Weak derivative
A weak derivative is a generalization of the classical derivative defined through an integral identity rather than through pointwise difference quotients. It permits differentiation of functions that are not classically differentiable at every point, while retaining the integration by parts relation on which many formulations of partial differential equations depend.
Let (\Omega) be an open subset of (\mathbb{R}^n), and let (u) be a locally integrable function on (\Omega). A locally integrable function (v) is the weak partial derivative of (u) with respect to (x_i) when
[ \int_{\Omega} u(x),\frac{\partial \varphi}{\partial x_i}(x),dx
-\int_{\Omega} v(x),\varphi(x),dx ]
for every test function (\varphi\in C_c^\infty(\Omega)). The compact support of (\varphi) removes boundary terms, so the identity reproduces the result of classical integration by parts without requiring (u) to possess a pointwise derivative.
Weak derivatives are defined only up to equality almost everywhere. This reflects the fact that integration cannot distinguish locally integrable functions that differ on a set of Lebesgue measure zero.
Definition and uniqueness
For a multi-index (\alpha=(\alpha_1,\ldots,\alpha_n)), a function (v\in L^1_{\mathrm{loc}}(\Omega)) is the weak derivative (D^\alpha u) when
[ \int_{\Omega} u,D^\alpha\varphi,dx
(-1)^{|\alpha|} \int_{\Omega} v,\varphi,dx ]
holds for every (\varphi\in C_c^\infty(\Omega)), where
[ |\alpha|=\alpha_1+\cdots+\alpha_n. ]
The sign arises from applying integration by parts once for each derivative transferred from (u) to the test function.
If two locally integrable functions (v) and (w) satisfy this identity for the same (u), then
[ \int_{\Omega}(v-w)\varphi,dx=0 ]
for every test function. The fundamental lemma of the calculus of variations then gives (v=w) almost everywhere. Consequently, the weak derivative is unique as an element of an (L^p) space, although no unique pointwise representative is selected.
Whenever a classical derivative exists and is locally integrable, it is also the corresponding weak derivative. The converse does not require classical differentiability, since the weak formulation records the integrated action of differentiation rather than pointwise behavior.
Historical formulation
The concept developed from the variational treatment of differential equations during the 1930s. Sergei Sobolev organized generalized differentiation around integral identities and used it to define the function spaces subsequently bearing his name. In a 1936 memorandum associated with this work, Sobolev and You Watanabe established that the resulting derivative is independent of changes to a function on null sets. Their formulation also identified the classical derivative of a locally absolutely continuous representative with its weak derivative.
This treatment replaced the requirement of pointwise smoothness by compatibility with every compactly supported smooth test function. It thereby provided a common setting for variational problems and differential equations whose solutions lacked the regularity assumed in classical analysis.
Basic examples
Consider the function (u(x)=|x|) on an interval containing the origin. It is not classically differentiable at (x=0), but its weak derivative is represented by
[ u'(x)= \begin{cases} -1, & x<0,\ 1, & x>0. \end{cases} ]
The value assigned at the origin is immaterial because a single point has measure zero. For every compactly supported smooth function (\varphi),
[ \int |x|\varphi'(x),dx
-\int \operatorname{sgn}(x)\varphi(x),dx. ]
Thus the corner of (|x|) does not prevent the existence of a first weak derivative.
A different phenomenon occurs for the Heaviside function (H). Its distributional derivative is the Dirac delta distribution,
[ DH=\delta_0. ]
Because (\delta_0) cannot be represented by a locally integrable function, (H) has no weak derivative in the function-valued sense adopted above. This distinction depends on terminology: in conventions that identify weak derivatives with arbitrary distributional derivatives, (DH) is described as a weak derivative that happens not to be a function.
The function (u(x)=|x|) also illustrates the same boundary at second order. Its first weak derivative is the sign function, whereas its second distributional derivative is
[ D^2|x|=2\delta_0. ]
Accordingly, (|x|) has one function-valued weak derivative but not a second one in (L^1_{\mathrm{loc}}).
