Characterisation of the weak lower semicontinuity for a type of nonlocal integral functional: The n-dimensional scalar case

Múñoz, Julio

doi:10.1016/j.jmaa.2009.06.068

Cited by 14 publications

(19 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This condition was quite implicit, but could be shown to be equivalent to the separate convexity of the integrand W a decade later by Bevan and Pedregal [14]. Also in the vectorial case, W being separately convex is the characterizing property to ensure weak lower semicontinuity of I , as Muñoz proved in [37]; the latter is formulated in the gradient setting, using W 1, p -weak convergence of scalar valued functions, but the statement and the ideas of the proof carry over to functionals of the form (2.9), cf. [41].…”

Section: Double-integral Functionals and Separate Convexitymentioning

confidence: 94%

“…Nonlocal functionals in the form of double integrals appear naturally in different applications; examples include peridynamics [13,34,47], image processing [16,27] or the theory of phase transitions [20,22,46]. In the homogeneous case, separate convexity of the integrands has been identified as a necessary and sufficient condition for the weak lower semicontinuity of such functionals [14,37,39]. When it comes to relaxation, meaning the characterization of weak lower semicontinuous envelopes, though, the problem is still largely open.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Kreisbeck

Zappale

2020

Calc. Var.

View full text Add to dashboard Cite

We study variational problems involving nonlocal supremal functionalswhere ⊂ R n is a bounded, open set and W : R m × R m → R is a suitable function. Motivated by existence theory via the direct method, we identify a necessary and sufficient condition for L ∞ -weak * lower semicontinuity of these functionals, namely, separate level convexity of a symmetrized and suitably diagonalized version of the supremands. More generally, we show that the supremal structure of the functionals is preserved during the process of relaxation. The analogous statement in the related context of double-integral functionals was recently shown to be false. Our proof relies substantially on the connection between supremal and indicator functionals. This allows us to recast the relaxation problem into characterizing weak * closures of a class of nonlocal inclusions, which is of independent interest. To illustrate the theory, we determine explicit relaxation formulas for examples of functionals with different multi-well supremands.

show abstract

Section: Double-integral Functionals and Separate Convexitymentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

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

Kreisbeck

Zappale

2020

Calc. Var.

View full text Add to dashboard Cite

show abstract

“…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, have attracted a great attention in the mathematical community in the last two decades, coming from fields such as statistical mechanics [5], abstract results involving nonlocality of gradients [34,33], ferromagnetism [38], nonlocal p-Laplacian [8], imaging [28,24,12], characterization of Sobolev spaces [13,14,36,37,32], as well as, of course, peridynamics [2,18,21,26,25,4].The outline of the paper is as follows. In Section 2 we present the mechanical model, make the general assumptions of the paper, and explain the notation used.…”

mentioning

confidence: 99%

“…Nonlocal variational problems, of which (1.1) is a particular case, have attracted a great attention in the mathematical community in the last two decades, coming from fields such as statistical mechanics [5], abstract results involving nonlocality of gradients [34,33], ferromagnetism [38], nonlocal p-Laplacian [8], imaging [28,24,12], characterization of Sobolev spaces [13,14,36,37,32], as well as, of course, peridynamics [2,18,21,26,25,4].…”

mentioning

confidence: 99%

Existence for Nonlocal Variational Problems in Peridynamics

Bellido

Mora-Corral²

2014

SIAM J. Math. Anal.

View full text Add to dashboard Cite

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...

show abstract

“…Bevan and Pedregal [4] had found a necessary and sufficient condition for the weak lower semicontinuity of I , and recently, Muñoz [18] have extended the result for the n-dimensional case. Namely, they have proved that I is weak lower semicontinuous if and only if the symmetric part of W is separately convex.…”

Section: Introductionmentioning

confidence: 96%

The method of moments for some one-dimensional, non-local, non-convex variational problems

Aranda

Meziat

2011

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

We focus on the analysis of homogeneous, non-convex, non-local variational principles given in the general form:where W can be expressed as a polynomial. This work extends previous result of Pedregal (1997) [19] by using moments of parametrized measures to transform the generalized form of the problem in terms of Young measures into a more convenient mathematical program with quadratic and conic structure.

show abstract

Characterisation of the weak lower semicontinuity for a type of nonlocal integral functional: The n-dimensional scalar case

Cited by 14 publications

References 11 publications

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

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

Existence for Nonlocal Variational Problems in Peridynamics

The method of moments for some one-dimensional, non-local, non-convex variational problems

Contact Info

Product

Resources

About