Abstract
We review the current state of the art concerning the characterization of traces of the spaces $W^{1, p} (\mathbb{B}^{m-1}\times (0,1), \mathcal{N})$ of Sobolev mappings with values into a compact manifold $\mathcal{N}$. In particular, we exhibit a new analytical obstruction to the extension, which occurs when $p < m$ is an integer and the homotopy group $\pi_p (\mathcal{N})$ is non trivial. On the positive side, we prove the surjectivity of the trace operator when the fundamental group $\pi_1 (\mathcal{N})$ is finite and $\pi_2 (\mathcal{N}) \simeq \dotsb \simeq \pi_{\lfloor p - 1 \rfloor} (\mathcal{N}) \simeq \{0\}$. We present several open problems connected to the extension problem.
Highlights
The classical trace theory characterizes the boundary values of functions in the linear Sobolev spaces W 1,p(Rm−1 × (0, 1), R), with m 2 and 1 p < ∞
W s,p(R, R) {u : R → R; u ∈ Lp and Es,p(u) < ∞}, where the fractional Gagliardo energy Es,p(u) of a measurable function u : R → R is given by E s,p(u)
If u ∈ W 1−1/p,p(Rm−1, R), its harmonic extension U to Rm−1 × (0, ∞), restricted to Rm−1 × (0, 1), is an extension of u in the sense that it belongs to W 1,p(Rm−1 × (0, 1), R) and has trace u on Rm−1 × {0}
Summary
The classical trace theory characterizes the boundary values of functions in the linear Sobolev spaces W 1,p(Rm−1 × (0, 1), R), with m 2 and 1 p < ∞. An equivalent formulation of the above topological obstruction is the following: there exists a map f ∈ C1(S p−1 , N ) that cannot be extended continuously to the ball B p Given such an f , an explicit example of a map u ∈ W 1−1/p,p(Bm−1, N ) with no extension U ∈ W 1,p(Bm−1 × (0, 1), N ) is given by u(x , x ) f (x /|x |),. In view of the quantitative obstructions to the extension problem [4], the (5) Here and in what follows, the subscript b denotes classes of maps with trace b on the boundary. (7) A homotopy equivalence argument shows that the condition does not depend on the triangulation
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