Odtnhj

This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations. (May 2014) (Learn how and when to remove this template message)

In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i.e., to lie in the L^p space L1([a,b])displaystyle mathrm L ^1([a,b]). See distributions for a more general definition.

Definition

Let udisplaystyle u be a function in the Lebesgue space L1([a,b])displaystyle L^1([a,b]). We say that vdisplaystyle v in L1([a,b])displaystyle L^1([a,b]) is a weak derivative of udisplaystyle u if,

∫abu(t)φ′(t)dt=−∫abv(t)φ(t)dtdisplaystyle int _a^bu(t)varphi '(t)dt=-int _a^bv(t)varphi (t)dt

for all infinitely differentiable functions φdisplaystyle varphi with φ(a)=φ(b)=0displaystyle varphi (a)=varphi (b)=0. This definition is motivated by the integration technique of Integration by parts.

Generalizing to ndisplaystyle n dimensions, if udisplaystyle u and vdisplaystyle v are in the space Lloc1(U)displaystyle L_loc^1(U) of locally integrable functions for some open set U⊂Rndisplaystyle Usubset mathbb R ^n, and if αdisplaystyle alpha is a multi-index, we say that vdisplaystyle v is the αthdisplaystyle alpha ^th-weak derivative of udisplaystyle u if

∫UuDαφ=(−1)|α|∫Uvφ,displaystyle int _UuD^alpha varphi =(-1)^int _Uvvarphi ,

for all φ∈Cc∞(U)displaystyle varphi in C_c^infty (U), that is, for all infinitely differentiable functions φdisplaystyle varphi with compact support in Udisplaystyle U. Here Dαφdisplaystyle D^alpha varphi is defined as

Dαφ=∂|α|φ∂x1α1…∂xnαn.displaystyle qquad D^alpha varphi =frac partial ^varphi partial x_1^alpha _1dots partial x_n^alpha _n.

If udisplaystyle u has a weak derivative, it is often written Dαudisplaystyle D^alpha u since weak derivatives are unique (at least, up to a set of measure zero, see below).

Examples

• The absolute value function u : [−1, 1] → [0, 1], u(t) = |t|, which is not differentiable at t = 0, has a weak derivative v known as the sign function given by

v:[−1,1]→[−1,1];t↦v(t)={1,if t>0;0,if t=0;−1,if t<0.displaystyle vcolon [-1,1]to [-1,1];quad tmapsto v(t)=begincases1,&mboxif t>0;\0,&mboxif t=0;\-1,&mboxif t<0.endcases

This is not the only weak derivative for u: any w that is equal to v almost everywhere is also a weak derivative for u. Usually, this is not a problem, since in the theory of L^p spaces and Sobolev spaces, functions that are equal almost everywhere are identified.

• The characteristic function of the rational numbers 1Qdisplaystyle 1_mathbb Q is nowhere differentiable yet has a weak derivative. Since the Lebesgue measure of the rational numbers is zero,

∫1Q(t)φ(t)dt=0.displaystyle int 1_mathbb Q (t)varphi (t)dt=0.

Thus v(t)=0displaystyle v(t)=0 is the weak derivative of 1Qdisplaystyle 1_mathbb Q . Note that this does agree with our intuition since when considered as a member of an Lp space, 1Qdisplaystyle 1_mathbb Q is identified with the zero function.

• The Cantor function c does not have a weak derivative, despite being differentiable almost everywhere. This is because any weak derivative of c would have to be equal almost everywhere to the classical derivative of c, which is zero almost everywhere. But the zero function is not a weak derivative of c, as can be seen by comparing against an appropriate test function φdisplaystyle varphi . More theoretically, c does not have a weak derivative because its distributional derivative, namely the Cantor distribution, is a singular measure and therefore cannot be represented by a function.

Properties

If two functions are weak derivatives of the same function, they are equal except on a set with Lebesgue measure zero, i.e., they are equal almost everywhere. If we consider equivalence classes of functions such that two functions are equivalent if they are equal almost everywhere, then the weak derivative is unique.

Also, if u is differentiable in the conventional sense then its weak derivative is identical (in the sense given above) to its conventional (strong) derivative. Thus the weak derivative is a generalization of the strong one. Furthermore, the classical rules for derivatives of sums and products of functions also hold for the weak derivative.

Extensions

This concept gives rise to the definition of weak solutions in Sobolev spaces, which are useful for problems of differential equations and in functional analysis.

See also

• Subderivative

References

• 
  Gilbarg, D.; Trudinger, N. (2001). Elliptic partial differential equations of second order. Berlin: Springer. p. 149. ISBN 3-540-41160-7..mw-parser-output cite.citationfont-style:inherit.mw-parser-output qquotes:"""""""'""'".mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em
  


• 
  Evans, Lawrence C. (1998). Partial differential equations. Providence, R.I.: American Mathematical Society. p. 242. ISBN 0-8218-0772-2.
  


• 
  Knabner, Peter; Angermann, Lutz (2003). Numerical methods for elliptic and parabolic partial differential equations. New York: Springer. p. 53. ISBN 0-387-95449-X.