# American Institute of Mathematical Sciences

• Previous Article
Strong instability of standing waves for nonlinear Schrödinger equations with a partial confinement
• CPAA Home
• This Issue
• Next Article
Steady-state mode interactions of radially symmetric modes for the Lugiato-Lefever equation on a disk
July  2018, 17(4): 1651-1669. doi: 10.3934/cpaa.2018079

## Logarithmic Sobolev and Shannon's inequalities and an application to the uncertainty principle

 1 Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan 2 Tamachi Branch, Risona Bank Co., Ltd., Tokyo 108-014, Japan

Received  January 2017 Revised  December 2017 Published  April 2018

The uncertainty principle of Heisenberg type can be generalized via the Boltzmann entropy functional. After reviewing the $L^p$ generalization of the logarithmic Sobolev inequality by Del Pino-Dolbeault [6], we introduce a generalized version of Shannon's inequality for the Boltzmann entropy functional which may regarded as a counter part of the logarithmic Sobolev inequality. Obtaining best possible constants of both inequalities, we connect both the inequalities to show a generalization of uncertainty principle of the Heisenberg type.

Citation: Takayoshi Ogawa, Kento Seraku. Logarithmic Sobolev and Shannon's inequalities and an application to the uncertainty principle. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1651-1669. doi: 10.3934/cpaa.2018079
##### References:

show all references

##### References:
Young Functions
 [1] Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296 [2] Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050 [3] Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048 [4] Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 [5] 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 [6] Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250 [7] Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049 [8] Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

2019 Impact Factor: 1.105
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7744070887565613, "perplexity": 5387.648103845988}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141193221.49/warc/CC-MAIN-20201127131802-20201127161802-00555.warc.gz"}