A Two Well Liouville Theorem

Lorent, Andrew

doi:10.1051/cocv:2005009

Cited by 14 publications

(14 citation statements)

References 15 publications

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“…With respect to the quantitative results of [4,8,11,14], we stress that, in spite of being also rigidity results for nonzero energy states, they are essentially different from the ones in the present paper. First, they differ in spirit, because the former ones rely on estimates involving the distances between ru and K, and ru and a particular energy well (estimates which we don't attempt).…”

Section: Introductioncontrasting

confidence: 99%

“…The first quantitative version of Dolzmann and Müller's result, for the nonlinear two-well problem, was proved by Lorent [14] for bi-Lipschitz maps and two wells with equal determinant. More precisely, he proved that if the perimeter of the transition in a ball, as controlled by kD 2 uk.B 1 /, is sufficiently small, there exist a certain power < 1 of the L 1 -norm of the distance between ru and the set K that controls the L 1 -norm of the distance of ru to one of the wells in some smaller ball.…”

Section: Introductionmentioning

confidence: 95%

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A rigidity result for a perturbation of the geometrically linear three‐well problem

Capella

Otto

2009

Comm Pure Appl Math

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We study a three-dimensional model for alloys that undergo a cubic-to-tetragonal phase transition in the martensitic phase. Any pair of the three martensitic variants can form a stress-free laminate. However, this laminate is only compatible on average with the remaining variant. The resulting local stresses favor a microstructure if all three variants are present (for instance, because of an externally imposed average strain).Next to the linearized elastic energy, the variational model features an interfacial energy between the three variants. This introduces a material length scale, which together with the sample size (which we mimic by periodic boundary conditions) gives rise to a nondimensional parameter Á.We rigorously establish the scaling of the minimal energy e per volume in case of externally imposed volume fractions of the martensitic variants in Á. More precisely, we show e Á 2=3 . The upper bound construction is achieved by a few patches of branched laminates; the lower bound relies on suitable interpolation inequalities. This is in the spirit of a celebrated work by Kohn and Müller and relies on techniques developed for domain branching in micromagnetics.We also prove a rigidity result in the sense that if the energy per volume e of a configuration is much smaller than Á 2=3 , the configuration is approximately a simple unbranched laminate. In particular, one of the three volume fractions has to be small. This is related to similar rigidity results by Dolzmann and Müller and by Kirchheim.

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Section: Introductioncontrasting

confidence: 99%

Section: Introductionmentioning

confidence: 95%

A rigidity result for a perturbation of the geometrically linear three‐well problem

Capella

Otto

2009

Comm Pure Appl Math

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“…This improves a previous result by Lorent [32], who, under the additional assumptions that u is bi-Lipschitz and det A = det B, obtained that the minimum of dist(∇u, F) L 1 ( ) over F ∈ K is controlled by a power of dist(∇u, K ) L 1 ( ) . In the limiting case where u satisfies ∇u ∈ K a.e.…”

Section: Introductionsupporting

confidence: 86%

“…A first result in this direction was obtained independently by Lorent [32] for the case that det A = det B and u is bi-Lipschitz (i.e., Lipschitz with Lipschitz inverse). Specifically, Lorent proved that (2.1) implies…”

Section: Introductionmentioning

confidence: 97%

Rigidity and gamma convergence for solid‐solid phase transitions with SO(2) invariance

Conti

Schweizer

2006

Comm Pure Appl Math

160

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The singularly perturbed two-well problem in the theory of solid-solid phase transitions takes the formwhere u : ⊂ R n → R n is the deformation, and W vanishes for all matrices in K = SO(n)A ∪ SO(n)B. We focus on the case n = 2 and derive, by means of Gamma convergence, a sharp-interface limit for I ε . The proof is based on a rigidity estimate for low-energy functions. Our rigidity argument also gives an optimal two-well Liouville estimate: if ∇u has a small BV norm (compared to the diameter of the domain), then, in the L 1 sense, either the distance of ∇u from SO(2)A or the one from SO(2)B is controlled by the distance of ∇u from K . This implies that the oscillation of ∇u in weak L 1 is controlled by the L 1 norm of the distance of ∇u to K .

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“…For the special case of a functional whose wells are given by two rank-1 connected matrices a complete understanding of the scaling has been achieved in [15], [6]. Our main tool for studying this question is a two well Liouville Theorem proved in [17] (see Theorem 1.1). In order to use it we will have to minimise over a subset of A F .…”

Section: The Question: How Does I Scale ?mentioning

confidence: 99%