It is generally considered that the linear filamentation instability encountered when two counter streaming electron beams interpenetrate is purely transverse. Exact and approximated results are derived in the relativistic fluid approximation showing that within some parameter range, filamentation can be indeed almost longitudinal with cos(k,Ê)≲1−3.1∕γb, where γb is the relativistic factor of the beam. Temperature effects are then evaluated through relativistic kinetic theory and yield even fewer transverse filamentation modes. In the cold case, the transverse approximation overestimates the growth rate by a factor ∝γb.

We study nonlocal variational problems in L p , like those that appear in peridynamics. The functional object of our study is given by a double integral. We establish characterizations of weak lower semicontinuity of the functional in terms of nonlocal versions of either a convexity notion of the integrand, or a Jensen inequality for Young measures. Existence results, obtained through the direct method of the Calculus of variations, are also established. We cover different boundary conditions, for which the coercivity is obtained from nonlocal Poincaré inequalities. Finally, we analyze the relaxation (that is, the computation of the lower semicontinuous envelope) for this problem when the lower semicontinuity fails. We state a general relaxation result in terms of Young measures and show, by means of two examples, the difficulty of having a relaxation in L p in an integral form. At the root of this difficulty lies the fact that, contrary to what happens for local functionals, non-positive integrands may give rise to positive nonlocal functionals.

We present an existence theory based on minimization of the nonlocal energies appearing in peridynamics, which is a nonlocal continuum model in Solid Mechanics that avoids the use of deformation gradients. We employ the direct method of the calculus of variations in order to find minimizers of the energy of a deformation. Lower semicontinuity is proved under a weaker condition than convexity, whereas coercivity is proved via a nonlocal Poincaré inequality. We cover Dirichlet, Neumann and mixed boundary conditions. The existence theory is set in the Lebesgue L p spaces and in the fractional Sobolev W s,p spaces, for 0 < s < 1 and 1 < p < ∞.for some 1 < p < ∞ and 0 ≤ α < n + p. For this special growth, we distinguish the weakly singular case 0 ≤ α < n and the strongly singular case n < α < n + p. When 0 ≤ α < n, the analysis of the lower semicontinuity is reduced to the recent study carried out by Elbau [22] and lies in the functional framework of Lebesgue L p spaces. The weak lower semicontinuity is proved in [22] to be equivalent to an interesting convexity property of the integrand w, of a different nature that those convexity properties equivalent to weak lower semicontinuity for local problems (see, e.g., [16, Ch. 8]); we will discuss this issue in Section 3 in our particular peridynamics framework. The coercivity for the Dirichlet problem was proved by Andreu et al. [7] in their study of nonlocal diffusion problems, and later used by [3,26] in the context of peridynamics. The coercivity for the Neumann and mixed problem was proved by Aksoylu & Mengesha [2] using a Poincaré-type inequality proved by Ponce [36] in his study of nonlocal characterizations of Sobolev spaces (see also [13]). As a matter of fact, we shall need some adaptations of those results to our context. At this point, we ought to mention that Dirichlet and mixed boundary value problems have a slightly different meaning than for local problems, one the reasons being that L p functions do not have traces of the boundary ∂Ω. In contrast, Dirichlet conditions in the context of peridynamics prescribe the value of the deformation in a set of positive measure.The lower semicontinuity in the case n < α < n + p is in fact trivial, since the functional framework is that of the fractional Sobolev spaces W s,p with s = α−n p , and weak convergence in W s,p implies (for a subsequence) convergence a.e. The coercivity, on the other hand, is a consequence of an improved Poincaré-type inequality in fractional Sobolev spaces recently proved in Hurri-Syrjänen & Vähäkangas [27]. It is worth mentioning that the need of improved Poincaré-type inequalities is a result of the assumption that w(x, ·) vanishes for |x| large.The existence theory for the critical case α = n is also covered by reducing it to the case 0 ≤ α < n and to the functional framework of L p spaces. In doing that, we do not provide a full characterization of the lower semicontinuity, so that our conditions on w may not be optimal.Nonlocal variational problems, of which (1.1) is a particular case, h...

In this paper we propose a nonlocal model of hyperelasticity obtained by substitution of the classical gradient by the Riesz fractional gradient. We show existence of solutions for those nonlocal models in Bessel fractional spaces under the main assumption of polyconvexity of the energy density. The main ingredient is the fractional Piola identity, which establishes that the fractional divergence of the cofactor matrix of the fractional gradient vanishes. This identity implies the weak convergence of the determinant of the fractional gradient, and, in turn, the existence of minimizers of the nonlocal hyperelastic energy.

Eringen's model is one of the most popular theories in nonlocal elasticity. It has been applied to many practical situations with the objective of removing the anomalous stress concentrations around geometric shape singularities, which appear when the local modelling is used. Despite the great popularity of Eringen's model in mechanical engineering community, even the most basic questions such as the existence and uniqueness of solutions have been rarely considered in the research literature for this model. In this work we focus on precisely these questions, proving that the model is in general ill-posed in the case of smooth kernels, the case which appears rather often in numerical studies. We also consider the case of singular, non-smooth kernels, and for the paradigmatic case of the Riesz potential we establish the well-posedness of the model in fractional Sobolev spaces. For such a kernel, in dimension one the model reduces to the well-known fractional Laplacian. Finally, we discuss possible extensions of Eringen's model to spatially heterogeneous material distributions.