Trace Operator

PDEsFunctionalAnalysis In order to define Sobolev Spaces that encode boundary conditions of PDE, consider the subspace: so that every function has an approximating sequence of compactly supported, smooth functions, w.r.t. the Sobolev norm.

encodes the boundary condition that the first weak derivatives vanish on the boundary of . Take , so that and which holds only when the restriction is a continuous linear operator. This can fail in two cases:

1. is defined a.e. in but for the -dimensional Lebesgue measure
2. is possibly not continuous up to the boundary

A trace operator is defined as a bounded, linear operator s.t.

2. for

These conditions can be summarized as ensuring:

1. The integral of the restriction, is well-defined in the sense
2. The restriction is defined continuously up to and including the boundary

The proof of existence is done in the following steps:

1. Restrict a smooth function to a small, local section of the boundary using a partition of unity of a ball containing the boundary
2. Bound the norm on a portion of the boundary in the ball by the derivative on the ball
3. Use compactness of the boundary to get the bound for finitely many balls that cover the boundary
4. Use bound to show that the restrictions form a Cauchy sequence and have a limit in