### A local minimum energy condition of hexagonal circle packing.

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In this paper we analyse the structure of approximate solutions to the compatible two well problem with the constraint that the surface energy of the solution is less than some fixed constant. We prove a quantitative estimate that can be seen as a two well analogue of the Liouville theorem of Friesecke James Müller. Let $H=\left({\textstyle \begin{array}{cc}\sigma & 0\\ 0& {\sigma}^{-1}\end{array}}\right)$ for $\sigma \>0$. Let $0\<{\zeta}_{1}\<1\<{\zeta}_{2}\<\infty $. Let $K:=SO\left(2\right)\cup SO\left(2\right)H$. Let $u\in {W}^{2,1}\left({Q}_{1}\left(0\right)\right)$ be a ${\mathrm{C}}^{1}$ invertible bilipschitz function with $\mathrm{Lip}\left(u\right)\<{\zeta}_{2}$, $\mathrm{Lip}\left({u}^{-1}\right)\<{\zeta}_{1}^{-1}$. There exists positive constants ${\U0001d520}_{1}\<1$ and ${\U0001d520}_{2}\>1$ depending only on $\sigma $, ${\zeta}_{1}$, ${\zeta}_{2}$ such that if $\u03f5\in \left(0,{\U0001d520}_{1}\right)$ and $u$ satisfies the...

In this paper we analyse the structure of approximate solutions to the compatible two well problem with the constraint that the surface energy of the solution is less than some fixed constant. We prove a quantitative estimate that can be seen as a two well analogue of the Liouville theorem of Friesecke James Müller. Let $H=\left({\textstyle \begin{array}{ccc}\sigma & 00& {\sigma}^{-1}\end{array}}\right)$ for $\sigma >0$. Let $0<{\zeta}_{1}<1<{\zeta}_{2}<\infty $. Let $K:=SO\left(2\right)\cup SO\left(2\right)H$. Let $u\in {W}^{2,1}\left({Q}_{1}\left(0\right)\right)$ be a $$ invertible bilipschitz function with $\mathrm{Lip}\left(u\right)<{\zeta}_{2}$, $\mathrm{Lip}\left({u}^{-1}\right)<{\zeta}_{1}^{-1}$. There exists positive constants ${\U0001d520}_{1}<1$ and ${\U0001d520}_{2}>1$ depending only on σ, ${\zeta}_{1}$, ${\zeta}_{2}$ such that if $\u03f5\in \left(0,{\U0001d520}_{1}\right)$ and u satisfies...

Rate-independent evolution for material models with nonconvex elastic energies is studied without any spatial regularization of the inner variable; due to lack of convexity, the model is developed in the framework of Young measures. An existence result for the quasistatic evolution is obtained in terms of compatible systems of Young measures. We also show as this result can be equivalently reformulated with probabilistic language and leads to the description of the quasistatic evolution in terms...

We establish an approximation theorem for a sequence of linear elastic strains approaching a compact set in ${L}^{1}$ by the sequence of linear strains of mapping bounded in Sobolev space ${W}^{1,p}$. We apply this result to establish equalities for semiconvex envelopes for functions defined on linear strains via a construction of quasiconvex functions with linear growth.

We establish an approximation theorem for a sequence of linear elastic strains approaching a compact set in L1 by the sequence of linear strains of mapping bounded in Sobolev space W1,p . We apply this result to establish equalities for semiconvex envelopes for functions defined on linear strains via a construction of quasiconvex functions with linear growth.

We prove by giving an example that when $n\ge 3$ the asymptotic behavior of functionals ${\int}_{\Omega}\epsilon |{\nabla}^{2}{u|}^{2}{+(1-|\nabla u|}^{2}{)}^{2}/\epsilon $ is quite different with respect to the planar case. In particular we show that the one-dimensional ansatz due to Aviles and Giga in the planar case (see [2]) is no longer true in higher dimensions.

We prove by giving an example that when n ≥ 3 the asymptotic behavior of functionals ${\int}_{\Omega}\epsilon |{\nabla}^{2}{u|}^{2}{+(1-|\nabla u|}^{2}{)}^{2}/\epsilon $ is quite different with respect to the planar case. In particular we show that the one-dimensional ansatz due to Aviles and Giga in the planar case (see [2]) is no longer true in higher dimensions.

In this note we give sharp lower bounds for a non-convex functional when minimised over the space of functions that are piecewise affine on a triangular grid and satisfy an affine boundary condition in the second lamination convex hull of the wells of the functional.

The equilibrium configurations of a one-dimensional variational model that combines terms expressing the bulk energy of a deformable crystal and its surface energy are studied. After elimination of the displacement, the problem reduces to the minimization of a nonconvex and nonlocal functional of a single function, the thickness. Depending on a parameter which strengthens one of the terms comprising the energy at the expense of the other, it is shown that this functional may have a stable absolute...

In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have recently been proposed. They aim at coupling an atomistic model (discrete mechanics) with a macroscopic model (continuum mechanics). We provide here a theoretical ground for such a coupling in a one-dimensional setting. We briefly study the general case of a convex energy, and next concentrate on a specific example of a nonconvex energy, the Lennard-Jones case....