Dimension reduction for functionals on solenoidal vector fields

We study integral functionals constrained to divergence-free vector fields in $L^p$ on a thin domain, under standard $p$-growth and coercivity assumptions, $1<p<\infty$. We prove that as the thickness of the domain goes to zero, the Gamma-limit with respect to weak convergence in $L^p$ is always given by the associated functional with convexified energy density wherever it is finite. Remarkably, this happens despite the fact that relaxation of nonconvex functionals subject to the limiting constraint can give rise to a nonlocal functional as illustrated in an example.Comment: 25 page

Heterogeneous thin films: Combining homogenization and dimension reduction with directors

We analyze the asymptotic behavior of a multiscale problem given by a sequence of integral functionals subject to differential constraints conveyed by a constant-rank operator with two characteristic length scales, namely the film thickness and the period of oscillating microstructures, by means of $\Gamma$-convergence. On a technical level, this requires a subtile merging of homogenization tools, such as multiscale convergence methods, with dimension reduction techniques for functionals subject to differential constraints. One observes that the results depend critically on the relative magnitude between the two scales. Interestingly, this even regards the fundamental question of locality of the limit model, and, in particular, leads to new findings also in the gradient case.Comment: 28 page

Scan matching by cross-correlation and differential evolution

Scan matching is an important task, solved in the context of many high-level problems including pose estimation, indoor localization, simultaneous localization and mapping and others. Methods that are accurate and adaptive and at the same time computationally efficient are required to enable location-based services in autonomous mobile devices. Such devices usually have a wide range of high-resolution sensors but only a limited processing power and constrained energy supply. This work introduces a novel high-level scan matching strategy that uses a combination of two advanced algorithms recently used in this field: cross-correlation and differential evolution. The cross-correlation between two laser range scans is used as an efficient measure of scan alignment and the differential evolution algorithm is used to search for the parameters of a transformation that aligns the scans. The proposed method was experimentally validated and showed good ability to match laser range scans taken shortly after each other and an excellent ability to match laser range scans taken with longer time intervals between them.Web of Science88art. no. 85

Hofmannsthal und Azorín zwischen Asthetizismus und Zeitproblemen.

Sin resume

Scaling of the elastic energy of small balls for maps between manifolds with different curvature tensors

Motivated by experiments and formal asymptotic expansions in the physics literature, Maor and Shachar (J. Elasticity 134 (2019), 149-173) studied the behaviour of a model elastic energy of maps between manifolds with incompatible metrics. For thin objects they analysed the scaling of the minimal elastic energy as a function of the thickness. In particular they showed that for maps from a ball of radius h in an oriented Riemannian manifold to Euclidean space, the infimum of a model elastic energy per unit volume scales like the fourth power of h and after rescaling one gets convergence to a quadratic expression in the curvature tensor R(p), where p denotes the centre of the ball. In this paper we show the same result for general compact oriented Riemannian targets with R(p) replaced by a suitable difference of the curvature tensors in the target and the domain, thus answering Open Question 1 in the paper by Maor and Shachar. The result extends noncompact targets provided they satisfy a uniform regularity condition. A key idea in the proof is to use Lipschitz approximations to define a suitable notion of convergence.Comment: Typos corrected, exposition slightly expanded, reference [2] adde