Communications in Mathematical Analysis

A Convex Minorant Problem Arising in Electron Density Theory

Gisèle R. Goldstein, Jerome A. Goldstein, and Naima Naheed

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We find the largest convex minorant of the function

\begin{equation*} F\left( x,y\right) =ax^{2}+xy+by^{2} \end{equation*}

where $a,b$ are positive constants and $x\geq 0,\ y\geq 0$. We explain how the problem is closely connected with finding the ground state Thomas-Fermi electron density for a spin polarized quantum mechanical system with the Fermi-Amaldi correction.

Article information

Commun. Math. Anal., Volume 8, Number 2 (2010), 92-102.

First available in Project Euclid: 21 April 2010

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Zentralblatt MATH identifier

Primary: 52A41: Convex functions and convex programs [See also 26B25, 90C25] 26B25: Convexity, generalizations
Secondary: 81Q99: None of the above, but in this section 81V55: Molecular physics [See also 92E10] 92E10: Molecular structure (graph-theoretic methods, methods of differential topology, etc.)

Convex minorant Thomas-Fermi theory Fermi-Amaldi correction ground state electron density


Goldstein, Gisèle R.; Goldstein, Jerome A.; Naheed, Naima. A Convex Minorant Problem Arising in Electron Density Theory. Commun. Math. Anal. 8 (2010), no. 2, 92--102.

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  • Ph. Bénilan and H. Brezis, Nonlinear problems related to the Thomas-Fermi equation. J. Evol. Eqns. 3 (2003), pp 637-652.
  • Ph. Bénilan, J. A. Goldstein and G. R. Rieder$ \footnote{ *G.\ R.\ Rieder\ is now G.\ R.\ Goldstein.}$, A nonlinear elliptic system arising in electron density theory. Comm. PDE. 17 (1992), pp 2079-2092.
  • Ph. Bénilan, J. A. Goldstein and G. R. Rieder, The Fermi-Amaldi correction in spin polarized Thomas-Fermi Theory, in Differential Equations and Mathematical Physics (ed. by C. Bennewitz), Academic Press. (1991), pp 25-37.
  • H. Brezis, Some variational problems of Thomas-Fermi type , in Variational Inequalities and Complementary Problems: Theory and Applications (ed. by R.W. Cottle, F. Giannessi, and J. L. Lions), Wiley (1980), pp 53-73.
  • H. Brezis, Nonlinear problems related to the Thomas-Fermi equation, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, (ed. by G. M. de la Penha and L.A. Medeiros), North Holland, Amsterdam, (1978), pp 81-89.
  • E. Fermi, Un metodo statistico per la determinazione di alcune prioret$\grave{a}$ dell'atome$,\ $Rend. Acad. Naz. Lincei 6 (1927), pp 602-607.
  • E. Fermi and E. Amaldi, Le orbit $\infty $s degli elementi, Mem. Accad. d'Italia 6 (1934), pp 119-149.
  • G. R. Goldstein, J. A. Goldstein and W. Jia, Thomas-Fermi theory with magnetic fields and the Fermi-Amaldi correction, Diff. & Int. Eqns. 8 (1995), pp 1305-1316.
  • G. R. Goldstein, J. A. Goldstein and N. Naheed, in preparation.
  • J. A. Goldstein and G. R. Rieder, Spin-polarized Thomas-Fermi Theory, J. Math. Phys. 29 (1988), pp 709-716.
  • J. A. Goldstein and G. R. Rieder, Recent rigorous results in Thomas-Fermi theory, in Lecture Notes in Math No. 1394 (ed. by T. L. Gill and W. W. Zachary), Springer (1989), pp 68-82.
  • C. LeBris, On the spin-polarized Thomas-Fermi model with the Fermi-Amaldi correction, Nonlinear Anal., TMA 25 (1995), pp 669-679.
  • E. Lieb and B. Simon, Thomas-Fermi Theory revisited , Phys. Rev. Lett. 31 (1975), pp 681-683.
  • E. H. Lieb and B. Simon, The Thomas-Fermi Theory of atoms, molecules and solids, Adv. Math. 23 (1977), pp 22-116.
  • N. Naheed, Mathematical Contributions to Spin-polarized Thomas-Fermi Theory, Ph. D. Thesis, Univ. of Memphis, 2009.
  • G. R. Rieder, Mathematical contributions to Thomas-Fermi theory, Houston J. Math. 16 (1990), pp 407-430.
  • L. H. Thomas, The calculation of atomic fields, Proc. Camb. Phil. Soc. 23 (1927), pp 542-548.