Lower Semicontinuity and Relaxation Via Young Measures for Nonlocal Variational Problems and Applications to Peridynamics

Abstract

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.

Lower semicontinuity and relaxation of nonlocal $$L^\infty $$-functionals

“…(2.6); indeed, if (u j ) j generates the Young measure ν = {ν x } x∈ , the sought-after set consists of all the product measures = { (x,y) } (x,y)∈ × = {ν x ⊗ ν y } (x,y)∈ × with supp contained almost everywhere in a Cartesian subset of K , see Theorem 5.12 for the precise statement. Interpreted in the context of indicator functionals, the latter yields a Young measure relaxation result for a class of unbounded functionals (defined precisely in (6.6)), extending part of a recent work by Bellido and Mora-Corral [12,Section 6], cf. Sect.…”

“…Results about inhomogeneous double-integral functionals, meaning with integrands W depending also explicitly on x, y ∈ , can be found e.g. in [12,37,41].…”

“…As discussed recently in [12], there are different 'nonlocal' definitions of convexity related to the weak lower semicontinuity of I , which coincide under suitable assumptions. In Sect.…”

“…The question of relaxation of functionals I as in (2.9) for which the density W fails to be separately convex is still mostly open. It may seem counter-intuitive, but there are examples [12,14,41] indicating that separate convexification of W does in general not give rise to the right candidate for the weakly lower semicontinuous envelope of I . Even more remarkably, as recently proven in [32,35], relaxation in the weak L p -topology of double-integrals functionals cannot always be expected to be structure-preserving.…”

“…When it comes to relaxation, meaning the characterization of weak lower semicontinuous envelopes, though, the problem is still largely open. The difficulty lies in the fact that, counterintuitively, relaxation formulas in general cannot be obtained via separate convexification of the integrands, as explicit examples in [12,14,41] indicate. As first shown in [32], and with different techniques in [35], even a representation of the relaxation with a double integral of the same type is not always possible.…”