Talk:Sobolev space

message: The notation I use is not generally Wpk but usually Wk,p. I wrote large chunk with the W_p^k notation because the initial stub used that notation, but I must say I prefer to have both p and k in superscript. Loisel 06:08, 19 Aug 2004 (UTC)
thread_title: Placement of p

message: Was it really necessary to remove the unit circle example?
Charles Matthews 20:00, 18 Aug 2004 (UTC)
replies: text: Well, em, no, of course, but since the whole article is about the line, I thought it would be more clear if the examples were on the line too. So I changed sum to integral, basically. Or did I miss some subtlety? Gadykozma 22:04, 18 Aug 2004 (UTC)

text: I am actually a "circle person" myself (out of perhaps 8 papers I have in analysis, only one is on the line). Maybe it's worth to add a paragraph there about Sobolev spaces on the circle? What do you think? Should such a paragraph appear before or after the discussion of R? Gadykozma 13:38, 19 Aug 2004 (UTC)
thread_title: Unit circle example

message: Hi Sobolev guys. Could you take an interest in fractional calculus - see the talk page?
Charles Matthews 21:42, 24 Sep 2004 (UTC)
thread_title:  Fractional Calculus 

message: This has been in the article a long time, so it occurred to me after I deleted it that I should copy it here.  The problem is it is completely ahistorical.  Sobolev spaces were invented to solve PDEs, and that is still their major application today. Of course things like stability and error estimates were and are important, but that is not the same thing as 'the butterfly effect'. Brian Tvedt 02:24, 11 January 2006 (UTC)
replies: text: If the initial data and/or the model are slightly wrong, how wrong can my prediction be?
By turning to this question, mathematicians eventually gave precise descriptions of "slightly wrong data" and "wrong prediction". In so doing, it became apparent that the natural space of C^1 functions was inadequate. As mathematicians found what the meaning of "slightly wrong data" and "wrong prediction" ought to be, it became obvious that sometimes the "predictions" would not be C^1. This required a careful investigation of the meaning of a differential equation, when the solution is not even differentiable.
The Sobolev spaces are the modern replacement for the space C^1 of solutions of partial differential equations. In these spaces, we can estimate the size of the butterfly effect or, if it cannot be estimated, we can often prove that the butterfly effect is too strong to be controlled.
It would be great to see some discussion of why S. spaces are important restored to this article. Bungo77 12:59, 12 March 2007 (UTC)

text: Indeed, please see the discussion at Wikipedia talk:WikiProject Mathematics/Archive 23#Sobolev space. Jmath666 15:28, 22 March 2007 (UTC)
thread_title:  Text removed from introduction 

message: In the article we started with definition of H^k for natural k. We never defined H^{-k} (for k>0) and never mentioned the dual relationship between H^k and H^{-k}.(Igny 13:12, 19 March 2007 (UTC)
thread_title:  noninteger k 

message: Please see the discussion at Wikipedia talk:WikiProject Mathematics/Archive 23#Sobolev space. Shouldn't the discussion be moved here? Jmath666 01:32, 22 March 2007 (UTC)
thread_title:  Rewrite proposed 

message: While Sobolev spaces on Riemannian manifolds are not defined, it is used in the embedding theorems section. Temur (talk) 07:51, 9 January 2008 (UTC)
replies: text: Good point.  There are a few things to say about Sobolev spaces on Riemannian manifolds.  If I have the time in the near future, I can start a new section.  Silly rabbit (talk) 15:31, 10 February 2008 (UTC)
thread_title:  Sobolev spaces on Riemannian manifolds 

message: I don't understand how:
* a function can be in W^{1, 1} but not continuous or even bounded (section before ACL; I agree with that)
* any function in W^{1, p} is absolutely continuous on every line parallel to axes (ACL section; I might believe in almost every line).
Bdmy (talk) 20:32, 27 December 2008 (UTC)
replies: text: Yes, it should be almost every line.  Moreover, it may also be necessary to modify the function on a set of measure zero first.  siℓℓy rabbit (talk) 21:48, 27 December 2008 (UTC)
thread_title: I don't understand how...

message: Section "Sobolev embedding": "(For p = ∞ the Sobolev space W^{k,\infty} is defined to be the Hölder space Cn,α where k = n + α and 0 < α ≤ 1.)" A line before n is the dimension. But here probably n is not at all the dimension. --Boris Tsirelson (talk) 20:58, 10 January 2011 (UTC)
replies: text: I really miss the part about negative ks. It seems like it has been there before. --93.104.36.169 (talk) 01:01, 21 September 2011 (UTC)
thread_title:  Wrong formula? 

message: We made some changes to the article. We are a team of people who have held and attended lectures on function spaces. We wanted to realize some of the suggestions in the discussion here. Here is a list of what we changed:
*  the article has a new structure, roughly following the proposal for structural change above: no special treatment for unit ball/real line, double results (e.g. trace and extension sections) are eliminated, ...
* we moved the focus away from the p=2 case. Everything on Sobolev spaces (traces, interpolation, fractional order, approximation,….) presented in the original article works in the same way for 1 ≤  p ≤ ∞ or 1 <  p  <  ∞ and does not depend on the Hilbert space structure. It is still mentioned that some things are special for p = 2.
* we changed/eliminated the part on interpolation. It is treated (link now exists) in a different article. We still state the connection. Several results in the original article were imprecise and incorrect (e.g. reiteration theorem for complex interpolation)
* we changed the sections on fractional order spaces, following the suggestions above. There were a few substantial mistakes and insufficient literature before. We hope that the general ideas behind these spaces become clear, but we would also appreciate more text and explanations for the layperson. 
For the future, we would like to include negative order spaces and duality, as in the rewrite by Igny.
Please comment and correct our version. Thank you!  — Preceding unsigned comment added by FunctionspaceInvader (talk • contribs) 12:46, 17 March 2011 (UTC)
replies: text: I think you are doing a good job: the emphasis on Sobolev spaces on the unit circle was not useful as a simplification, since it has not the wealth of applications the multi-dimensional theory has. Also, on the unit circle, other more general function spaces are well understood and manageable, therefore rewriting the entry for the more traditional multi-dimensional point of view deserves a praise. Daniele.tampieri (talk) 19:18, 3 April 2011 (UTC)
thread_title: Changes