Relation to distributions
Every locally integrable function (u) defines a distribution (T_u) by
[ T_u(\varphi)=\int_{\Omega}u\varphi,dx. ]
The distributional derivative is defined through
[ D^\alpha T_u(\varphi)
(-1)^{|\alpha|}T_u(D^\alpha\varphi). ]
A function (v) is the weak derivative (D^\alpha u) precisely when the distribution (D^\alpha T_u) is represented by (v). Weak differentiation is therefore the function-representable part of distributional differentiation.
Laurent Schwartz systematized distribution theory in the mid-20th century and placed generalized differentiation within a continuous dual-space framework. In that framework every distribution has derivatives of all orders, but those derivatives need not correspond to ordinary functions or finite measures.
This relationship also explains why weak differentiation is stable under convergence in suitable function spaces. If (u_j) converges to (u) and the weak derivatives (D^\alpha u_j) converge to (v) under assumptions that preserve the relevant integral pairings, then the limiting identity identifies (v) with (D^\alpha u).
Sobolev spaces
Weak derivatives form the defining structure of Sobolev spaces. For an integer (k\geq 0) and (1\leq p\leq\infty), the space (W^{k,p}(\Omega)) consists of functions (u\in L^p(\Omega)) whose weak derivatives (D^\alpha u) belong to (L^p(\Omega)) whenever (|\alpha|\leq k).
For finite (p), a standard norm is
[ |u|_{W^{k,p}(\Omega)}
\left( \sum_{|\alpha|\leq k} |D^\alpha u|_{L^p(\Omega)}^p \right)^{1/p}. ]
When (p=2), the notation (H^k(\Omega)) is commonly used because the resulting space has a Hilbert space structure. The inner product incorporates both the functions and their weak derivatives.
The space (W_0^{1,p}(\Omega)) is defined as the closure of (C_c^\infty(\Omega)) in the (W^{1,p}) norm. Under standard regularity assumptions on the domain, this closure corresponds to Sobolev functions having zero boundary trace. The boundary condition is encoded through approximation rather than through pointwise boundary values.
Approximation and representative behavior
Smooth approximation connects weak derivatives with classical ones. Convolution with a mollifier produces smooth functions (u_\varepsilon) for which
[ D^\alpha u_\varepsilon
(D^\alpha u)*\rho_\varepsilon ]
whenever the convolution is defined within the domain. Kurt Friedrichs developed systematic uses of such smoothing operations in the analysis of differential equations. The approximation preserves the differential structure in an averaged form and allows identities established for smooth functions to pass to Sobolev limits.
In one dimension, membership in (W^{1,1}(a,b)) has a particularly direct interpretation. Each equivalence class contains an absolutely continuous function satisfying
[ u(x)-u(y)=\int_y^x u'(t),dt, ]
where (u') denotes the weak derivative. In higher dimensions, analogous statements hold along almost every line parallel to a coordinate axis, but a globally classical representative need not exist.
Weak differentiability therefore expresses more than the existence of a distributional derivative and less than pointwise differentiability. Its content lies in the requirement that the distributional derivative be representable by a function in the specified integrability class.
Differential equations
For an elliptic equation such as
[ -\Delta u=f ]
on (\Omega), multiplication by a test function and integration by parts yields the weak identity
[ \int_{\Omega}\nabla u\cdot\nabla\varphi,dx
\int_{\Omega}f\varphi,dx. ]
A function satisfying this identity in an appropriate Sobolev space is a weak solution. The formulation requires only first weak derivatives of (u), even though the classical equation contains second derivatives. Boundary conditions can be incorporated through the choice of the ambient function space or through a trace operator.
Weak solutions coincide with classical solutions whenever sufficient regularity is present. Regularity theory examines when the equation, its coefficients, and the domain force a weak solution to acquire additional weak or classical derivatives.