On how Poincaré inequalities imply weighted ones
Abstract
Keywords
Weighted Poincaré inequality Poincaré constant Sobolev inequality Superlevel setsMathematics Subject Classification
26D10 35A23 46E351 Introduction
- (i)
Dirichlet boundary conditions, i.e., \(\mathcal {C}\) consisting of functions vanishing on the boundary \(\partial \varOmega \) or
- (ii)vanishing mean, i.e.,$$\begin{aligned} \int _\varOmega f w = 0\quad \text {for all } f\in \mathcal {C}. \end{aligned}$$
Due to its importance in the theory of partial differential equations a vast amount of work and effort has been put in the study of Poincaré type inequalities. The following overview is by far not a complete collection of the available research on this topic.
For star shaped domains—again under the assumptions that w is constant and \(p=q\)—an explicit bound on the Poincaré constant is given in [18].
In the onedimensional situation [9] provides a bound on the Poincaré constant for arbitrary p, q and w, which in certain cases is sharp in some sense. Computing the bound amounts to determining the supremum of a possibly complicated expression and therefore—depending on w—may not be feasible. In [10] the same authors give explicit bounds on the Poincaré constant when \(q\le p\), \(\varOmega \) is bounded and convex and w is a positive power of a concave function.
Gaussian or more generally so called log-concave weights on the full space \(\varOmega =\mathbb {R}^d\) are considered in [5, 7] for the case \(p=q=2\).
For arbitrary weight w and domain \(\varOmega \) the Poincaré constant can be estimated from above in terms of the so called Cheeger constant—a well studied concept in Riemannian geometry—see [8] and [19, Appendix]. Again computation of the Cheeger constant may not be feasible depending on w.
Explicit bounds for the constant in weighted Poincaré inequalities are only known in very specific scenarios. The main result of this paper shows that the Poincaré constant w.r.t. a weight w can be controlled in terms of the Poincaré constants w.r.t. the constant weight 1 on the superlevel sets of w. Combined with knowledge of Poincaré constants for the unweighted case this yields a powerful tool for estimating the Poincaré constant on a general domain equipped with a weight whose superlevel sets only have to be connected. The key ideas are inspired by the work of Dyda and Kassmann in [13], where radially symmetric weights were considered. As we will see a similar approach can be taken in a much more general setting.
2 Preliminaries
In the following \(\varOmega \) will be called a domain in \(\mathbb {R}^d\), if \(\varOmega \) is a nonempty, open and connected subset of \(\mathbb {R}^d\). We call a function \(w:\varOmega \rightarrow \mathbb {R}\) a weight on \(\varOmega \subset \mathbb {R}^d\) if w is measurable and nonnegative.
Remark 1
The central topic of this work are so called Poincaré inequalities:
Definition 1
3 Main theorem
Lemma 1
The following lemma provides a formula on how to write a weighted integral as a double integral.
Lemma 2
Proof
Next we show a simple relation between Lipschitz function spaces as defined in (5).
Lemma 3
Proof
We are set to state the main result:
Theorem 1
- 1.
If none of the superlevel sets \(\varOmega _t^w\) possesses a bottleneck w.r.t. to the constant weight 1 then neither will \(\varOmega \) w.r.t. w. In that case \(C_q^p(\varOmega ,w,w)\) is small.
- 2.
However if \(C_q^p(\varOmega ,w,w)\) is large then some of the superlevel sets \(\varOmega _t^w\) will have large Poincaré constants w.r.t. the constant weight 1 and therefore a bottleneck, see Fig. 1.
The left figure shows a typical example of a bottlenecked weight on a plane domain. The two main parts of its superlevel sets—as sketched on the right—are connected by a thin bridge and therefore we observe the presence of a bottleneck
Proof of Theorem 1
Finally we consider a concrete example. Our aim here is not to find the smallest possible bound on the Poincaré constant but rather to indicate how the result of Theorem 1 can be applied in practice. We choose a weight that is neither a power of a concave function nor log-concave in order to emphasize that our method yields results in situations where existing techniques are not applicable.
Example 1
Footnotes
- 1.
We say a bottleneck is present if the domain can be partitionated into two subdomains of roughly equal measure w.r.t. w such that the weight is small on the seperating boundary of the two subdamains.
