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I wonder what are the "moral" differences between Poincare and Sobolev inequalities. Let me state them (hopefully without errors) using Wikipedia as source:

Poincare inequalities Let $\Omega$ be a domain in $\mathbb R^n$ bounded in at least one direction, then there exists some $C = C(p, \Omega)$ such that for all $u \in W^{1,p}_0(\Omega)$ ($1\leq p < \infty$) $$ \Vert u \Vert_{L^p(\Omega)} \leq C \Vert \nabla u \Vert_{L^p(\Omega)} $$ We can also change the condition of $u$ belonging in $W^{1,p}_0(\Omega)$ by $u \in W^{1,p}(\Omega)$ and $\int_\Omega u dx = 0$ (only for bounded $\Omega$). This is called Poincare-Wirtinger.

Gagliardo-Nirenberg-Sobolev inequality Assume $u \in W^{1,p}(\mathbb R^n)$ (in Wikipedia they say $u\in C^1(\mathbb R^n)$ with compact support but by density I think it doesn't matter, correct me if I'm wrong please). Then for $p \in [1,n)$, there exists some $C=C(p)>0$ such that $$ \Vert u \Vert_{p^*} \leq C \Vert \nabla u \Vert_{p} $$ where $p^*$ is some real number larger than $p$ and depending on $n$ and $p$.

I can see that Poincare inequalities are intimately related to domains, whereas GNS isn't. But at first glance, GNS is stronger because if we assume $\Omega$ bounded then $u \in W^{1,p}_0(\Omega) \subset W^{1,p}(\mathbb R^n)$ (by extending $u$ by $0$ outside of $\Omega$) and moreover $\Vert \nabla u \Vert_p \lesssim \Vert \nabla u \Vert_{p^*}$ by Holder ineq.

Summing up, it seems to me that in GNS we don't ask any condition on $u$ besides being in $W^{1,p}$ and we get an (intuitively) better bound on its $p$-norm because it "uses" lower integrability on $\nabla u$.

A related very vague question, whenever I am working in a domain, when should I be thinking about Poincare inequality and when about GNS inequality?

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    $\begingroup$ The main difference is the $p$. In the first inequality it is the same, in the second it is not. $\endgroup$ Commented Feb 9, 2021 at 21:47
  • $\begingroup$ GNS usually deal with whole space as domain and have $p^*$ on the left. There is also a Sobolev-Poincare which has $p^*$ on the left and on (nice) bounded domains. The proof of latter follows from GNS + existence of an extension operator (to allow use of global GNS). $\endgroup$ Commented Jul 29, 2022 at 10:56
  • $\begingroup$ @GiuseppeNegro that's the whole point of the question: Poincare has a lower value of $p$ on the LHS, so on a bounded domain that's a weaker result. Why bother with Poincare then? $\endgroup$
    – Bananach
    Commented May 30, 2023 at 8:26
  • $\begingroup$ @BehnamEsmayli that's whe whole point of the question: since, by extension, you can apply GNS to the bounded domain setting and get a stronger result than the classical Poincare inequality, why is the latter even thought and presented as anything but an imperfect result that's sometimes good enough? $\endgroup$
    – Bananach
    Commented May 30, 2023 at 8:28
  • $\begingroup$ By 'moral' do you mean 'practical'? gio67's answer below is valuable. $\endgroup$ Commented May 30, 2023 at 22:05

1 Answer 1

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A cheap answer. Poincaré’s inequality works for all $1\le p\le\infty$ while GNS only for $p<n$. Since Sobolev spaces are used a lot in pdes and for linear pdes one of the important cases is the $L^2$ case, you have $p=2$ and $n\ge2$ so you are in the case where you cannot apply GNS.

A second cheap answer. Usually, Poincaré’s inequality is used to say that in $W^{1,p}_0$ you can use $\|\nabla u\|_p$ as an equivalent norm. It is true that when $p<n$, you can derive it from the GNS inequality, but the proof of Poincaré’s inequality takes half a page and uses just the ftc and Holder’s inequality and not much else. The proof of GNS inequality is much more involved although it does give more information when $p<n$. So if you are just trying to say that you can take the $L^p$ norm of the gradient as an equivalent norm, using GNS is like killing a fly with a cannon.

If you had asked the difference between Poincare-Wirtinger and the GNS inequality, my answer would have been much longer. In calculus of variations we use Poincare-Wirtinger all the time. If you have an energy of the type $$\int_{\Omega}f(\nabla u)\, dx$$ on a bounded domain, and you want to minimize it (together with some constraints or boundary conditions), then from the energy you get a control on the derivative but not on the function. You then use some kind of Poincaré or Poincare-Wirtinger to obtain control of the function. For example you could have a boundary condition only on a portion of the boundary. Then you could use a type of Poincaré inequality but not GNS, because to extend your function to the entire space you need the full norm and not just the norm of the gradient.

On the other hand, if you have the same energy but $\Omega$ is unbounded in every direction, then you cannot use Poincaré’s inequality. In this case GNS is one of the few tools that can help getting some kind of control on the function using what you know about the derivative. However, you can only do that if $\Omega$ is the entire space or a half space or the super graph of a nice function and if $p<n$.

But I am digressing. I hope this helps a bit.

EDIT There is no moral equivalent of Poincaré inequality for $p\ge n$. When $p=n$, you have that $$\|u\|_q\le C(\|u\|_{p}+\|\nabla u\|_p)$$ for $n\le q<\infty$ while for $p>n$ you have $$\|u\|_\infty\le C(\|u\|_{p}+\|\nabla u\|_p)$$ (this is part of Morrey’s inequality). The problem is that both inequalities have the norm of $u$ on the right hand side, so you cannot use to control the norm of $u$ with the norm of the gradient.

You can combine these inequalities with either Poincaré or Poincaré Wirtinger inequalities to improve the exponent. You can put $p^*$ for $p<n$, and any $q<\infty$ for $p\ge n$. For Poincaré you need the domain bounded by two parallel hyperplanes and for Poincaré Wirtinger you need a bounded smooth domain.

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  • $\begingroup$ Great answer. Pedagogy encourages the use of Poincare because it makes elliptic problems so easy to work with, but more sophisticted tools are needed in real applications. $\endgroup$
    – whpowell96
    Commented May 30, 2023 at 22:01
  • $\begingroup$ Do you know anything about the optimality of the exponent in Poincare(-Wirtinger)? $\endgroup$
    – Bananach
    Commented May 31, 2023 at 8:31
  • $\begingroup$ Also, the "moral equivalent of" Sobolev for $p\geq n$ does exist, is called Morrey's inequality, and guarantees boundedness, which in the in the case of unbounded domain is all the integrability you could hope for. $\endgroup$
    – Bananach
    Commented May 31, 2023 at 8:41
  • $\begingroup$ It seems to me that the Poincare inequality on bounded domains is strictly weaker than (GN)S. Could you confirm whether the exponents in the (1) Poincare-Wirtinger inequality for oscillations around the mean on bounded domains (2) Poincare inequality for functions on domains bounded in only one direction, are optimal (for smooth domains even?)? For example can't at least the former be derived and improved from extension and Sobolev theorems? $\endgroup$
    – Bananach
    Commented May 31, 2023 at 8:55
  • $\begingroup$ I replied to the questions (see the edit in the post). $\endgroup$
    – Gio67
    Commented May 31, 2023 at 9:48

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