RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Algebra i Analiz: Year: Volume: Issue: Page: Find

 Algebra i Analiz, 2012, Volume 24, Issue 4, Pages 137–155 (Mi aa1295)

Research Papers

An operator equation characterizing the Laplacian

H. Königa, V. Milmanb

a Mathematisches Seminar, Universität Kiel, Kiel, Germany
b School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

Abstract: The Laplace operator on $\mathbb R^n$ satisfies the equation
$$\Delta(fg)(x)=(\Delta f)(x)g(x)+f(x)(\Delta g)(x)+2\langle f'(x),g'(x)\rangle$$
for all $f,g\in C^2(\mathbb R^n,\mathbb R)$ and $x\in\mathbb R^n$. In the paper, an operator equation generalizing this product formula is considered. Suppose $T\colon C^2(\mathbb R^n,\mathbb R)\to C(\mathbb R^n,\mathbb R)$ and $A\colon C^2(\mathbb R^n,\mathbb R)\to C(\mathbb R^n,\mathbb R^n)$ are operators satisfying the equation
$$T(fg)(x)=(Tf)(x)g(x)+f(x)(Tg)(x)+\langle(Af)(x),(Ag)(x)\rangle \tag{1}$$
for all $f,g\in C^2(\mathbb R^n,\mathbb R)$ and $x\in\mathbb R^n$. Assume, in addition, that $T$ is $O(n)$-invariant and annihilates the affine functions, and that $A$ is nondegenerate. Then $T$ is a multiple of the Laplacian on $\mathbb R^n$, and $A$ a multiple of the derivative,
$$(Tf)(x)=\frac{d(\|x\|)^2}2(\Delta f)(x),\quad (Af)(x)=d(\|x\|)f'(x),$$
where $d\in C(\mathbb R_+,\mathbb R)$ is a continuous function. The solutions are also described if $T$ is not $O(n)$-invariant or does not annihilate the affine functions. For this, all operators $(T,A)$ satisfying (1) for scalar operators $A\colon C^2(\mathbb R^n,\mathbb R)\to C(\mathbb R^n,\mathbb R)$ are determined. The map $A$, both in the vector and the scalar case, is closely related to $T$ and there are precisely three different types of solution operators $(T,A)$.
No continuity or linearity requirement is imposed on $T$ or $A$.

Keywords: Laplace operator, second order Leibniz rule, operator functional equations.

Full text: PDF file (243 kB)
References: PDF file   HTML file

English version:
St. Petersburg Mathematical Journal, 2013, 24:4, 631–644

Bibliographic databases:

Language:

Citation: H. König, V. Milman, “An operator equation characterizing the Laplacian”, Algebra i Analiz, 24:4 (2012), 137–155; St. Petersburg Math. J., 24:4 (2013), 631–644

Citation in format AMSBIB
\Bibitem{KonMil12} \by H.~K\"onig, V.~Milman \paper An operator equation characterizing the Laplacian \jour Algebra i Analiz \yr 2012 \vol 24 \issue 4 \pages 137--155 \mathnet{http://mi.mathnet.ru/aa1295} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3088010} \zmath{https://zbmath.org/?q=an:1273.47035} \elib{http://elibrary.ru/item.asp?id=20730169} \transl \jour St. Petersburg Math. J. \yr 2013 \vol 24 \issue 4 \pages 631--644 \crossref{https://doi.org/10.1090/S1061-0022-2013-01257-X} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000331548500006} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84878627261} 

• http://mi.mathnet.ru/eng/aa1295
• http://mi.mathnet.ru/eng/aa/v24/i4/p137

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Proc. Steklov Inst. Math., 280 (2013), 191–207
•  Number of views: This page: 255 Full text: 79 References: 20 First page: 13