In this paper a we derive by means of Γ-convergence a macroscopic strain-gradient plasticity from a semi-discrete model for dislocations in an infinite cylindrical crystal. In contrast to existing work, we consider an energy with subquadratic growth close to the dislocations. This allows to treat the stored elastic energy without the need to introduce an ad-hoc cut-off radius. As the main tool to prove a complementing compactness statement, we present a generalized version of the geometric rigidity result for fields with non-vanishing curl. A main ingredient is a fine decomposition result for L 1 -functions whose divergence is in certain critical Sobolev spaces.If β is a gradient then this estimates reduces to the geometric rigidity result in [29]. This estimate allows us to control the rotational invariance of the energy in order to obtain a compactness result for a sequence of suitably rescaled elastic strains which induce a uniformly bounded rescaled elastic energy, Theorem 1.2. The proof of the generalized (nonlinear) Korn's inequality in [20] and [30] rely on a fine estimate due to Bourgain and Brézis (see [3,4] and also [5]). It states that an L 1 -function in two dimensions, whose divergence is in H −2 , is already in H −1 andFor our mixed-growth situation we prove a corresponding result, namely we show that an L 1 -function whose divergence is in the space H −2 + W −2,p , for 1 < p < 2, belongs to the space H −1 + W −1,p . Corresponding estimates hold, Theorem 1.4.The paper is organized as follows. In Subsection 1.1 we introduce notation and the mathematical setting of the problem. The main results are presented in Subsection 1.2. In Section 2 we prove the generalized Bourgain-Brézis decomposition result which we use in Section 3 to show the generalized geometric rigidity result in the mixed growth case. Finally, the proof of the Γ-convergence result can be found in Section 4. Setting of the ProblemLet Ω ⊆ R 2 be a simply-connected, bounded domain with Lipschitz boundary representing the cross section of an infinite cylindrical crystal. The set of (normalized) minimal Burgers vectors for the given crystal is denoted by S = {b 1 , b 2 } for two linearly independent vectors b 1 , b 2 ∈ R 2 . Moreover, set S = span Z S = {λ 1 b 1 + λ 2 b 2 : λ 1 , λ 2 ∈ Z} to be the the set of (renormalized) admissible Burgers vectors. Let ε > 0 the interatomic distance for the given crystal. We define the set of admissible dislocation densities as a subset of the R 2 -valued Radon measures M(Ω; R 2 ), namelywhere we assume that ρ ε satisfies 1. lim ε→0 ρ ε /ε s = ∞ for all fixed s ∈ (0, 1) and

We study pattern formation in magnetic compounds near the helimagnetic/ferromagnetic transition point in case of Dirichlet boundary conditions on the spin field. The energy functional is a continuum approximation of a $$J_1-J_3$$ J 1 - J 3 model and was recently derived in Cicalese et al. (SIAM J Math Anal 51: 4848–4893, 2019). It contains two parameters, one measuring the incompatibility of the boundary conditions and the other measuring the cost of changes between different chiralities. We prove the scaling law of the minimal energy in terms of these two parameters. The constructions from the upper bound indicate that in some regimes branching-type patterns form close to the boundary of the sample.

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The estimation of motion in an image sequence is a fundamental task in image processing. Frequently, the image sequence is corrupted by noise and one simultaneously asks for the underlying motion field and a restored sequence. In smoothly shaded regions of the restored image sequence the brightness constancy assumption along motion paths leads to a pointwise differential condition on the motion field. At object boundaries which are edge discontinuities both for the image intensity and for the motion field this condition is no longer well defined. In this paper a total-variation type functional is discussed for joint image restoration and motion estimation. This functional turns out not to be lower semicontinuous, and in particular fine-scale oscillations may appear around edges. By the general theory of vector valued BV functionals its relaxation leads to the appearance of a singular part of the energy density, which can be determined by the solution of a local minimization problem at edges. Based on bounds for the singular part of the energy and under appropriate assumptions on the local intensity variation one can exclude the existence of microstructures and obtain a model well-suited for simultaneous image restoration and motion estimation. Indeed, the relaxed model incorporates a generalized variational formulation of the brightness constancy assumption. The analytical findings are related to ambiguity problems in motion estimation such as the proper distinction between foreground and background motion at object edges.