Scattering for wave maps exterior to a ball. (English) Zbl 1264.35045

Lawrie, A.; Schlag, W.

Adv. Math. 232, No. 1, 57-97 (2013).

This paper deals with equivariant wave maps \(U : \mathbb R\times(\mathbb R^3\setminus B)\rightarrow S^3\) with the Dirichlet condition \(U(\partial B) = {N}\), where \(N\) is a fixed point on \(S^3\) and \(B\subset \mathbb{R}^3\) is the unit ball centered at 0 in \(\mathbb{R}^3\). The main result states that 1-equivariant maps of degree zero scatter to zero irrespective of their energy. For positive degrees, asymptotic stability of the unique harmonic map in the energy class determined by the degree is proved.

Reviewer: Cornelia-Livia Bejan (Iaşi)

MSC:

35B40	Asymptotic behavior of solutions to PDEs
35P25	Scattering theory for PDEs
53C43	Differential geometric aspects of harmonic maps
58E20	Harmonic maps, etc.
35L71	Second-order semilinear hyperbolic equations

Keywords:

sigma model; concentration compactness; dispersive theory; Dirichlet condition; asymptotic stability