Talk:Elliptic partial differential equation

At very least this topic deserves a talk page.

There is a complaint about the nature of the current coverage (this redirects to elliptic operator) given to the subject in WP ( proposition at Talk:Elliptic_operator#split). I find I agree. Also the style of the article is too abstract, and does not concretely instatiate any elliptic PDE. It does not, but think it should, follow the style of parabolic and hyperbolic equations. 41.58.15.94 (talk) 21:55, 30 December 2012 (UTC)

B^2 - AC

I believe it should be B^2 - 4AC 38.121.241.183 (talk) 04:32, 7 March 2024 (UTC)

What about non-homogeneous equations ?

Non-homogeneous equations, e.g. D^2 u=f are also considered to be elliptic in most textbooks, yet they are not mentioned at all in this article AneasSu (talk) 04:15, 22 October 2024 (UTC)

The coverage in this article is extraordinarily lacking, I would encourage you to add material. I disagree with the editor who removed your previous edits; that was all material that I as a reader would completely expect to find (in some form) on this page. (However I do think it could have been phrased more clearly.) Gumshoe2 (talk) 04:48, 22 October 2024 (UTC)

The reason I reverted the edit was because it was semi-duplicate content WP:SED. It's perfectly appropriate to address things like existence or regularity, but only if it shows how it specifically applies to elliptic PDEs. Otherwise a simple hyperlink would suffice.

Also, I think it's strange to start a new subsection called "weak solutions" while there isn't even a subsection discussing the solutions yet. Furthermore, it's just a copy/paste of page 296 of the Lawrence 2010 reference which applies to most PDEs, not just elliptic ones. Roffaduft (talk) 05:07, 22 October 2024 (UTC)

No, almost all of the "weak solutions" content in the edit I'm looking at is about elliptic equations and would definitely not be valid for non-elliptic equations. It is from Chapter 6 of Evans' book, which is specifically about second-order elliptic equations, and does not seem to be a copy/paste to me. (However, unfortunately I have the second edition, in which page 296 is in Section 5.8 and is basically unrelated.) I agree that it's strange to jump to weak solutions, but there is so little content presently on this page that I think any addition of fundamental material is definitely worthwhile.

It would be worthwhile to figure out what should go here and what should go at elliptic operator. To me, "elliptic operator" suggests a type of differential operator acting on sections of a vector bundle, with natural relations to elliptic systems. The content from the above edit isn't what I'd expect to see there, although some of it should also be present there in a more general formulation. Gumshoe2 (talk) 05:19, 22 October 2024 (UTC)

I was referring to Chpt 6.1.2. Weak solutions, equation (7), "Motivation for definition of weak solution". It doesn't specifically address the elliptic PDE. Also, there is already a wikipedia article on weak solutions.

To put it differently, equation (7) from Evans addresses PDEs of the form $Lu=f$ without taking the ellipticity into consideration. (Ellipticity is only being adressed in the subsequent "Definition", (8) in Evans) Roffaduft (talk) 05:29, 22 October 2024 (UTC)

Through the entire chapter, L is taken to be an elliptic operator. Evans doesn't say so explicitly, although he should have. This is apparent from the fact that pretty much none of the results in the chapter hold without assuming ellipticity, and ellipticity is used in all of the proofs. The definition implicit in equation (7) is the only thing that can plausibly be said to have broader context than ellipticity.

There is minor overlap here with the weak solution wikipage, only in as much as the definition of weak solution needs to be given here briefly. The results on existence, regularity, and maximum principle are particular to elliptic equations and don't presently appear on the weak solution page. Gumshoe2 (talk) 05:41, 22 October 2024 (UTC)

The definition implicit in equation (7) is the only thing that can plausibly be said to have broader context than ellipticity.

Thereby proving my point exactly. The edit only addresses equation (7) in it's "definition" of a weak solution. This "definition" would be a great addition to the weak solution article instead. Roffaduft (talk) 05:48, 22 October 2024 (UTC)

I don't follow what you're saying. That edit also contains material on existence, regularity, and maximum principles for weak solutions of elliptic equations, which you also removed. It's not only stating the definition of a weak solution. (If it were, I would agree with your edit.) Gumshoe2 (talk) 05:55, 22 October 2024 (UTC)

I don’t agree with your removal of the section “weak solutions”, as

1. the meaning of term may differ with the type of the equation.

