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Local minimizers and planar interfaces in a phase-transition model with interfacial energy

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Abstract

Interfacial energy is often incorporated into variational solid-solid phase transition models via a perturbation of the elastic energy functional involving second gradients of the deformation. We study consequences of such higher-gradient terms for local minimizers and for interfaces. First it is shown that at slightly sub-critical temperatures, a phase which globally minimizes the elastic energy density at super-critical temperatures is an L 1-local minimizer of the functional including interfacial energy, whereas it is typically only a W 1,∞-local minimizer of the purely elastic functional. The second part deals with the existence and uniqueness of smooth interfaces between different wells of the multi-well elastic energy density. Attention is focussed on so-called planar interfaces, for which the deformation depends on a single direction x · N and the deformation gradient then satisfies a rank-one ansatz of the form \({Dy(x) = A + u(x \cdot N) \otimes N}\) , where A and \({B=A+a \otimes N}\) are the gradients connected by the interface.

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Correspondence to J. M. Ball.

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Communicated by L. Ambrosio.

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Ball, J.M., Crooks, E.C.M. Local minimizers and planar interfaces in a phase-transition model with interfacial energy. Calc. Var. 40, 501–538 (2011). https://doi.org/10.1007/s00526-010-0349-8

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