Sequential weak approximation for maps of finite Hessian energy

Abstract

Consider the space W 2,2 ( ; N ) of second order Sobolev mappings v from a smooth domain ⊂ R m to a compact Riemannian manifold N whose Hessian energy |∇ 2 v| 2 dx is finite. Here we are interested in relations between the topology of N and the W 2,2 strong or weak approximability of a W 2,2 map by a sequence of smooth maps from to N . We treat in detail W 2,2 (B 5 , S 3 ) where we establish the sequential weak W 2,2 density of W 2,2 (B 5 , S 3 ) ∩ C ∞ . The strong W 2,2 approximability of higher order Sobolev maps has been studied in the recent paper of Bousquet et al. (J. Eur. Math. Soc. (JEMS) 17 (4), 2015). For an individual map v ∈ W 2,2 (B 5 , S 3 ), we define a number L(v) which is approximately the total length required to connect the isolated singularities of a strong approximation u of v either to each other or to ∂ Bx 5 . Then L(v) = 0 if and only if v admits W 2,2 strongly approximable by smooth maps. Our critical result, obtained by constructing specific curves connecting the singularities of u, is the bound L(u) ≤ c + c B 5 |∇ 2 u| 2 dx . This allows us to construct, for the given Sobolev map v ∈ W 2,2 (B 5 , S 3 ), the desired W 2,2 weakly approximating sequence of smooth maps. To find suitable connecting curves for u, one uses the twisting of a u pull-back normal framing on a suitable level surface of u.Mathematics Subject Classification 56D15 · 46E35 · 46T20

A counterexample to the weak density of smooth maps between manifolds in Sobolev spaces

“…In a related direction, a positive answer was given in [16,28] for (p−1)-connected manifolds N and in [28] in the case p = 2, whatever manifold N, similar results involving the H 2 energy are given in [24]. The main result of this paper presents an obstruction to sequential weak density of smooth maps when (10) holds and deals with the special case N = S 2 and p = 3, for which π 3 (S 2 ) = Z.…”

“…The functional L branch , which, as seen, is related to the defect energy by (28), is therefore not controlled by the Dirichlet energy E 3 , in contrast with inequality (24) for N = S 3 . This property is at the heart of the paper.…”

“…We have seen in the case of S 3 -valued that we may bound the defect energy of a map in u ∈ R ct (B 4 , S 3 ) by the 3-energy of the map itself (see inequality (24)) and that this upper bound, combined with the strong density of R ct (B 4 , S 3 ) directly leads to the weak density of smooth maps. If an estimate similar to (24) would exist for S 2 -valued maps, then the same line of thoughts would yield weak approximability as well. Our next result states precisely that there is no analog of (24) for S 2 -valued maps.…”

Strong density for higher order Sobolev spaces into compact manifolds

“…This result was known for an arbitrary manifold N n only in the case k = 1 [3, Theorem 2] (see also [28,Theorem 1.3]). It is a fundamental tool in the study of the weak density of smooth maps in Sobolev spaces and in the study of topological singularities of Sobolev maps [5,22,23,29,31,45]. Counterparts of Theorems 1 and 2 for fractional Sobolev spaces W s,p (Q m ; N n ) such that 0 < s < 1 have been investigated by Brezis and Mironescu [12].…”