= Solution
The zero <Dirichlet boundary condition> must be part of the definition. Thus $u\in H^1(B)$ is a <weak solution> precisely when $u\in H_0^1(B)$, equivalently its <Sobolev trace> vanishes on $\partial B$, and
$$
\boxed{\int_B\nabla u\cdot\nabla\phi\,dx
=\int_B\chi_E\phi\,dx=\int_E\phi\,dx
\qquad\text{for every }\phi\in H_0^1(B).}
$$
Here $\chi_E$ is the <indicator function> of the <measurable set> $E$. One may initially require the identity only for <test functions> $\phi\in C_c^\infty(B)$: both sides are continuous in the <Sobolev norm>, and the definition of the <zero-trace Sobolev space> then extends it to all $H_0^1(B)$. This formulation uses first <weak derivatives> and encodes $-\Delta u=\chi_E$ without assuming classical second <derivatives>.
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