Interpolation Inequality and Sobolev Spaces

Integrate by parts to prove the interpolation inequality:
1 1
2 2
2 2 2 2
|Du| dx C u dx |D u| dx
U U U


for all u Cc (U ). Assume U is smooth, and prove this inequality if u
2 1
H (U ) H0 (U ).

Hint: Take {vk } Cc (U ) converging to u in H0 (U ), and {wk }
k=1
1
k=1
2 k k,2
C (U ) converging to u in H (U ). Note, H = W .




1