Linear dependence of bivariate Minimal Support and Locally Refined B-splines over LR-meshes

The focus on locally refined spline spaces has grown rapidly in recent years due to the need in Isogeoemtric analysis (IgA) of spline spaces with local adaptivity: a property not offered by the strict regular structure of tensor product B-spline spaces. However, this flexibility sometimes results in collections of B-splines spanning the space that are not linearly independent. In this paper we address the minimal number of B-splines that can form a linear dependence relation for Minimal Support B-splines (MS B-splines) and for Locally Refinable B-splines (LR B-splines) on LR-meshes. We show that the minimal number is six for MS B-splines, and eight for LR B-splines. The risk of linear dependency is consequently significantly higher for MS B-splines than for LR B-splines. Further results are established to help detecting collections of B-splines that are linearly independent

Crystal dislocations with different orientations and collisions

We study a parabolic differential equation whose solution represents the atom dislocation in a crystal for a general type of Peierls-Nabarro model with possibly long range interactions and an external stress. Differently from the previous literature, we treat here the case in which such dislocation is not the superpositions of transitions all occurring with the same orientations (i.e. opposite orientations are allowed as well)

A Neumann eigenvalue problem for fully nonlinear operators

In this paper we study the asymptotic behavior of the principal eigenvalues associated to the Pucci operator in bounded domain $\Omega$ with Neumann/Robin boundary condition i.e. $\partial_n u=\alpha u$ when $\alpha$ tends to infinity. This study requires Lipschitz estimates up to the boundary that are interesting in their own rights.Comment: 19 page

Obstacle Mean-Field Game Problem

In this paper, we introduce and study a first-order mean-field game obstacle problem. We examine the case of local dependence on the measure under assumptions that include both the logarithmic case and power-like nonlinearities. Since the obstacle operator is not differentiable, the equations for first-order mean field game problems have to be discussed carefully. Hence, we begin by considering a penalized problem. We prove this problem admits a unique solution satisfying uniform bounds. These bounds serve to pass to the limit in the penalized problem and to characterize the limiting equations. Finally, we prove uniqueness of solutions

Relaxation times for atom dislocations in crystals

We study the relaxation times for a parabolic differential equation whose solution represents the atom dislocation in a crystal. The equation that we consider comprises the classical Peierls-Nabarro model as a particular case, and it allows also long range interactions. It is known that the dislocation function of such a model has the tendency to concentrate at single points, which evolve in time according to the external stress and a singular, long range potential. Depending on the orientation of the dislocation function at these points, the potential may be either attractive or repulsive, hence collisions may occur in the latter case and, at the collision time, the dislocation function does not disappear. The goal of this paper is to provide accurate estimates on the relaxation times of the system after collision. More precisely, we take into account the case of two and three colliding points, and we show that, after a small transition time subsequent to the collision, the dislocation function relaxes exponentially fast to a steady state. We stress that the exponential decay is somehow exceptional in nonlocal problems (for instance, the spatial decay in this case is polynomial). The exponential time decay is due to the coupling (in a suitable space/time scale) between the evolution term and the potential induced by the periodicity of the crystal

Homogenization of the Peierls-Nabarro model for dislocation dynamics

This paper is concerned with a result of homogenization of an integro-differential equation describing dislocation dynamics. Our model involves both an anisotropic L\'{e}vy operator of order 1 and a potential depending periodically on u/\ep. The limit equation is a non-local Hamilton-Jacobi equation, which is an effective plastic law for densities of dislocations moving in a single slip plane.Comment: 39 page