Talk:Uniqueness theorem for Poisson's equation

All right, so I rewrote the uniqueness theorem (as known in physics) here as the previous page did not contain any formulas but had a somewhat confusing text-only proof. If anyone wants to clean up and make things prettier, please do go ahead! Also, I changed the page type from math to physics, as this is what this page discusses anyways. For people who want just the math definition, there already exists uniqueness page which is just that. IlyaV (talk) 19:21, 3 April 2010 (UTC)

Boundary at infinity

16.7.2014 Hey, my name is Ofek, I'm completing a PhD in Physics. Looking over the article (and the discussion below), I definitely think there should be some, however short, mention of boundaries at infinity, and when they do or do not make uniqueness hold. It's an important issue, which appears neither here nor under the Boundary_condition page. Should I edit the page, or wait a week for responses? — Preceding unsigned comment added by 132.64.2.181 (talk) 12:45, 16 July 2014 (UTC)

I'm reverting the edits about boundary at infinities.

I think boundaries at infinities are already covered in the 4 boundary conditions aforementioned. If $\phi$ or $\mathbf {\nabla } \phi$ don't converge at infinity rapidly enough then the uniqueness theorem doesn't hold. There is nothing in the solution of the Laplace equation that necessitates that in

$\varphi =\sum \limits _{n,m}{\left({{a}_{n,m}}{{r}^{n}}+{\frac {{b}_{n,m}}{{r}^{n+1}}}\right)}P_{n}^{m}\left(\cos \theta \right)\cos m\phi +\sum \limits _{n,m}{\left({{c}_{n,m}}{{r}^{n}}+{\frac {{d}_{n,m}}{{r}^{n+1}}}\right)}P_{n}^{m}\left(\cos \theta \right)\sin m\phi$

${a}_{n,m}$ or ${c}_{n,m}$ should be 0. Simply, if they are zero, then the uniqueness theorem holds, if they aren't, then it doesn't hold. Which is to say, $\phi =r$ is a valid solution to the Laplace equation with non-zero (actually infinite potential) boundaries at infinity. I suspect one can also construct (unphysical) cases where $\phi$ and it's gradient both go to non-zero constants asymptotically AND satisfy Laplace equation (again, uniqueness theorem doesn't hold then).

I think the only thing that can be added about "boundaries at infinity" is that since energy of the fields is proportional to $\mathbf {\nabla } \phi ^{2}$, if $\mathbf {\nabla } \phi$ doesn't go rapidly enough to 0 as $r\to \infty$ this would imply infinite energy, so the difference in energies of the two solutions would be infinite which is unphysical (the two systems would have to have very different Hamiltonians, indeed, difference in Hamiltonians of two solutions is sufficient for the uniqueness theorem to not hold; of course it's also very unphysical). Again, this is not a "new" boundary condition, it's simply a statement that for infinitely large systems, the electric field at infinity must be 0 (Neumann boundary condition).

All the best

IlyaV (talk) 00:23, 25 April 2010 (UTC)

Reply:

In real world,the boundary conditions are not necessary because all real charges are allocated within bounded domain and the potential hence decreases rapidly enough.But here we are talking about mathematical model,like an infinitely large grounded conductor plate,which is NOT finite.Another example is an infinitely long charged conductor pipe,which has infinite energy and its field doesn't even go to zero.It's unphysical,but it is still a good model under certain circumstances.If you want everything to be purely PHYSICAL,then there should not be the concept of "plane" and "particle" alike.

Netheril96 (talk)

I think you misunderstood my argument. What you are saying is that there has to be ${a}_{n,m}=0$ or ${c}_{n,m}=0$ but there is absolutely no reason for this to be true unless boundaries (at infinity) force that. But the boundaries can only be one of the 4 boundaries already mentioned. If boundaries at infinity are not established, the Uniqueness Theorem doesn't hold. To avoid over-reverting, I won't revert until our argument is settled, but please do reply. If I don't get a reply within a week I'll revert back (no hard feelings). IlyaV (talk) 17:53, 29 April 2010 (UTC)

Reply:

OK,I did misunderstand you.The boundary conditions at infinity ARE included in the 4 mentioned,but you didn't understand what I meant,either.What I emphasize is that it requires a different demonstration at infinity.If the solution of Laplace's equation could be something that is proportional to ${\frac {1}{\sqrt {r}}}$,then the uniqueness theorem would still hold for bounded domain,but for a infinitely large domain

$\oint _{S}{\varphi {\frac {\partial \varphi }{\partial n}}}dS\propto {\frac {1}{\sqrt {r}}}\cdot {\frac {1}{\sqrt {{r}^{3}}}}\cdot {{r}^{2}}=1$

This wouldn't converge to zero when r approaches infinity.Then uniqueness theorem would not hold for infinite domain with boundary conditions set.Therefore it is necessary to prove the infinite one separately.