2. The existence and regularity theorem does not apply to other PDEs in general. AneasSu (talk) 05:27, 22 October 2024 (UTC)

Regarding your second point: If that's true then why are they discussed in Chpt. 7 of Evans (2010) in the cases of parabolic and hyperbolic PDEs? Roffaduft (talk) 05:37, 22 October 2024 (UTC)

There are also existence and regularity theorems for parabolic and hyperbolic differential equations, but they're different than those for elliptic equations. They belong on the wikipages parabolic partial differential equation and hyperbolic partial differential equation. Gumshoe2 (talk) 05:42, 22 October 2024 (UTC)

Chpt. 7.1.3. Regularity. [and i quote]

"In this section we discuss the regularity of our weak solutions $\mathbf {u}$ to the initial/boundary-value problem for second-order parabolic equations. Our eventual goal is to prove that $u$ is smooth, provided the coefficients of the PDE, the boundary of the domain, etc. are smooth. The following presentation mirrors that from §6.3."

Ergo, regularity, as addressed in chpt. 6.3, would be a great subsection in the partial differential equation article. The specific application to elliptic PDEs would be a great extention to the elliptic PDE article (including a nice hyperlink to the - to be introduced - "regularity" subsection in the PDE article). Roffaduft (talk) 05:58, 22 October 2024 (UTC)

Section 7.1 is about parabolic partial differential equations. The regularity theory for parabolic PDE (in 7.1.3) does have strong parallels to the regularity theory for elliptic PDE (in 6.3), just as there are similar (and similarly proved) maximum principles for both parabolic and elliptic PDE. However the actual theorems are, fundamentally, distinct. What is stated and proved in Section 6.3 is specifically for elliptic PDE, there cannot be any question about this. It is not something about PDE in general. Gumshoe2 (talk) 06:05, 22 October 2024 (UTC)

I've clearly said that I don't have an issue with adding information that is specific to elliptic PDEs. My issue is that in the edit "Regularity" comes seemingly out of nowhere. It lacks any context, i.e., WP:PCR.

• What is regularity?
• Why is this releveant to elliptic PDEs?
• What are $H^{m},H_{0}^{1},C^{m+1}$ etc.?
• What is the relevancy of Interior and/or Boundary regularity?

That is why I suggested to introduce the concept in the main partial differential equation article, and the specific theorems in the elliptic article. Alternatively, you could provide (a lot) more context. Roffaduft (talk) 06:25, 22 October 2024 (UTC)

I'm sorry, I must have misunderstood your point. I thought you had been suggesting that this material about elliptic PDE is not actually particular to elliptic PDE.

I absolutely agree that more context and explanation would be extremely valuable, and all notation has to be explained. I think the material AneasSu added was good as a start to build upon, certainly not as a finished version. I would suggest to AneasSu (talk · contribs) to revert your removal of his edit, and we can all work on making it clearer and more accessible. Gumshoe2 (talk) 06:36, 22 October 2024 (UTC)

I think a “weak solutions” section should be added to the partial differential equation article, and the existence, regularity, maximum theorems about elliptic equations should be added here AneasSu (talk) 06:47, 22 October 2024 (UTC)

I agree, there should certainly be material about weak solutions at partial differential equation. Gumshoe2 (talk) 06:49, 22 October 2024 (UTC)

I'm not going to do that. I actually did assume WP:GF and I've initially tried to fix the mathematics and provide more context myself. But the elliptic PDE article is just not the place to introduce e.g. the divergence form. The main reason I've reverted the edit was simply because too much of it was out of place, ambiguously phrased or lacking context.