Notes
Acknowledgements
Open access funding provided by University of Vienna. I would like to thank Philipp Grohs for inspiring discussions.
Funding
Funding was provided by Universität Wien.
References
- 1.Acosta, G., Durán, R.G.: An optimal Poincaré inequality in \(L^1\) for convex domains. Proc. Am. Math. Soc. 132(1), 195–202 (2004)CrossRefzbMATHGoogle Scholar
- 2.Alaifari, R., Daubechies, I., Grohs, P., Yin, R.: Stable phase retrieval in infinite dimensions. Found. Comput. Math. (2018). https://doi.org/10.1007/s10208-018-9399-7 Google Scholar
- 3.Amick, C.J.: Some remarks on Rellich’s theorem and the Poincaré inequality. J. Lond. Math. Soc. 18(3), 319–328 (1978)MathSciNetzbMATHGoogle Scholar
- 4.Bebendorf, M.: A note on the Poincaré inequality for convex domains. Z. Anal. Anwend. 22(4), 751–756 (2003)CrossRefzbMATHGoogle Scholar
- 5.Bobkov, S.G.: Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27(4), 1903–1921 (1999)MathSciNetzbMATHGoogle Scholar
- 6.Bojarski, B.: Remarks on Sobolev imbedding inequalities. In: Laine, I., Rickman, S., Sorvali, T. (eds.) Complex Analysis Joensuu 1987, pp. 52–68. Springer, Berlin (1988)CrossRefGoogle Scholar
- 7.Brandolini, B., Chiacchio, F., Henrot, A., Trombetti, C.: An optimal Poincaré–Wirtinger inequality in Gauss space. Math. Res. Lett. 20(3), 449–457 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
- 8.Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Gunning, R.C. (ed.) Problems Analysis, p. 195. Princeton University Press, Princeton (1970)Google Scholar
- 9.Chua, S.K., Wheeden, R.L.: Sharp conditions for weighted 1-dimensional Poincaré inequalities. Indiana Univ. Math. J. 49(1), 143–175 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
- 10.Chua, S.K., Wheeden, R.L.: Estimates of best constants for weighted Poincaré inequalities on convex domains. Proc. Lond. Math. Soc. 93(1), 197 (2006)CrossRefzbMATHGoogle Scholar
- 11.Deny, J., Lions, J.L.: Les espaces du type de beppo levi. Annales de l’institut Fourier 5, 305–370 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
- 12.Drelichman, I., Durán, R.G.: Improved Poincaré inequalities with weights. J. Math. Anal. Appl. 347(1), 286–293 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
- 13.Dyda, B., Kassmann, M.: On weighted Poincaré inequalities. Ann. Acad. Sci. Fenn. Math. 38, 721–726 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
- 14.Edmunds, D.E., Opic, B.: Weighted Poincaré and Friedrichs inequalities. J. Lond. Math. Soc. 47(2), 79–96 (1992)zbMATHGoogle Scholar
- 15.Ern, A., Guermond, J.: Theory and Practice of Finite Elements. Applied Mathematical Sciences. Springer, New York (2004)CrossRefzbMATHGoogle Scholar
- 16.Esposito, L., Nitsch, C., Trombetti, C.: Best constants in Poincaré inequalities for convex domains. J. Convex Anal. 20(1), 253–264 (2011)zbMATHGoogle Scholar
- 17.Fabes, E.B., Kenig, C.E., Serapioni, R.P.: The local regularity of solutions of degenerate elliptic equations. Commun. Partial Differ. Equ. 7(1), 77–116 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
- 18.Farwig, R., Rosteck, V.: Note on Friedrichs’ inequality in n-star-shaped domains. J. Math. Anal. Appl. 435, 1514–1524 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
- 19.Grohs, P., Rathmair, M.: Stable Gabor phase retrieval and spectral clustering. Commun. Pure Appl. Math. (To appear) Google Scholar
- 20.Martio, O.: John domains, bilipschitz balls and Poincaré inequalitiy. Rev. Rom. Math. Pures Appl. 33, 107–112 (1988)zbMATHGoogle Scholar
- 21.Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
- 22.Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Ration. Mech Anal. 5, 286–292 (1960)CrossRefzbMATHGoogle Scholar
Copyright information
OpenAccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.