Unique Solvability of a System of Nonlinear Elliptic PDEs Arising in Solid State Physics

This paper presents a uniqueness theorem for a system of nonlinear elliptic PDEs. This system is related to a quantum chemistry model, namely the Thomas–Fermi–von‐Weizsäcker model. The present result is concerned with the solution of a system modeling a semicrystal, i.e., a periodic structure filling a half‐space. This kind of uniqueness theorem has been studied in a previous work [I. Catto, C. Le Bris, and P.‐L. Lions, Mathematical Theory of Thermodynamic Limits: Thomas–Fermi Type Models, Clarendon Press, Oxford University Press, New York, 1998] for the case of a whole crystal. We give here a simpler version of their proof, and then an adaptation to the case of a semicrystal. The methods we use are the maximum principle, supersolution methods, and elliptic estimates.