However, I totally agree that the article should be expanded. I'd like to suggest to first improve the main partial differential equation article together, adding things like the $Lu=f$, the divergence forms, weak solutions etc. Alternatively we could also extend the weak solution article. If we've done that, it becomes very easy to introduce, e.g., Interior/Boundary regularity to the elliptic PDE article without risking writing semi-duplicate subsections everywhere. Roffaduft (talk) 06:54, 22 October 2024 (UTC)

I agree that this article certainly could use some work. But elaborating on things like homogeneous versus non-homogeneous equations would be less appropriate. As this classification applies to all PDEs, it should be addressed in the partial differential equation article.

ps. Please use math display="block">" rather than writing all equations inline. I really helps with the layout and readability.

Kind regards,Roffaduft (talk) 04:52, 22 October 2024 (UTC)

Lu=f + (boundaries conditions) is the more general form, so Lu=f, instead of Lu=0 should be mentioned in the description part of the article. AneasSu (talk) 05:29, 22 October 2024 (UTC)

On the first page of Chpt 6. of Evans (2010) it states that $L$ is a second-order partial differential equation. Therefore addressing the distinction between the homogeneous and inhomogeneous case belongs to the partial differential equation article. That's all I'm saying.

Kind regards, Roffaduft (talk) 05:33, 22 October 2024 (UTC)

Even though the distinction does not belong here, the statement “any 2nd-order elliptic PDEs can be written in the form …=0” is not correct, and it should be fixed as “…=f”. AneasSu (talk) 06:25, 22 October 2024 (UTC)

If you're referring to the introduction, I believe that: $f=-G$, so technically the statement is correct. Roffaduft (talk) 06:31, 22 October 2024 (UTC)

Alright. AneasSu (talk) 06:32, 22 October 2024 (UTC)

Order of the elliptic PDE

Given that we've reached consensus, it seemed appropriate to start a new topic.

@Gumshoe2, I've got a question regarding your latest edit. What do you consider to be the "order" by saying that elliptic PDEs do not have to be second order? Are you perhapse confusing the numer of variables and/or dimensions of systems of equations with the order of the equation?

Secondly, you state:

• As for the smaller class of linear equations, ellipticity here means that..

which contradicts the (correct) statement that elliptic PDEs need not be linear. That is, for linear, semi-linear and quasi-linear second order PDEs, ellipticity means B^2 - AC < 0. Which is also addressed at the bottom of Partial_differential_equation#Linear_equations_of_second_order.

Kind regards, Roffaduft (talk) 07:42, 22 October 2024 (UTC)

In introductory books, elliptic PDE are almost always second-order, but they can be higher-order as well. For example, the biharmonic equation is a fourth-order elliptic PDE. See for example these references. Gumshoe2 (talk) 08:12, 22 October 2024 (UTC)

Ok, I'm not going to check the references from stackexchange. I did, however, get my hand on the references you've cited in your latest edit. If you could cite the page numbers, that would be really helpful.

If I take a look at Monge–Ampère_equation#Ellipticity_results, then it's pretty clear that the "ellipticity" refers to the linearized solution method. I strongly suspect that "higher-order" elliptic PDEs refer to "ellipticity" in a similar manner. In other words, I think it's better to stick to the second-order statement in the intro and add a new "Generalization" subsection starting from the last paragraph of the "Definition" subsection.

Kind regards, Roffaduft (talk) 08:35, 22 October 2024 (UTC)

After looking at the Wloka reference, it is clear that it's about the elliptic operator. Sticking to the definition of Evans chpt. 6.1.1.: The elliptic PDE is second-order. Roffaduft (talk) 08:51, 22 October 2024 (UTC)

Yes, I will definitely include page numbers when I can. How exactly are you drawing a line between "elliptic PDE" and "elliptic operator"? The latter is the more unambiguous concept (in basically the same way that $x\mapsto x^{2}$ is unambiguously quadratic while perhaps it's not completely clear whether one would want to call $(x^{2}-y)^{4}=0$ a quadratic equation or just obviously equivalent to a quadratic equation), but L being an elliptic operator corresponds directly to Lu=f being an elliptic equation.

