(Trace theorem). Assume that
is a bounded set and
admits a locally continuously differentiable parametrization. The
is restricted
.
There exists a bounded linear
operator
such that for any

Proof

The statement is true for a flat boundary
and
.
If the boundary is not flat then there is a change of variables that makes it
locally flat. Then procedure extends globally by partition of unity (see the
proof of the proposition
(
Global approximation by
smooth functions
) for an example of the technique). The procedure extends
to
by closure of
in
.

Proposition

(Trace-zero functions in
)
Assume that
is a bounded set and
admits a locally continuously differentiable parametrization. The
is restricted
.
Let
.
Then