Zero extension of H01
= Zero extension of H01
By the definition of $H_0^1(\Omega)$ as the closure of compactly supported smooth functions, extension by zero maps it continuously into $H^1(\mathbb R^d)$ without creating a boundary distribution.
= Zero extension of H01
By the definition of $H_0^1(\Omega)$ as the closure of compactly supported smooth functions, extension by zero maps it continuously into $H^1(\mathbb R^d)$ without creating a boundary distribution.