, H. Le Dret (2002)

Bollettino dell'Unione Matematica Italiana

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We compute the quasiconvex envelope of certain functions defined on the space $M_{m n}$ of real $m \times n$ matrices. These functions are basically functions of a quadratic form on $M_{m n}$ . The quasiconvex envelope computation is applied to densities that are related to the James-Ericksen elastic stored energy function.