Weak derivatives and distributional identities

Let $\Omega\subset\mathbb R^n$ be open and let $u,g_i\in L^1_{\mathrm{loc}}(\Omega)$. We call $g_i$ the weak derivative $D_i u$ when integrating $uD_i\varphi$ equals integrating $-g_i\varphi$ for every $\varphi\in C_c^\infty(\Omega)$. Compact support keeps the test function away from the boundary, so no boundary term appears and no boundary condition has been imposed. The definition is local: these integrals only inspect compact subsets of the open set. Global integrability will be a separate Sobolev-space requirement.

If a weak derivative exists, it is unique almost everywhere. Indeed, subtract the identities for two candidates: their difference integrates to zero against every test function, so it is the zero locally integrable distribution. Altering a function or derivative at a null set does not change this conclusion. For a continuously differentiable function the weak and classical derivatives agree by integration by parts. Agreement in that direction is easy; merely possessing a classical derivative almost everywhere does not by itself establish the weak identity.

A continuous piecewise smooth function can have a corner without a singular first derivative. Split the integral at each interface and integrate by parts on the pieces. The interface terms cancel when the values of the function match, leaving the piecewise classical slope as the weak derivative. This is why continuous piecewise linear finite elements belong locally to first-order Sobolev spaces when their slopes are integrable. Discontinuities in their slopes can still contribute singularities to their second distributional derivatives, so first-order membership does not automatically extend to second order.

For a function with a jump, the interface terms do not cancel. A jump of size $J$ at an interior point in one dimension contributes $J\delta_a$ to the distributional derivative. The Dirac distribution evaluates the test function at that point and cannot be represented by a locally integrable density. One can see this using smooth tests bounded by one, equal to one at the point, and supported in successively smaller neighborhoods: integrals against any fixed locally integrable density tend to zero, whereas point evaluation stays one. A distributional derivative always exists, but it need not be a weak derivative represented by a function.

For a multi-index $\alpha$, the corresponding identity has sign $(-1)^{|\alpha|}$ after all derivatives move onto the test function. Distributional mixed derivatives commute because the derivatives of smooth tests commute. Product rules remain available under suitable assumptions, for example multiplying a weakly differentiable function by a smooth cutoff. The cutoff derivative contributes an additional term and must not be dropped when localizing a PDE. More general products require enough integrability for both the product and its proposed derivative to define distributions.

On a bounded interval, a function in $W^{1,1}$ has an absolutely continuous representative whose ordinary derivative agrees almost everywhere with its weak derivative. This representative satisfies the fundamental theorem of calculus and provides meaningful endpoint values. The statement is stronger than saying the ordinary derivative exists almost everywhere; singular continuous functions show why that weaker condition is insufficient. In higher dimensions, absolute continuity along almost every coordinate line is a useful related viewpoint, while pointwise continuity of the entire function requires additional integrability or embedding hypotheses.