The stationary equation x dot Du = u is solved by radial characteristics

Extend $u$ independently of a time variable: $\tilde{u}(x,s)=u(x)$ on $({\mathbb{R}}^n\setminus \{0\})\times \mathbb{R}$. It satisfies ${\tilde{u}}_s+x\cdot {\mathrm{\nabla }}_x\tilde{u}-\tilde{u}=0$. Regard this as transport with characteristic ODE $X^{\mathrm{\prime }}(s)=X(s)$; its solutions are the rays $X(s)=e^s{x}_0$ by [L2] applied componentwise, and by [L1] the restricted function $v(s)=u(X(s))$ satisfies $v^{\mathrm{\prime }}(s)=v(s)$. [L2] then gives $v(s)=e^{s}v(0)$, since the constant coefficients are continuous on the whole real characteristic interval.

Writing $x=r\omega$ with $r=e^s$ and $\omega \in S^{n-1}$ gives $u(r\omega )=ru(\omega )$, so every solution has the claimed form with $g(\omega )=u(\omega )$; conversely, for $u(r\omega )=rg(\omega )$ the radial derivative is $u\mathrm{/}r$, hence $x\cdot \mathrm{\nabla }u=u$.