= Solution
By the definition in (i), $u$ itself is an admissible test in the <weak formulation>. Hence
$$
\|\nabla u\|_{L^2(B)}^2=\int_Eu\,dx\leq\int_E|u|\,dx=\|u\|_{L^1(E)}.
$$
The absolute value makes a proof of positivity unnecessary. <Holder inequality> with exponents $6$ and $6/5$, followed by the zero-boundary <Sobolev inequality>, gives
$$
\|u\|_{L^1(E)}\leq |E|^{5/6}\|u\|_{L^6(B)}
\leq C_S|E|^{5/6}\|\nabla u\|_{L^2(B)}.
$$
Taking square roots of both estimates proves exactly the requested chain:
$$
\boxed{\|\nabla u\|_2\leq\|u\|_{L^1(E)}^{1/2}
\leq C_S^{1/2}|E|^{5/12}\|\nabla u\|_2^{1/2}.}
$$
If $\|\nabla u\|_2=0$, the zero-boundary <Poincaré inequality> gives $u=0$ and the conclusion follows. Otherwise divide the first-to-last estimate by $\|\nabla u\|_2^{1/2}$ and square, obtaining $\|\nabla u\|_2\leq C_S|E|^{5/6}$. Finally the same <Poincaré inequality> gives
$$
\boxed{\|u\|_{H^1(B)}\leq C|E|^{5/6}.}
$$
All constants depend only on the fixed unit ball and the dimension, and are independent of $E$ and $u$. If $|E|=0$, testing the homogeneous equation with $u$ gives the same zero solution directly.
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