(Boundary elliptic regularity) Let be a bounded open set, , , and be a weak solution of the elliptic boundary problem (see the definition ( Elliptic differential operator )) Let be .

Then and where the constant depends only on and .

The above proposition is usually combined with the proposition ( Elliptic boundedness of inverse ) to get rid of the term.

The proof is a combination of proofs of the propositions ( Second order internal elliptic regularity ), ( High order internal elliptic regularity ) and ( Trace theorem ).