Liquid Crystals and Energy Estimates for S2- Valued Maps

Optimal Shapes and Masses, and Optimal Transportation Problems

“…It may happen that the class T (f + , f − ) is empty; this occurs for instance when f − = 1 2 (δ x0 +δ x1 ) and f + = δ x2 in a space X which contains at least 3 points. Furthermore, even if T (f + , f − ) = ∅, in general the minimum in (30) is not achieved as shown in the next example. The behaviour of minimizing sequences of transport maps in the previous example shows also that the class of transport maps is not closed with respect to a topology sufficiently strong to deal with the nonlinearity of functional (29).…”

The relaxed energy for \( S^{2} \)-valued maps and measurable weights

“…Let P and N be two distinct points in Ω. We take g ≡ (0, 0, 1) and consider u ∈ H 1 g (Ω, S 2 ) ∩ C 1 ( Ω \ {P , N}) (such a map is constructed for instance in [6,8]). By Theorem 1.2, we have N ).…”

Energy with weight for S2-valued maps with prescribed singularities

“…Coron and E. Lieb have proved that for w ≡ 1 this quantity is equal to 8πL where L is the length of a minimal connection associated to the configuration (a i , d i ) N i=1 and the Euclidean geodesic distance d Ω on Ω (see also [1,6,7,17]). The first motivation for studying such a problem comes from the theory of liquid crystals (see [14,15]).…”

“…Another way to compute L w is to use the following formula (see [9]), 6) where the supremum is taken over all functions ζ : Ω → R which are 1-Lipschitz with respect to d w i.e., |ζ(x) − ζ(y)| ≤ d w (x, y) for all x, y ∈ Ω. In Section 2.3, we give a characterization of 1-Lipschitz functions for the distance d w .…”

“…Since δ is arbitrary, we obtain that Following the computations in [6] and using the properties of Φ, we check that u n ∈ W 1,∞ loc Ω \ {P, N }, S 2 , deg(u n , P ) = +1, deg(u n , N ) = −1. Choosing n sufficiently large and smoothening u n around γ by the procedure in [2], we get a new map u δ ∈ E which satisfies (5.4).…”