I frankly don't understand what you have in mind. Are you suggesting that the biharmonic equation $\Delta \Delta u=0$ is not considered to be an elliptic PDE since it isn't second-order? Or that an equation like $\Delta u=u^{2}$ is not an elliptic PDE since it's nonlinear? It is true that neither of these fall into the context of Chapter 6 of Evans' book, but that book is only written as an introductory overview. Elliptic PDE has a much broader scope than what's given there. Gumshoe2 (talk) 16:41, 22 October 2024 (UTC)

I agree with you almost entirely, and I would like emphasize that the following is mostly a semantic discussion.

From a mathematical point of view it's makes complete sense to start with the elliptic operator and then discuss specific (non)linear cases, orders, dimensionalities, etc. In fact, all the references I looked at take this approach. However, this does lead to some ambiguity in practice. For example:

• the elliptic operator acting on a multivariable function $u$
• the "elliptic partial differential equation" as described in the intro
• Systems of first order equations being elliptic

While one could say that, literally speaking, these are all elliptic partial differential equations, it isn't very helpful from an encyclopedic point of view.

One should ask the question why the elliptic partial differential equation article isn't just a subsection of the elliptic operator article. IMHO that is because of the context: In applied mathematics it's a classification of a second-order, (semi/quasi)linear PDE which, due to it's close connection with physics and engineering, grants its own article. Wikipedia written for everybody, not just mathematicians.

The reason I said "just add a Generalization subsection" to this article, is because I'm lazy. The alternative would require a complete overhaul of the partial differential equation, hyperbolic partial differential equation and parabolic partial differential equation articles as well: starting from the general partial differental operator and step by step add distinctions and special cases. The problem is that this would be dangerously close to WP:NOTTEXTBOOK.

To summerize, my main issue is the ambiguity that arises when "generalizations" are addressed as "elliptic PDEs" in an article discussing the elliptic PDE as a specific classification.

Kind regards, Roffaduft (talk) 04:54, 23 October 2024 (UTC)

Where to add maximum principles ?

Maximum principles are also important conclusions about elliptic PDEs, should I write them here, or in Elliptic operators ? AneasSu (talk) 12:44, 22 October 2024 (UTC)

Looking at the already existing maximum principle article, I wouldn't elaborate too much on it. I think a short mention would suffice and/or probably some extra context rather than a partial copy/paste of equations from the main article. Let's refrain from WP:SED.

Instead it might be better to first focus on introducing the (non) divergence form of the boundary value-problem in the partial differential equation article.

Kind regards, Roffaduft (talk) 13:04, 22 October 2024 (UTC)

Speaking of semi-duplicate, I just found the Elliptic boundary value problem article. It basically covers everything we've discussed so far, citing Evans as well.

This is the problem with adding relatively general information to articles of specific cases. Chances are an article already exists that covers the exact same material.

I'm not quite sure how to streamline/integrate this properly yet. It'll probably boil down to hyperlinking though.

Kind regards, Roffaduft (talk) 05:12, 23 October 2024 (UTC)

Thanks for pointing this out, I wasn't aware of this page. I agree with your concern of not duplicating content but I think that, at least in terms of this concern, the page elliptic PDE is (for now) satisfactory. I don't have a good idea for now about how material (once added) should be split up between these three pages. In the end, possibly some will have to be renamed or refocused, or merged with others. Gumshoe2 (talk) 17:28, 23 October 2024 (UTC)

Maybe adding a “Boundary Value problem” with the main article link and a summary of it can be helpful. AneasSu (talk) 17:49, 23 October 2024 (UTC)

Thank you for all your work! I did have some questions regarding the "Characteristics and regularity" subsection though:

• Does "null directions" have the correct hyperlink?
• While the topic of characteristics being addressed in (subsections of the) articles PDE, Differential operator, MoC etc., you kinda have to read between the lines to get a proper definition. Therefore, I think it would be helpful to give the characteristic form of the principal symbol directly, instead of just the principal symbol itself.
• I had a hard time understanding:

characteristics are significant in understanding how irregular points of f propagate to the solution u of the PDE

Do you mean: characteristics are significant in understanding the propagation of singular points of the solution u of the PDE?

or are we specifically discussing the inhomogeneous case?

