    2011, 18: 54-60. doi: 10.3934/era.2011.18.54

## Derivative and entropy: the only derivations from $C^1(RR)$ to $C(RR)$

 1 Mathematisches Seminar, Universität Kiel, 24098 Kiel, Germany 2 School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel

Received  February 2011 Revised  April 2011 Published  July 2011

Let $T:C^1(RR)\to C(RR)$ be an operator satisfying the derivation equation

$T(f\cdot g)=(Tf)\cdot g + f \cdot (Tg),$

where $f,g\in C^1(RR)$, and some weak additional assumption. Then $T$ must be of the form

$(Tf)(x) = c(x) \, f'(x) + d(x) \, f(x) \, \ln |f(x)|$

for $f \in C^1(RR), x \in RR$, where $c, d \in C(RR)$ are suitable continuous functions, with the convention $0 \ln 0 = 0$. If the domain of $T$ is assumed to be $C(RR)$, then $c=0$ and $T$ is essentially given by the entropy function $f \ln |f|$. We can also determine the solutions of the generalized derivation equation

$T(f\cdot g)=(Tf)\cdot (A_1g) + (A_2f) \cdot (Tg),$

where $f,g\in C^1(RR)$, for operators $T:C^1(RR)\to C(RR)$ and $A_1, A_2:C(RR)\to C(RR)$ fulfilling some weak additional properties.

Citation: Hermann Köenig, Vitali Milman. Derivative and entropy: the only derivations from $C^1(RR)$ to $C(RR)$. Electronic Research Announcements, 2011, 18: 54-60. doi: 10.3934/era.2011.18.54
##### References:
  J. Aczél, "Lectures on Functional Equations and their Applications,", Mathematics in Science and Engineering, 19 (1966). Google Scholar  S. Artstein-Avidan, H. König and V. Milman, The chain rule as a functional equation,, Journ. Funct. Anal., 259 (2010), 2999.  doi: 10.1016/j.jfa.2010.07.002.  Google Scholar  H. König and V. Milman, Characterizing the derivative and the entropy function by the Leibniz rule,, preprint., ().   Google Scholar

show all references

##### References:
  J. Aczél, "Lectures on Functional Equations and their Applications,", Mathematics in Science and Engineering, 19 (1966). Google Scholar  S. Artstein-Avidan, H. König and V. Milman, The chain rule as a functional equation,, Journ. Funct. Anal., 259 (2010), 2999.  doi: 10.1016/j.jfa.2010.07.002.  Google Scholar  H. König and V. Milman, Characterizing the derivative and the entropy function by the Leibniz rule,, preprint., ().   Google Scholar
  Dariusz Idczak. A global implicit function theorem and its applications to functional equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2549-2556. doi: 10.3934/dcdsb.2014.19.2549  Christian Wolf. A shift map with a discontinuous entropy function. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 319-329. doi: 10.3934/dcds.2020012  Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025  Jean Ginibre, Giorgio Velo. Modified wave operators without loss of regularity for some long range Hartree equations. II. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1357-1376. doi: 10.3934/cpaa.2015.14.1357  Mohamed Assellaou, Olivier Bokanowski, Hasnaa Zidani. Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3933-3964. doi: 10.3934/dcds.2015.35.3933  Shitao Liu, Roberto Triggiani. Determining damping and potential coefficients of an inverse problem for a system of two coupled hyperbolic equations. Part I: Global uniqueness. Conference Publications, 2011, 2011 (Special) : 1001-1014. doi: 10.3934/proc.2011.2011.1001  Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307  Tomás Caraballo, Gábor Kiss. Attractivity for neutral functional differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1793-1804. doi: 10.3934/dcdsb.2013.18.1793  Boris Paneah. On the over determinedness of some functional equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 497-505. doi: 10.3934/dcds.2004.10.497  Vitalii G. Kurbatov, Valentina I. Kuznetsova. On stability of functional differential equations with rapidly oscillating coefficients. Communications on Pure & Applied Analysis, 2018, 17 (1) : 267-283. doi: 10.3934/cpaa.2018016  Yongqiang Suo, Chenggui Yuan. Large deviations for neutral stochastic functional differential equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2369-2384. doi: 10.3934/cpaa.2020103  Olesya V. Solonukha. On nonlinear and quasiliniear elliptic functional differential equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 869-893. doi: 10.3934/dcdss.2016033  Pierluigi Benevieri, Alessandro Calamai, Massimo Furi, Maria Patrizia Pera. On general properties of retarded functional differential equations on manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 27-46. doi: 10.3934/dcds.2013.33.27  John A. D. Appleby, Denis D. Patterson. Subexponential growth rates in functional differential equations. Conference Publications, 2015, 2015 (special) : 56-65. doi: 10.3934/proc.2015.0056  Hermann Brunner, Chunhua Ou. On the asymptotic stability of Volterra functional equations with vanishing delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 397-406. doi: 10.3934/cpaa.2015.14.397  Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167  Prof. Dr.rer.nat Widodo. Topological entropy of shift function on the sequences space induced by expanding piecewise linear transformations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 191-208. doi: 10.3934/dcds.2002.8.191  Zhiyou Wu, Fusheng Bai, Guoquan Li, Yongjian Yang. A new auxiliary function method for systems of nonlinear equations. Journal of Industrial & Management Optimization, 2015, 11 (2) : 345-364. doi: 10.3934/jimo.2015.11.345  Ansgar Jüngel, Josipa-Pina Milišić. Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution. Kinetic & Related Models, 2011, 4 (3) : 785-807. doi: 10.3934/krm.2011.4.785  Patrick Ballard, Bernadette Miara. Formal asymptotic analysis of elastic beams and thin-walled beams: A derivation of the Vlassov equations and their generalization to the anisotropic heterogeneous case. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1547-1588. doi: 10.3934/dcdss.2019107

2019 Impact Factor: 0.5