In all problems, is a bounded domain, and derivatives are understood as weak derivatives.
The space is a Hilbert space with respect to the inner product
The and norms are equivalent in the space .
The restriction is bounded from into . (Hint: Extend the normal field from to and use the divergence theorem, together with the Cauchy inequality.)
Consider, in the disk , the functions ,
- The are orthogonal in , with respect to the product. (Hint: Use integration in polar coordinates; note that you also need to calculate the gradient of a function in in polar coordinates.)
- , and thus if and only if .
- Construct a function that is not the restriction of an function to the circle .