Differential and Integral Equations, Volume 8, Number 6, July 1995, pp. 1305 – 1316. THOMAS-FERMI THEORY WITH MAGNETIC FIELDS AND THE FERMI-AMALDI CORRECTION Gis` ele Ruiz Goldstein and Jerome A. Goldstein Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803 Wenyao Jia Department of Neurobiology and Anatomy, University of Texas Health Center, Houston, TX 77030 Abstract. Of concern is a quantum mechanical system having N 1 (resp. N 2 ) spin up (resp. spin down) electrons, in the presence of a potential V and a magnetic field B. When the Fermi-Amaldi correction is incorporated into the Thomas-Fermi energy functional, convexity is lost and the computation of the ground state spin up and down electron densities becomes nontrivial. We discuss the existence of these densities and various approximation procedures for them, via variational calculus, differential equations, and numerical procedures. 1. Introduction. In earlier work of the first two authors with Philippe B´ enilan [4], [5], a study was initiated of Thomas-Fermi theory for spin polarized systems with the Fermi-Amaldi correction. (For earlier work on the Fermi-Amaldi correction in a spin unpolarized context, see [7], [13].) The spin polarized theory is indeed necessary for finding ground states when an external magnetic field is present; see [10]. The approach used in [5] is based on the Euler-Lagrange equation method. The Euler-Lagrange equation in this case reduced to a coupled system of semilinear equations on R 3 with signed measure inhomogeneities. But solving these equations did not necessarily give a ground state. The problem is that the energy functional is not convex, so that a solution of the Euler-Lagrange equations gives a stationary solution but not necessarily one of minimum energy. The results of [5] will give the ground state provided that one can establish uniqueness for the elliptic system. Once one knows both that a minimum exists and that there is only one stationary solution, then that solution minimizes the energy functional. This paper represents a step in the following research program. A quantum mechanical system in a magnetic field has Z protons and N 1 (resp. N 2 ) spin up (resp. spin down) electrons. The Thomas-Fermi energy functional for this problem (see [4] or [5]) requires fixing N 1 and N 2 (both greater than one). Part A. Fix N 1 ,N 2 and find the ground state density, i.e., the one corresponding to minimum energy when the total number of spin up (resp. spin down) electrons is N 1 (resp. N 2 ). Let E(N 1 ,N 2 ) be the corresponding (minimum) value of the energy. Part B. Specify N = N 1 +N 2 , the total number of electrons; and minimizeE(N 1 ,N 2 ) subject to the constraint N 1 + N 2 = N ( and, of course, find the corresponding spin up and down electron densities.) For both parts we require N  Z +1. Received for publication June 1994. This work was reported on at the International Conference on Differential Equations in August, 1993. AMS Subject Classifications: 49S05, 49J10, 49J52, 49K20, 35J45, 81V45, 81V55, 81V70, 81Q05. 1305