ABSTRACT
We next develop approaches for obtaining other patterns, which are not detected in Proposition 1.4 by the L-S and fibering methods. In order to construct other families of solutions (see Section 1.6 for illus-
trations of those for m = 3), we need an auxiliary approximation of patterns. Namely, we first perform a Cartesian decomposition via
F = h+ w, (140)
where h ∈ W 2,m0 (BR) is a smooth “step-like function” that takes the equilibrium values ±1 and 0 on some disjoint subsets of BR (with a smooth connection in between). Sufficiently close to the boundary points, we always have h(y) = 0. For instance, in 1D, for getting the patterns in Figure 1.22, we take h(y) as a smooth approximation of the step function, taking values ±1 and 0 on the intervals of oscillations of the solution about these equilibria. In other words, we are going to perform the radial fibering not about the
origin, but about a non-trivial point h, which plays the role of an initial approximation of the pattern that we are interested in. Obviously, the choice of such h’s is of principal importance, which thus should be done very carefully. Substituting (140) into the functional yields the new one,
Eˆ(w) = E(h+ w) = − 12 ∫ BR
h2 + L0(h)w
|h+ w|β , (141)
where, by L0, we denote the linear functional
L0(h)w = − ∫ BR
D˜mh · D˜mw + ∫ BR
hw.
