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In this paper we establish relaxation of an arbitrary 1-equivariant wave map from \({\mathbb{R}^{1+3}_{t,x}{\setminus} (\mathbb{R}\times B(0,1))\to S^3}\) of finite energy and with a Dirichlet condition at r = 1, to the unique stationary harmonic map in its degree class. This settles a recent conjecture of Bizoń, Chmaj, Maliborski who observed this asymptotic behavior numerically.

Relaxation of Wave Maps Exterior to a Ball to Harmonic Maps for All Data