Approximation of Sobolev Functions

FunctionalAnalysisApproximationTheoryPDEs One has the following density result for functions in Sobolev Space for by smooth functions:

1. Local approximation: Given
   • : as , where is the standard Mollifier and for all . This relies on the fact that .
2. Global approximation: Given
   • bounded
   • :
3. Global approximation up to boundary:
   • bounded and

In particular note that:

• is dense in (in the sense of their restrictions to )
• is NOT dense in (due to boundary conditions)