Two slope functions minimizing fractional seminorms and applications to misfit dislocations

TL;DR

This paper analyzes periodic piecewise affine functions with two slopes to identify minimizers of fractional seminorms, revealing their periodicity and asymptotic behavior, with applications to misfit dislocations in materials science.

Contribution

It proves that minimizers of fractional seminorms are periodic with minimal period and determines their asymptotic energy behavior, applying these results to misfit dislocation configurations.

Findings

Minimizers are periodic with minimal period based on slopes and length scale.

Asymptotic energy density is characterized as interval length ratio vanishes.

Optimal dislocation configurations are proven to be periodic.

Abstract

We consider periodic piecewise affine functions, defined on the real line, with two given slopes and prescribed length scale of the regions where the slope is negative. We prove that, in such a class, the minimizers of $s$-fractional Gagliardo seminorm densities, with $0<s<1$, are in fact periodic with the minimal possible period determined by the prescribed slopes and length scale. Then, we determine the asymptotic behavior of the energy density as the ratio between the length of the two intervals where the slope is constant vanishes. Our results, for $s=\frac 1 2$, have relevant applications to the van der Merwe theory of misfit dislocations at semi-coherent straight interfaces. We consider two elastic materials having different elastic coefficients and casting parallel lattices having different spacing. As a byproduct of our analysis, we prove the periodicity of optimal dislocation configurations and we provide the sharp asymptotic energy density in the semi-coherent limit as the ratio between the two lattice spacings tends to one.