# American Institute of Mathematical Sciences

2017, 24: 28-37. doi: 10.3934/era.2017.24.004

## Rigidity of the ${{L}^{p}}$-norm of the Poisson bracket on surfaces

 1 School of Mathematical Sciences, Faculty of Exact Sciences, Tel Aviv University 2 Department of Mathematics, Faculty of Natural Sciences, University of Haifa

Received  September 29, 2016 Published  May 2017

Fund Project: We wish to thank Lev Buhovsky and Leonid Polterovich for reading a preliminary version of the paper and making useful comments and for their interest. We thank the anonymous referee for reviewing the paper. KS is partially supported by the Israel Science Foundation grant number 178/13, and by the European Research Council Advanced grant number 338809. FZ is partially supported by grant number 1281 from the GIF, the German–Israeli Foundation for Scientific Research and Development, and by grant number 1825/14 from the Israel Science Foundation.

For a symplectic manifold $(M,ω)$, let $\{·,·\}$ be the corresponding Poisson bracket. In this note we prove that the functional $(F,G) \mapsto \|\{F,G\}\|_{L^p(M)}$ is lower-semicontinuous with respect to the $C^0$-norm on $C^∞_c(M)$ when $\dim M = 2$ and $p < ∞$, extending previous rigidity results for $p = ∞$ in arbitrary dimension.

Citation: Karina Samvelyan, Frol Zapolsky. Rigidity of the ${{L}^{p}}$-norm of the Poisson bracket on surfaces. Electronic Research Announcements, 2017, 24: 28-37. doi: 10.3934/era.2017.24.004
##### References:
 [1] L. Buhovsky, The $2/3$ -convergence rate for the Poisson bracket, Geom. Funct. Anal., 19 (2010), 1620-1649.  doi: 10.1007/s00039-010-0045-z.  Google Scholar [2] S. S. Cairns, A simple triangulation method for smooth manifolds, Bull. Amer. Math. Soc., 67 (1961), 389-390.  doi: 10.1090/S0002-9904-1961-10631-9.  Google Scholar [3] F. Cardin and C. Viterbo, Commuting Hamiltonians and Hamilton-Jacobi multi-time equations, Duke Math. J., 144 (2008), 235-284.  doi: 10.1215/00127094-2008-036.  Google Scholar [4] M. Entov and L. Polterovich, $C^0$-rigidity of Poisson brackets, in Symplectic Topology and Measure Preserving Dynamical Systems, Contemp. Math., 512, Amer. Math. Soc., Providence, RI, 2010, 25–32. doi: 10.1090/conm/512/10058.  Google Scholar [5] M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., 3 (2007), 1037-1055.  doi: 10.4310/PAMQ.2007.v3.n4.a9.  Google Scholar [6] H. Federer, Geometric Measure Theory Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.  Google Scholar [7] K. Samvelyan, Rigidity Versus Flexibility of the Poisson Bracket with Respect to the ${L}_p$ -Norm Master's thesis, Tel Aviv University, 2015. Google Scholar [8] F. Zapolsky, Quasi-states and the Poisson bracket on surfaces, J. Mod. Dyn., 1 (2007), 465-475.  doi: 10.3934/jmd.2007.1.465.  Google Scholar

show all references

##### References:
 [1] L. Buhovsky, The $2/3$ -convergence rate for the Poisson bracket, Geom. Funct. Anal., 19 (2010), 1620-1649.  doi: 10.1007/s00039-010-0045-z.  Google Scholar [2] S. S. Cairns, A simple triangulation method for smooth manifolds, Bull. Amer. Math. Soc., 67 (1961), 389-390.  doi: 10.1090/S0002-9904-1961-10631-9.  Google Scholar [3] F. Cardin and C. Viterbo, Commuting Hamiltonians and Hamilton-Jacobi multi-time equations, Duke Math. J., 144 (2008), 235-284.  doi: 10.1215/00127094-2008-036.  Google Scholar [4] M. Entov and L. Polterovich, $C^0$-rigidity of Poisson brackets, in Symplectic Topology and Measure Preserving Dynamical Systems, Contemp. Math., 512, Amer. Math. Soc., Providence, RI, 2010, 25–32. doi: 10.1090/conm/512/10058.  Google Scholar [5] M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., 3 (2007), 1037-1055.  doi: 10.4310/PAMQ.2007.v3.n4.a9.  Google Scholar [6] H. Federer, Geometric Measure Theory Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.  Google Scholar [7] K. Samvelyan, Rigidity Versus Flexibility of the Poisson Bracket with Respect to the ${L}_p$ -Norm Master's thesis, Tel Aviv University, 2015. Google Scholar [8] F. Zapolsky, Quasi-states and the Poisson bracket on surfaces, J. Mod. Dyn., 1 (2007), 465-475.  doi: 10.3934/jmd.2007.1.465.  Google Scholar
Illustrating $K_n\subseteq U$ and an element $\Phi (Q') = Q$ in its subdivision
Producing ${\tilde{F}}$ and ${\tilde{G}}$ (the dashed curves)
 [1] Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020353 [2] Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269 [3] Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249 [4] Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071 [5] Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292 [6] Aihua Fan, Jörg Schmeling, Weixiao Shen. $L^\infty$-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363 [7] Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447 [8] Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052 [9] Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352 [10] Yichen Zhang, Meiqiang Feng. A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 [11] Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $p$-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445 [12] Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $p$ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442 [13] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 [14] Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $L^2$-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298 [15] Lei Liu, Li Wu. Multiplicity of closed characteristics on $P$-symmetric compact convex hypersurfaces in $\mathbb{R}^{2n}$. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378 [16] Wenqiang Zhao, Yijin Zhang. High-order Wong-Zakai approximations for non-autonomous stochastic $p$-Laplacian equations on $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2021, 20 (1) : 243-280. doi: 10.3934/cpaa.2020265

2019 Impact Factor: 0.5