Scaling laws for linear-elastic two-variant martensitic phase transitions in general domains
Abstract
We consider a two-well problem within geometrically linear elasticity, on a bounded Lipschitz set in dimension n≥ 2. We assume that the two wells are compatible through a symmetrized rank-one connection and that the displacement matches their average on the boundary, and we include a regularization based on the total surface energy. We show that there are two distinct regimes, with the energy scaling differently in terms of the surface-energy coefficient eps: it is proportional to eps2/3 if there are two distinct compatible twin-normal directions, and to eps4/5 if there is only one. This matches previous findings in rectangles and other special geometries, mostly in two dimensions, and the lower bounds obtained by a Fourier method by Raiţă, Rüland, Tissot and Tribuzio (2024). Our results are obtained by explicit constructions, which contain a self-similar refinement of oscillations close to the boundary, and by a short real-space proof of the matching lower bounds.
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BibTeX
@misc{conti2026scaling,
title = {{Scaling laws for linear-elastic two-variant martensitic phase transitions in general domains}},
author = {Sergio Conti and Oleksandr Misiats},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04329},
eprint = {2611.04329},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04329},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Scaling laws for linear-elastic two-variant martensitic phase transitions in general domains
AU - Sergio Conti
AU - Oleksandr Misiats
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04329
UR - https://arxiv.org/abs/2611.04329
AB - We consider a two-well problem within geometrically linear elasticity, on a bounded Lipschitz set in dimension n≥ 2. We assume that the two wells are compatible through a symmetrized rank-one connection and that the displacement matches their average on the boundary, and we include a regularization based on the total surface energy. We show that there are two distinct regimes, with the energy scaling differently in terms of the surface-energy coefficient eps: it is proportional to eps^2/3 if there are two distinct compatible twin-normal directions, and to eps^4/5 if there is only one. This matches previous findings in rectangles and other special geometries, mostly in two dimensions, and the lower bounds obtained by a Fourier method by Raiţă, Rüland, Tissot and Tribuzio (2024). Our results are obtained by explicit constructions, which contain a self-similar refinement of oscillations close to the boundary, and by a short real-space proof of the matching lower bounds.
ER -