= Solution
Choose a fixed smooth <cutoff function> with $\|\zeta'\|_\infty=C_0$ and set $J=(-1/2,1/2)$. The <Caccioppoli inequality> gives $\|u'\|_{L^2(J)}\le2RC_0\|u\|_{L^2(-1,1)}$. Since the length of $J$ is one, the corrected <interval Sobolev supremum estimate> yields $\|u\|_{L^\infty(J)}\le\|u\|_{L^2(J)}+\|u'\|_{L^2(J)}$, and the <Hölder seminorm> is at most $\|u'\|_{L^2(J)}$. Therefore
$$
\boxed{\|u\|_{L^\infty(J)}+[u]_{C^{0,1/2}(J)}\le(1+4RC_0)\|u\|_{L^2(-1,1)}.}
$$
For locally $H^1$ <weak solutions> with $u\in L^2(-1,1)$, <density of smooth functions in a Sobolev space> justifies the test $\zeta^2u$, and the previous representative estimates already apply to $H^1$. Mollifying a solution need not preserve the equation with the same measurable coefficient, so approximation is used for admissible tests and Sobolev estimates, not to assert that the mollified function solves the original equation. The required constant depends only on $R$.
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