Netheril96

There is another issue.The uniqueness theorem in fact doesn't not require the boundary conditions at all the surfaces set.It requires only the outermost boundary condtions.For inner interfaces,the two surface integrals from two sides automatically cancel out,that is to say

$\oint _{{S}_{1}}{{{\varepsilon }_{1}}{{\varphi }_{1}}}\nabla {{\varphi }_{1}}\cdot d\mathbf {S} +\oint _{{S}_{1}}{{{\varepsilon }_{2}}{{\varphi }_{2}}}\nabla {{\varphi }_{2}}\cdot d\mathbf {S} =\oint _{{S}_{1}}{\left({{\varepsilon }_{1}}{{\varphi }_{1}}{\frac {\partial {{\varphi }_{1}}}{\partial n}}-{{\varepsilon }_{2}}{{\varphi }_{2}}{\frac {\partial {{\varphi }_{2}}}{\partial n}}\right)}dS$

From the interface condition

${\begin{aligned}&{{\varphi }_{1}}={{\varphi }_{2}}\\&{{\varepsilon }_{1}}{\frac {\partial {{\varphi }_{1}}}{\partial n}}={{\varepsilon }_{2}}{\frac {\partial {{\varphi }_{2}}}{\partial n}}\\\end{aligned}}$

Thus the integrals cancel out.You need only to specify the boundary condition at the outermost surface.

(Here the two $\varphi$ means the difference of two possible solution at either side of an inner interface S₁)

Netheril96

OK, I think I understand what you're getting at. What you want to add to the article is an explicit statement that (for certain boundary conditions), uniqueness theorem holds even in an infinite domain. That's definitely something that has a place in this article, although perhaps it should be stated a bit more clearly. I'll try to edit it a bit, and hopefully we can get at something that we both like!

About boundaries, "Boundary Conditions", as used in ODEs/PDEs imply the boundary of the solution, which is synonymous to what you call "outermost" surfaces. Although, perhaps this is something that can be elucidated in the article.

Another thing I'm considering is removing units (epsilon, 4pi, etc) since the uniqueness theorem for Poisson's equation is used in a number of fields outside of electrostatics (eg magnetostatics and certain time independent Schrodinger equations). This way we'll just write it for some generic "field" and "source"... But this would also suggest adding a new section about how uniqueness theorem is used in different fields, which I don't really feel like writing it now :P IlyaV (talk) 21:19, 30 April 2010 (UTC)

I rewrote boundaries at infinities, hopefully you'll agree with this edit. I changed the form of the sum to be through spherical harmonics which made the equation look shorter. IlyaV (talk)

Reply:

Glad you finally agree with me and I appreciate you rewriting.My words are poorly organized since I'm a Chinese and not that good at English O(∩_∩)O~.As for a general uniqueness theorem,I will write a new one because that is slightly different (A general vector field may have not a scalar potential).I hope you will revise it because I'm not familiar with wiki markup and I don't have English textbooks to cite as references.Thanks.

As for the automatic cancelling of two integrals at either side of an interface,I think it is necessary to add to the original article because this is what electrostatic fields have in particular.For a general field,to solve it we need it to be smoothly enough.For instance,it must be twice differentiable for Laplace's equation to be meaningful.Thus we need to set the boundary conditions wherever it is discontinous.But for electrostatic field we need only the outermost ones.(A magnostatic field has similar interface condition and need only outermost too)

I also reorganized this article to distinguish your and my comments.

Netheril96

I'll certainly be glad to edit any additions to the article. If it is a large change, please post it here on the talk page first before adding it to the main page.

As for boundaries, the surface integral is already on the "outermost" boundaries, as in, we integrate over the volume (our entire domain) and require only one derivative of $\phi$ to exist, we then use Divergence Theorem. Since I explicitly keep $\epsilon$ everywhere, I don't think anything special is happening.

Again, perhaps if you write out the change you want to make, it'll make more sense to me.

All the best,

IlyaV (talk)

Uniqueness in linear electric circuit

Maybe we can have a new item for Uniqueness theorem? It is obviously different from uniqueness in electrostatics (no current) and uniqueness of the solution of the Maxwell's equations (charges and currents are usually treated as given boundary conditions). Though it may even seem to be trivial, a proof or some explanation is missing, it seems that the only place addressing it is on the page of Non linear networks.

Gamebm (talk) 14:22, 25 November 2012 (UTC)

Since there is no objection (after 5 years), I added a new item Electromagnetism uniqueness theorem for the solution of Maxwell's equation in the page uniqueness theorem. Some pages I edited indeed needed a link to a relevant theorem. Gamebm (talk) 18:55, 24 August 2017 (UTC)

proof

Is there a reference for the identity used in the proof? It might be the fifth one down here, but I'm not sureDiracsdeltafunction (talk) 21:24, 4 September 2016 (UTC)