\[ \begin{array} { r l r } { ( 4 . 7 ) } & { \displaystyle \int _ { \sqrt { x _ { 1 } ^ { 2 } + x _ { n } ^ { 2 } } = \delta } \frac { 1 } { 2 } | \nabla u | ^ { 2 } ( \psi \cdot \nu ) | x _ { n } | ^ { 1 - 2 s } - \int _ { \sqrt { x _ { 1 } ^ { 2 } + x _ { n } ^ { 2 } } = \delta } ( \nabla u \cdot \psi ) ( \nabla u \cdot \nu ) | x _ { n } | ^ { 1 - 2 s } } \\ & { \displaystyle } & { \displaystyle + \int _ { B _ { 1 } ^ { \prime } } 2 \chi \operatorname { d i v } \varphi = O ( \delta ) . } \end{array} \]

where \( \nu \) is the unit outer normal to \( \left\{ { \sqrt { x _ { 1 } ^ { 2 } + x _ { n } ^ { 2 } } } = \delta \right\} \). Next we estimate the first two integrals in the LHS of the above equality. Since u is homogeneous of degree s (and depends only on \( x _ { 1 } \) and \( x _ { n } \)), \( \nabla \boldsymbol { u } \cdot \boldsymbol { \nu } = s \boldsymbol { u } \). Thus,

\[ \int _ { \sqrt { x _ { 1 } ^ { 2 } + x _ { n } ^ { 2 } } = \delta } ( \nabla u \cdot \psi ) ( \nabla u \cdot \nu ) | x _ { n } | ^ { 1 - 2 s } = \int _ { \sqrt { x _ { 1 } ^ { 2 } + x _ { n } ^ { 2 } } = \delta } ( \nabla u \cdot \psi ) ( s u ) | x _ { n } | ^ { 1 - 2 s } = O ( \delta ) , \]

where we have used the growth estimate of u around the zero set as well as\|∇u\|\|x−2sn\|12 ∈ L2loc(Rn). To estimate the first integral we use the polar coordinates in \( ( x _ { 1 } , x _ { n } ) \)-plane: \( x _ { 1 } = r \cos ( \theta ) \), \( x _ { n } = r \sin ( \theta ) \). Then on \( \{ { \sqrt { x _ { 1 } ^ { 2 } + x _ { n } ^ { 2 } } } = \delta \} \),

\[ \begin{array} { l } { \displaystyle \frac { 1 } { 2 } | \nabla u | ^ { 2 } ( \psi \cdot \nu ) | x _ { n } | ^ { 1 - 2 s } } \\ { \displaystyle \qquad = \frac { s ^ { 2 } \alpha ^ { 2 } } { 2 \delta } ( \cos ( \theta / 2 ) ) ^ { - 1 + 2 s } ( | \sin ( \theta ) | ) ^ { 1 - 2 s } \left[ \cos ( \theta ) ( \psi \cdot e _ { 1 } ) + \sin ( \theta ) ( \psi \cdot e _ { n } ) \right] } \end{array} \]

Since \( \psi \cdot e _ { n } = 0 \) on \( B _ { 1 } ^ { \prime } \),

\[ \begin{array} { l } { \displaystyle \operatorname* { l i m } _ { \delta \to 0 } \int _ { \sqrt { x _ { 1 } ^ { 2 } + x _ { n } ^ { 2 } } = \delta } \frac { 1 } { 2 } | \nabla u | ^ { 2 } ( \psi \cdot \nu ) | x _ { n } | ^ { 1 - 2 s } d x = \displaystyle \operatorname* { l i m } _ { \delta \to 0 } \int _ { B _ { 1 } ^ { \prime \prime } } \frac { s ^ { 2 } \alpha ^ { 2 } } { 2 \delta } \cdot 2 \delta \times } \\ { \displaystyle \quad \times \int _ { 0 } ^ { \pi } ( \cos ( \theta / 2 ) ) ^ { - 1 + 2 s } ( \sin ( \theta ) ) ^ { 1 - 2 s } ( \cos ( \theta ) ) ^ { 2 } \psi _ { 1 } ( \delta \cos ( \theta ) , x ^ { \prime \prime } , \delta \sin ( \theta ) ) \ d \theta d x ^ { \prime \prime } } \\ { \displaystyle \quad = c _ { 0 } ( s ) \alpha ^ { 2 } \int _ { B _ { 1 } ^ { \prime \prime } } \psi _ { 1 } ( 0 , x ^ { \prime \prime } , 0 ) d x ^ { \prime \prime } , } \end{array} \]

where

\[ \begin{array} { l } { { c _ { 0 } ( s ) = s ^ { 2 } \displaystyle \int _ { 0 } ^ { \pi } ( \cos ( \theta / 2 ) ) ^ { - 1 + 2 s } ( \sin ( \theta ) ) ^ { 1 - 2 s } ( \cos ( \theta ) ) ^ { 2 } d \theta } } \\ { { \displaystyle = s ^ { 2 } 2 ^ { - 1 - 2 s } \frac { \sqrt { \pi } ( 7 + 4 s ( s - 2 ) ) \Gamma ( 1 - s ) } { \Gamma ( \frac { 7 } { 2 } - s ) } . } } \end{array} \]

Combining the above estimates and letting \( \delta \rightarrow \) 0, we have for all \( \varphi \in C _ { c } ^ { \infty } ( B _ { 1 } ^ { \prime } ; \mathbb { R } ^ { n - 1 } ) \)

\[ c _ { 0 } ( s ) \alpha ^ { 2 } \int _ { B _ { 1 } ^ { \prime \prime } } \psi _ { 1 } ( 0 , x ^ { \prime \prime } , 0 ) d x ^ { \prime \prime } + \int _ { B _ { 1 } ^ { \prime } } 2 \chi \, \mathrm { d i v } \, \varphi = 0 \]

Recalling \( \psi _ { 1 } = \varphi _ { 1 } \) on \( B _ { 1 } ^ { \prime } \), we have proved (4.3).

Step 2. We now show that \( \begin{array} { r } { \alpha = \sqrt { \frac { 2 M } { c _ { 0 } ( s ) } } } \end{array} \) and \( \chi = M \chi _ { \{ x _ { 1 } > 0 \} } \)\( \mathcal { H } ^ { n - 1 } \)-a.e. in \( \mathbb { R } ^ { n - 1 } \).

(a) \( \chi \equiv M \) in \( \{ x _ { 1 } > 0 \} \).

In fact, if \( y \in \{ x _ { 1 } > 0 \} \), then \( u ( y ^ { \prime } , 0 ) = \alpha ( y _ { 1 } ) ^ { s } > 0 \). Hence by the uniform Ho¨lder convergence of \( u _ { \varepsilon _ { j } } \) to u, we have \( u _ { \varepsilon _ { i } } \geq \varepsilon _ { j } \) in a neighborhood of \( ( y ^ { \prime } , 0 ) \)