Atefeh Hasan-Zadeh - Assistant Professor of Applied Mathematics, Fouman Faculty of Engineering, College of Engineering, University of Tehran, Iran
Hooman Fatoorehchi - Assistant Professor of Chemical Engineering, Tehran Faculty of Engineering, School of Chemical Engineering, College of Engineering, University of Tehran, Iran

This paper deals with Thomas-Fermi equation which is formulated as an Euler-Lagrange equation associated with the Fermi energy functional. Drawing upon advanced ingredients of Sobolev spaces and weak solutions, an analytic methodology is presented for the quantum correction near the origin of Thomas-Fermi equation. By this approach the existence and uniqueness of the minimizer for the energy functional of the Thomas-Fermi equation has been proved. It has been demonstrated that by the definition of such a functional and the relevant Sobolev spaces, the Thomas-Fermi equation, particularly of a neutral atom, extends to the nonlinear Poisson equation. Accordingly, weak solutions for more general Euler-Lagrange equation with more singularities are proposed

Thomas-Fermi equation, Fermi energy, Euler-Lagrange equation, Sobolev space, Weak solution