Analysis of a rigid ferromagnetic body under applied magnetic fields

Brandon, Deborah; Rogers, Robert C.

doi:10.1090/qam/1388016

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2006

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“…There are several examples of the use of non-local terms in concrete variational problems, ranging from non-local models of phase transitions (see [1,2]), fracture mechanics (see [3]) and ferromagnetism (see, for example, [4,6,10]) to a regularization of Young's tacking problem [5]; in each case the non-local term is tailor-made for the situation at hand. More general information on non-local variational problems can be found in [8], whose emphasis lies more on conditions necessary for the existence of minimizers than on the question of existence itself.…”

Section: Introductionmentioning

confidence: 99%

A necessary and sufficient condition for the weak lower semicontinuity of one-dimensional non-local variational integrals

Bevan

Pedregal

2006

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this short note we prove that the functional I : W1,p(J;R) → R defined by is sequentially weakly lower semicontinuous in W1,p(J,R) if and only if the symmetric part W+ of W is separately convex. We assume that W is real valued, continuous and bounded below by a constant, and that J is an open subinterval of R. We also show that the lower semicontinuous envelope of I cannot in general be obtained by replacing W by its separately convex hull Wsc.

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