Nonlocal regularization of L. C. Young's tacking problem

In this paper we study the existence of solutions for a type of nonlocal minimization problem. The minimization principle is relaxed by means of Young measures, and different generalized necessary conditions of optimality are used: the Euler-Lagrange equation and the extended version of the Weierstrass minimum principle. These conditions are applied to establish some results concerning the existence and the description of the simplest relaxation of the problem in terms of Young measures. The main result of the paper is the description of the simplest relaxation by considering only a combination of two Dirac measures. The research is focused on the homogeneous and nonhomogeneous 1-dimensional scalar cases. The results obtained are numerically exploited to compute the lower semicontinuous envelope for some academic examples. A detailed numerical analysis of the nonlocal regularization for Young's tacking problem formulated in [D. Brandon and R. Rogers, Appl. Math. Optim., 25 (1992), pp. 287-301] is given.

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“…[1], [7], [9], [8], [13], [2], [15]. Nonlocal elasticity is another field where this sort of principle plays a crucial role; cf.…”

Section: Our Nonlocal Problem Compared To the Classical Onementioning

confidence: 99%

Some Aspects of the Existence of Minimizers for a Nonlocal Integral Functional in Dimension One

Castellanos¹,

Múñoz²

2010

SIAM J. Control Optim.

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“…For a sufficiently smooth classical magnetization m, the magnetic field is defined to be the solution of the equations Curl h = 0, (2)(3)(4)(5) Div h = -Div m (2.6) and the jump conditions…”

Section: Definitionsmentioning

confidence: 99%

“…While this relieves us of the difficulty of dealing directly with the Young-measures, we must still deal with a point.wise inequality constraint (3.9). To get around this difficulty we follow the procedure of [2] and introduce slack variables Vi, i = 1,2,3, such that gi{x) = mj(x) + v? (x), i = 1,2,3.…”

Section: J Jnmentioning

confidence: 99%

Analysis of a rigid ferromagnetic body under applied magnetic fields

Brandon¹,

Rogers²

1996

Quart. Appl. Math.

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“…Concerning this type of problems several works have been published. In connection with variational problems of nonlocal nature the reader can consult [5] for problems related to Ferromagnetism, [6] about the regularization of a nonconvex problem, and [3,12] or [13] in order to analyze mechanical problems formulated in the general context of the Nonlocal Elasticity (see also [8]). In [1] and [15] some interesting tools to obtain a full relaxation of specific nonlocal variational problems have been analyzed, and [7] is also remarkable work for a general class of nonlocal integral functionals.…”

Section: Introductionmentioning

confidence: 99%