Talk:Weak derivative

Identical?

in the theory of Lp spaces and Sobolev spaces, functions that are equal almost everywhere are identified.

Did you mean to write "identical" in the sentence above?

Jörgen — Preceding unsigned comment added by 81.234.152.24 (talk) 20:16, 9 October 2008 (UTC)

No, the usual idiom is that the functions are identified. They are not identical functions, however. For instance, the characteristic function of a point and the zero function are different functions, but they are identified with each other in any discussion to do with Lp spaces or Sobolev spaces. siℓℓy rabbit (talk) 21:05, 9 October 2008 (UTC)

Identified in the sense of: are elements of the same equivalence class. 129.107.240.1 (talk) 18:26, 5 March 2009 (UTC)

Examples

Can we get a few more? The given example of the absolute value function seems pretty trivial. We could, in this case, just see that the signum is the derivative almost everywhere, and thus see that it is a weak derivative. What would be much better is an example of a function which is not differentiable throughout any interval (or even further, not differentiable anywhere) but which has a weak derivative. 129.107.240.1 (talk) 18:26, 5 March 2009 (UTC)

I added an example, however contrived it might be! --Paul Laroque (talk) 02:26, 13 April 2009 (UTC)

question

I am an undergrad, so forgive me if this is a stupid question. It wasn't really clear from the article. The Lebesgue Integral of a weak derivative is equal to the original function(not phrased correctly i think, but should be clear enough)? If so, almost everywhere, or everywhere?--79.235.185.160 (talk) 12:40, 6 February 2010 (UTC)

Applications

What are some of the applications of weak derivatives? Is it possible to build an optimisation theory based on weak differentiation? 203.167.251.186 (talk) 03:17, 24 June 2010 (UTC)

Lebesgue spaces

I was wondering why a function that is only weakly differential must be in $L^{1}$. I suppose that if the function is not strongly differentiable then one couldn't use the chain rule to get the derivative of say the $P^{2}$ norm for this function. However, perhaps a $P^{2}$ norm may still be integrable if we remove problematic points by using an improper integral.

On another note, why are we using the term Lebesgue space. Isn't the term Lp space more common? S243a (talk) 18:48, 18 October 2013 (UTC)

Why not this definition?

According to the article, we say that $v\in L^{1}([a,b])$ is a weak derivative of $u\in L^{1}([a,b])$ if

$\int _{a}^{b}u(t)\varphi '(t)dt=-\int _{a}^{b}v(t)\varphi (t)dt.$

for all infinitely differentiable functions $\varphi$ with $\varphi (a)=\varphi (b)=0$. However, my intuition of a weak derivative $v$ is that the original function $u$ is an antiderivative of $v$, i.e. that

$\int _{t_{1}}^{t_{2}}v(t)dt=u(t_{2})-u(t_{1}),$

for all $t_{1}$ and $t_{2}$ in $u$'s domain, which seems significantly simpler to me. But since it isn't the definition used in the article, I guess it is incorrect. In which ways do these two definitions differ, and what examples are there of functions $u$ and $v$ where the two definitions disagree on whether $v$ is a weak derivative of $u$? I.e. why is the former definition the one that is used and not the latter? —Kri (talk) 16:38, 16 January 2020 (UTC)

Sobolev space

The conventional definition of weak derivatives is used to define Sobolev space $W^{1,2}$ and uses proofs of completeness of convergent sequences (Because Hilbert space is complete, and because absolutely continuous functions are dense, Arzelà-Ascoli theorem and even the smooth functions are dense in the space. This article should state these assorted theorems about weak derivatives. The current version of the article Calculus on Euclidean space has some content on this more conventional, broader definition, which should be copied to, moved to this article. There's a bunch of stuff that can be said about weak derivatives; for example, weak solutions of the Poisson equation, etc. that are not even hinted at in this article. 67.198.37.16 (talk) 00:54, 22 March 2024 (UTC)