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.
References
Ph. Bénilan and H. Brezis, Nonlinear problems related to the Thomas-Fermi equation. J. Evol. Eqns. 3 (2003), pp 637-652. MR2058057 10.1007/s00028-003-0117-8 Ph. Bénilan and H. Brezis, Nonlinear problems related to the Thomas-Fermi equation. J. Evol. Eqns. 3 (2003), pp 637-652. MR2058057 10.1007/s00028-003-0117-8
Ph. Bénilan, J. A. Goldstein and G. R. Rieder$ \footnote{ *G.\ R.\ Rieder\ is now G.\ R.\ Goldstein.}$, A nonlinear elliptic system arising in electron density theory. Comm. PDE. 17 (1992), pp 2079-2092. MR1194750 10.1080/03605309208820914 Ph. Bénilan, J. A. Goldstein and G. R. Rieder$ \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. MR1194750 10.1080/03605309208820914
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. MR1126688 10.1016/S0076-5392(08)63373-1 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. MR1126688 10.1016/S0076-5392(08)63373-1
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. MR578739 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. MR578739
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. MR519638 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. MR519638
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. MR1329842 euclid.die/1368638167
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. MR1329842 euclid.die/1368638167
J. A. Goldstein and G. R. Rieder, Spin-polarized Thomas-Fermi Theory, J. Math. Phys. 29 (1988), pp 709-716. MR931476 10.1063/1.528011 J. A. Goldstein and G. R. Rieder, Spin-polarized Thomas-Fermi Theory, J. Math. Phys. 29 (1988), pp 709-716. MR931476 10.1063/1.528011
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. MR1021015 10.1007/BFb0086753 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. MR1021015 10.1007/BFb0086753
E. H. Lieb and B. Simon, The Thomas-Fermi Theory of atoms, molecules and solids, Adv. Math. 23 (1977), pp 22-116. MR428944 10.1016/0001-8708(77)90108-6 E. H. Lieb and B. Simon, The Thomas-Fermi Theory of atoms, molecules and solids, Adv. Math. 23 (1977), pp 22-116. MR428944 10.1016/0001-8708(77)90108-6
