Solution
= Solution
Linearity in $u$ follows from linearity of the <weak derivative> and <Lebesgue integration>. By the <Holder inequality>, the three-dimensional <Sobolev inequality>, and the preceding $L^3$ interpolation estimate,
$$
|\Phi(u)|
\leq\sqrt3\,\|\nabla u\|_2\|vw\|_2
\leq\sqrt3\,\|\nabla u\|_2\|v\|_6\|w\|_3
\leq C\|v\|_{H^1}\|w\|_{H^1}\|u\|_{H^1}.
$$
Thus $\Phi$ is a <linear functional> and a <continuous linear map> on $H^1(U)$.