Distributional derivative

For a distribution $u$ and a multi-index $\alpha$, its distributional derivative is defined by

The locally integrable function $\log |x|$ has distributional derivative equal to the principal-value reciprocal distribution. Symmetric integration by parts makes the boundary term $\log \epsilon [\phi (\epsilon )-\phi (-\epsilon )]$ tend to zero.

If $v^{\prime }=0$ on $\mathbb{R}$, then $v$ is a constant distribution. Every test function of integral zero is the derivative of its compactly supported primitive, so $v$ vanishes on those test functions. Choosing one test function $\eta$ of integral one gives $⟨v,\phi ⟩=⟨v,\eta ⟩\int \phi$. Iteration shows that a distribution with $m$th derivative zero is a polynomial of degree at most $m-1$.

A locally integrable function $g$ is the $i$th weak derivative of $u$ whenfor every test function $\phi \in C_c^{\infty }(U)$. Thus a weak derivative is a distributional derivative that is represented by a locally integrable function.

A bounded function which is $C^1$ away from one point in dimension $n\ge 2$, and whose classical gradient is locally integrable across that point, has that gradient as its weak derivative. Cut a test function off within distance $\epsilon$ of the point. The error from differentiating the cutoff is bounded by $C\Vert u{\Vert }_{\infty }{\epsilon }^{n-1}$ and vanishes. The other terms converge by local integrability. In dimension one the Heaviside step function is a counterexample: its classical derivative off the point is zero, but its distributional derivative is a Dirac delta.

A locally integrable function has weak derivatives on an open set exactly when it has them near every point. Local derivatives agree almost everywhere on overlaps because they define the same distributional derivative. A countable local cover therefore gives globally defined, locally integrable derivative functions. A smooth partition of unity subordinate to the cover splits each compactly supported test function; summing the local integration identities yields the global weak-derivative identity.

The weak gradient is the vector of first-order weak derivatives of a function. Its $i$th component is specified by $\int u{\partial }_i\phi =-\int ({\partial }_{i}u)\phi$ for compactly supported test functions. For a differentiable function with appropriate local integrability it agrees with the classical gradient.

If $u\in W_{\mathrm{loc}}^{1,2}$ and $\Phi$ is continuously differentiable with bounded derivative on the range in use, then $D(\Phi (u))={\Phi }^{\prime }(u)Du$ almost everywhere. One proves this first for smooth functions and then uses density of smooth functions in a Sobolev space and boundedness of ${\Phi }^{\prime }$. On compact sets one may truncate $\Phi$ away from the range of a bounded $u$.

If $u\in W_{\mathrm{loc}}^{1,2}$, then $Du=0$ almost everywhere on every level set $\{u=t\}$. Choose smooth truncations ${\Phi }_{\epsilon }(s)$ with $|{\Phi }_{\epsilon }|\le 2\epsilon$, derivative one near zero, bounded derivative, and derivative zero outside $(-2\epsilon ,2\epsilon )$. The Sobolev chain rule gives $D{\Phi }_{\epsilon }(u-t)\to 1_{\{u=t\}}Du$ in local $L^2$ by dominated convergence theorem, while ${\Phi }_{\epsilon }(u-t)\to 0$. The limiting weak derivative is therefore zero.