Zero-trace Sobolev space ($H_0^1(\Omega )$)

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.