We prove that for each positive integer \(N\) the set of smooth, zero degree maps \(\psi:\mathbb{S}^2\to \mathbb{S}^2\) which have the following three properties:
- there is a unique minimizing harmonic map \(u: \mathbb{B}^3\to S^2\) which satisfies the prescribed boundary condition \(u\mid_{\partial \mathbb{B}^3}=\psi\);
- this map \(u\) has at least \(N\) singular points in \(\mathbb{B}^3\);
- the Lavrentiev gap phenomenon holds for \(\psi\), i.e.,
\[
\min_{W^{1,2}_{\psi}(\mathbb{B}^3,\mathbb{S}^2)}E(u) < \inf_{W^{1,2}_{\psi}(\mathbb{B}^3,\mathbb{S}^2)\cap C^0(\overline{\mathbb{B}}^3)} E(u),
\]
where \(W^{1,2}_{\psi}(\mathbb{B}^3,\mathbb{S}^2)=\{v\in W^{1,2}(\mathbb{B}^3,\mathbb{S}^2):v\!\mid_{\partial\mathbb{B}^3}=\psi\text{ in the trace sense}\}\),
is dense in the set of all smooth zero degree maps \(\phi: \mathbb{S}^2\to\mathbb{S}^2\) endowed with the \(H^{1/2}\)-topology.
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