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Optimal design and relaxation of variational problems, III

✍ Scribed by Robert V. Kohn; Gilbert Strang

Publisher
John Wiley and Sons
Year
1986
Tongue
English
Weight
966 KB
Volume
39
Category
Article
ISSN
0010-3640

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✦ Synopsis

The goal was to establish its relaxation (ii) The functions u take values in the space R", and 0 c RZ. For any Jacobian -5 = v u E ( W 2 ) N , the numbers D and p are In the scalar case N = 1, where D = 0 and p = 1-51, the relaxed integrand B0 is the convexification of Go. Its graph has a cone with vertex at the origin, where the nonconvex function Go dropped down to the isolated value Go(0) = 0. In the vector case N > 1, the property required of (Do is quasiconvexity-so that problem (ii) is weakly lower semicontinuous and its minimum value is actually attained.

A solution u of (ii) leads to a near-minimizer iC for (i). To construct iC we approximated u by a piecewise affine function, and then introduced oscillations in the gradient. In the scalar case, 1021 can oscillate between 0 and l,,on sets with area fraction 1p and p. In the vector case, ail may take two nonzero values,

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