In fact, there is so much new terminology being introduced in the subsection (e.g. smoothness, irregular points, wave front set, propagation, frequency space etc.) of such idiosyncratic nature, that I question whether it belongs in this article. While the last paragraph provides some context, the sentence:

as steady states are generally smoothed out versions of truly dynamical solutions.

just sounds really vague to me. On top of that, the last sentence basically states that if modeling becomes a bit more complicated, all of the above is pretty much useless.

I think the first two paragraphs would be a nice addition to Elliptic boundary value problem#Regularity. Given that the "Characteristics" subsection is a subsection of the "Canonical form", it would be more appropriate to just address the characteristics of the elliptic PDE and subsequent distinction from parabolic and hyperbolic PDEs.

See it as a kind request though, as I myself am a bit out of my depth when it comes to characteristics. I really do appreciate all the work you have put into the article so far!

Kind regards, Roffaduft (talk) 07:01, 24 October 2024 (UTC)

I think these are all reasonable comments, I'll try to address them when I have time to edit. However I'm skeptical of adding material to elliptic boundary value problem, if anything I think that material should be transported here. (Though possibly with a different arrangement to this page.) It still isn't clear to me though how to divide material between elliptic PDE, elliptic boundary value problem, and elliptic operator. Gumshoe2 (talk) 15:22, 29 October 2024 (UTC)

|—elliptic PDE

| |—…

| |—boundary value (section)

| | |—main page: elliptic boundary value problems

| | |—summary of main page

| |—…

About elliptic operators, we’d better focus on its properties in functional analysis, e.g. compactness, spectrum, which would require rewriting the whole article.

This is my personal opinion.

Kind regards, AneasSu AneasSu (talk) 15:30, 29 October 2024 (UTC)

My premise would be to ask the question: "Why does [topic] have it's own article?"

• As I said in the previous topic, I think elliptic PDE, parabolic PDE and hyperbolic PDE have their own article due to it's close connection with physics and engineering. IMHO this should be the common thread throughout these articles when discussing the various properties and characteristics. For example, when adding a subsection on boundary value problems, rather than just giving a summary (i.e. semi-duplicate content), talk about the physics/engineering context.
• Regarding the elliptic boundary value problem, I'll have a look at it today; fixing the chronology of the article (which would be mostly cosmetic).

Personally I think the elliptic BVP could benefit by comparing it (in some way) to parabolic and hyperbolic BVPs, but I'm not sure how to do it without creating a big mess. In fact, this might also be the issue I have with the elliptic operator article. It relies too much on a single source (i.e. Evans) which really leaves a mark on the chronology and the way topics are discussed. Basically, Evans uses the elliptic case as a premise and introduces the parabolic and hyperbolic case as "extensions".

(I mean, you could also regard $t$ as any other variable, e.g., $\mathbf {x} =(x,y,z,t)\in \mathbb {R} ^{4}$)

Not sure how tackle this issue, but I think it would be good to look at different sources (e.g. about hyperbolic/parabolic BVDs/PDEs), the partial differential equation and boundary value problem articles as well. Take a step back, see the big picture first.

Kind regards, Roffaduft (talk) 05:51, 30 October 2024 (UTC)

Thank you. I was wondering if you could say a few words about the definition of an elliptic PDE as stated in chpt 6.1.1 of Evans. That is:

$$\sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}(x_{1},\ldots ,x_{n})\xi _{i}\xi _{j}\geq \theta (\xi _{1}^{2}+\cdots +\xi _{n}^{2}).$$

The "problem" I have is that it's near identical to the definition of the parabolic and hyperbolic PDE in chpt 7.1.1 and 7.2.1. The only difference being changing the operator to $\partial _{t}+L$ and $\partial _{tt}+L$ respectively.

Consequently, I don't think this is the most insightful definition of ellipticity. Instead, I'd like to propose using the definition from chpt 3.3 of the Zauder reference, as:

• the distinction between elliptic, parabolic and hyperbolic PDEs is more clear and easier to interpret
• it shows that $B^{2}-AC$ is just the determinant of the $2\times 2$ case
• it is more in line with the last paragraphs of Partial_differential_equation#Second_order_equations

Kind regards, Roffaduft (talk) 05:05, 30 October 2024 (UTC)