Energy with weight for S2-valued maps with prescribed singularities

Abstract

We generalize a result of H. Brezis, J.M. Coron and E. Lieb concerning the infimum of the Dirichlet energy over classes of S 2 -valued maps with prescribed singularities to an energy with measurable weight and we prove some geometric properties of such quantity. We also give some stability and approximation results.

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

“…In a previous paper [13], we have studied the following variational problem: given two distinct points P and N in Ω,…”

Estimates for the topological degree and related topics

“…The reader is warmly invited to consult the original paper [32] for the detailed discussions. Other types of results concerning S N -valued maps can be found in, e.g., [2,6,7,8,16,22,27,29,35,41,43,45,47,48,50,57,59,77,82] and the references therein.…”

Prescribed singularities with weights

“…The same approach was taken in [Mil05], where the Euclidean distance in R 3 was replaced by dist a whose definition we specified in the Introduction, formula (7). Considering the proposition below, it is clear that Theorem 1 gives, for the case m = 1, n = 2, the same formula, as does [Mil05, Theorem 1.1], for a(⋅) continuous.…”

“…As we already mentioned, the strategy used is the one from [ABO03]. However, in order to be able to take into consideration the fact that we are placed in a heterogeneous setting, we will have to define a new measure on Ω, by analogy with the distance dist a defined in [Mil05]. Naturally, the area of S in (3) must be replaced with the integral ∫ S a(x) dH 2 (x), so we will define a modified Hausdorff measure H h a (which in fact makes sense even if a(⋅) is only L n -measurable), and which for a ≡ 1 becomes the usual h-dimensional Hausdorff measure on R n .…”

“…The problem of the weighted energy was studied, in the classical context, by V. Millot in Energy with weight for S 2 -valued maps with prescribed singularities ( [Mil05]), where he considered the problem (1) with energy…”