Sobolev weak derivatives belong to pde

The derivative supplied here is the distribution derivative: ${\mathrm{\partial }}^{\alpha }u$ acts on a test by $(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}u({\mathrm{\partial }}^{\alpha }\phi )$, as in Distributional derivative. It exists for every distribution, without assuming that it is represented by a function. Asking whether such a derivative has an integrable function representative is an additional question.

Sobolev spaces, their norms, boundary traces and weak-solution estimates belong to the PDE track. This pair supplies the test-function and distribution-derivative language they require; it makes no assertion here about Sobolev embeddings, regularity estimates or boundary-value problems. This is a scope boundary, with no theorem or future result consumed as a prerequisite.