×

An operator equation characterizing the Laplacian. (English. Russian original) Zbl 1273.47035

St. Petersbg. Math. J. 24, No. 4, 631-644 (2013); translation from Algebra Anal. 24, No. 4, 137-155 (2012).
Summary: 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 present paper, an operator equation generalizing this product formula is considered. Suppose that \(T:C^2(\mathbb R^n,\mathbb R)\to C(\mathbb R^n,\mathbb R)\) and \(A:C^2(\mathbb R^n,\mathbb R)\to C(\mathbb R^n,\mathbb R^n)\) are operators satisfying the equation \[ T(f g)(x)=(Tf)(x)g(x)+f(x)(Tg)(x)+\langle (Af)(x), (Ag)(x)\rangle\tag{1} \] 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 \(A:C^2(\mathbb R^n,\mathbb R) \to C(\mathbb R^n,\mathbb R)\) 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 \(A\).

MSC:

47A62 Equations involving linear operators, with operator unknowns
39B52 Functional equations for functions with more general domains and/or ranges
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. Aczél, Lectures on functional equations and their applications, Mathematics in Science and Engineering, Vol. 19, Academic Press, New York-London, 1966. Translated by Scripta Technica, Inc. Supplemented by the author. Edited by Hansjorg Oser. · Zbl 0139.09301
[2] Semyon Alesker, Shiri Artstein-Avidan, and Vitali Milman, A characterization of the Fourier transform and related topics, Linear and complex analysis, Amer. Math. Soc. Transl. Ser. 2, vol. 226, Amer. Math. Soc., Providence, RI, 2009, pp. 11 – 26. · Zbl 1184.42009 · doi:10.1090/trans2/226/02
[3] Shiri Artstein-Avidan, Hermann König, and Vitali Milman, The chain rule as a functional equation, J. Funct. Anal. 259 (2010), no. 11, 2999 – 3024. · Zbl 1203.39014 · doi:10.1016/j.jfa.2010.07.002
[4] Shiri Artstein-Avidan and Vitali Milman, The concept of duality in convex analysis, and the characterization of the Legendre transform, Ann. of Math. (2) 169 (2009), no. 2, 661 – 674. · Zbl 1173.26008 · doi:10.4007/annals.2009.169.661
[5] Shiri Artstein-Avidan and Vitali Milman, A characterization of the concept of duality, Electron. Res. Announc. Math. Sci. 14 (2007), 42 – 59. · Zbl 1140.52300
[6] Shiri Artstein-Avidan and Vitali Milman, The concept of duality for measure projections of convex bodies, J. Funct. Anal. 254 (2008), no. 10, 2648 – 2666. · Zbl 1145.26003 · doi:10.1016/j.jfa.2007.11.008
[7] Helmut Goldmann and Peter Šemrl, Multiplicative derivations on \?(\?), Monatsh. Math. 121 (1996), no. 3, 189 – 197. · Zbl 0843.46018 · doi:10.1007/BF01298949
[8] Hermann König and Vitali Milman, Characterizing the derivative and the entropy function by the Leibniz rule, J. Funct. Anal. 261 (2011), no. 5, 1325 – 1344. · Zbl 1231.39011 · doi:10.1016/j.jfa.2011.05.003
[9] -, An operator equation generalizing the Leibniz rule for the second derivative (submitted, 2011).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.