Existence for Nonlocal Variational Problems in Peridynamics

Abstract

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...

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