Characterizations of second-order differential operators

arXiv2306.02788MaRDI QIDQ6439226

Aleksandra Świątczak, Eszter Gselmann, Włodzimierz Fechner

Publication date: 5 June 2023

Abstract: If $kgeq2$ is a positive integer, $Omegasubsetmathbb{R}^N$ is a domain then by the well-known properties of the Laplacian and the gradient, we have [ Delta(fcdot g)=g Delta f+f Delta g+2langle

abla f,

abla g angle ] for all $f,ginmathscr{C}^k(Omega,mathbbR)$. Due to the results of K"onig--Milman [7], the converse is also true under some assumptions. Thus the main aim is this paper is to study the equation [ T(fcdot g)= fT(g)+T(f)g+2B(A(f), A(g)) qquad left(f, gin P ight), ] where $Q$ and $R$ are commutative rings and $P$ is a subring of $Q$, further $TcolonPoQ$ and $AcolonPoR$ are additive mappings, while $BcolonRimesRoQ$ is a symmetric and bi-additive mapping. Related identities will also be considered.

Mathematics Subject Classification ID

Equations involving nonlinear operators (general) (47J05) Functional equations for real functions (39B22) Linear operators on function spaces (general) (47B38) Equations involving linear operators, with operator unknowns (47A62)

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