message: Right now, there is a theorem cited from Adams, 1975 that says we have approximation by smooth functions up to the boundary if the boundary is Lipschitz continuous.  However, Theorem 3 from section 5.3.3 of Evans Partial Differential Equation requires the boundary to be C1.  Any know if the Adams result is correct?  — Preceding unsigned comment added by 160.39.183.149 (talk) 01:29, 18 February 2013 (UTC)
thread_title:  Theorem on approximation by smooth functions 

message: The text of this section of the article seems to allow negative s in the definition of H^s_p, but in this case the elements are no longer functions in L^p, contrary to the definition displayed after mentioning real s.  The corresponding definition in Bergh and Löfström, p. 141, replaces f \in L^p by f being a tempered distribution. This must be clarified. Bdmy (talk) 21:46, 4 July 2013 (UTC)
thread_title:  Bessel potential spaces 

message: A rather minor point, but does a function in W1,p(ℝ) have to be continuous? I put in an example of a weak derivative, and purposely threw in a single point which makes a discontinuity there. But I see that User:Gadykozma wrote (back in 2004) that functions in W1,1(ℝ) are absolutely continuous. My discontinuity doesn't affect the L1 norm or the weak derivative. A more general question is whether the functions in a Sobolev space have to be defined everywhere or only almost everywhere. In other words, is the space actually a quotient space over "equivalence almost everywhere", like Lp space? Eric Kvaalen (talk) 11:16, 12 February 2014 (UTC)
replies: text: As far as I understand, a function in a Sobolev space is initially a special element of Lp, therefore, an equivalence class. (Therefore, no values at points.) For some values of the parameters (k,p,n) an embedding theorem guarantees existence of a continuous function within such equivalence class; then, naturally, other elements of this equivalence class are of no interest. Accordingly, it is usual to say that "functions in W1,1(ℝ) are absolutely continuous" etc. And your "u(x)=10 if x=0" is of little relevance to Sobolev spaces; no one bothers about u(0) when defining an equivalence class. Boris Tsirelson (talk) 11:33, 12 February 2014 (UTC)
replies: text: Thanks. Yes, I know that my "u(x)=10 if x=0" is of little relevance to Sobolev spaces. I just thought I'd throw that in to make the point that it doesn't matter! I think I will put in a little qualifier in the sentence about functions in W1,1(ℝ) being absolutely continuous. Eric Kvaalen (talk) 18:01, 12 February 2014 (UTC)
thread_title: Continuity question

message: In section "Approximation by smooth functions" I see the notion "smooth up to the boundary"; I wonder, what is its definition? Boris Tsirelson (talk) 09:52, 27 January 2015 (UTC)
replies: text: It means that the function and all derivatives extend continuously to the closure.  Sławomir Biały (talk) 13:16, 14 March 2015 (UTC)
replies: text: Thank you. It would be nice to write so in the article (or make a link). Boris Tsirelson (talk) 07:04, 15 March 2015 (UTC)

text: But I also bother about a non-regular open set (for instance, an open ball minus a closed disk). In this article (section), the domain has Lipschitz boundary, therefore regular. Is the notion "smooth up to the boundary" defined for such domains only? Boris Tsirelson (talk) 07:20, 15 March 2015 (UTC)
replies: text:  I think in this setting, Lipschitz means that in a neighborhood of every point of the boundary, the open set is the undergraph of a Lipschitz function.  This meaning of "Lipschitz boundary" is perhaps somewhat specific to this area of analysis and PDEs.  It rules out pathologies like the one you are describing.  Sławomir Biały (talk) 21:25, 15 March 2015 (UTC)
replies: text:  Yes. This is why I wrote "the domain has Lipschitz boundary, therefore regular". But I am not sure that my example is a pathology. Rather, different limits on the two sides of the disk could be allowed. Maybe this is known as "slit domain" in complex analysis. In fact, faced such situation, I hoped to find an appropriate definition, but was unhappy, not finding it. Well, for Wikipedia, my question is, what is the widely accepted definition (if any), including the indication of the class of domains on which the definition applies. Boris Tsirelson (talk) 21:46, 15 March 2015 (UTC)
replies: text:  Sorry, I parsed what you wrote incorrectly.  There are cone conditions (Wikipedia article missing) that allow domains like the kind you describe, with the interior lying on two sides of some closed subset.  Sławomir Biały (talk) 22:22, 15 March 2015 (UTC)
replies: text:  Interesting. But, has the notion "smooth up to the boundary" a counterpart for such domains? Boris Tsirelson (talk) 06:09, 16 March 2015 (UTC)
replies: text:  Something like smooth functions in Ω whose derivatives are bounded of all orders is a notion that is available with no conditions on Ω.  (This is the intersection of the Sobolev spaces W^{k,\infty}(\Omega) for all k.)  Sławomir Biały (talk) 12:35, 16 March 2015 (UTC)
replies: text:  Wow! Looks to be a clever idea.  I am afraid, brand new... Boris Tsirelson (talk) 17:52, 16 March 2015 (UTC)
thread_title:  smooth up to the boundary?