= Zero-trace Sobolev space
{title2=$H_0^1(\Omega)$}
The zero-trace first-order <Sobolev space> is the closure of $C_c^\infty(\Omega)$ in the $H^1$ norm. On a bounded Lipschitz domain it is the kernel of the <Sobolev trace operator>. On an interval, its elements have absolutely continuous representatives with both endpoint values zero. The <Poincaré inequality> makes the derivative norm equivalent to the full Sobolev norm. This is the natural form domain for an elliptic problem with homogeneous <Dirichlet boundary conditions>.
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