Trace Operator - Informal Discussion

Informal Discussion

Let be a bounded open set in the Euclidean space with C1 boundary If is a function that is (or even just continuous) on the closure of its function restriction is well-defined and continuous on If however, is the solution to some partial differential equation, it is in general a weak solution, so it belongs to some Sobolev space. Such functions are defined only up to a set of measure zero, and since the boundary does have measure zero, any function in a Sobolev space can be completely redefined on the boundary without changing the function as an element in that space. It follows that simple function restriction cannot be used to meaningfully define what it means for a general solution to a partial differential equation to behave in a prescribed way on the boundary of

The way out of this difficulty is the observation that while an element in a Sobolev space may be ill-defined as a function, can be nevertheless approximated by a sequence of functions defined on the closure of Then, the restriction of to is defined as the limit of the sequence of restrictions .