On The Lojasiewicz-Simon Inequality Of Hartree Equation

Lojasiewicz inequality,proved in a celebrated work of Leon Simon[26],has had a pro-found influence in various fields of analysis and geometry in the past 30 years,especially in the field of partial differential evolution equations,hence it is called Lojasiewicz-Simon in-equality.But it hasn't been mentioned in Hartree theory.The Hartree equation of helium atom is derived from the variational principle introduced by Hartree as an approximate method to describe the coulomb Hamiltonian of the interaction between electron and stat-ic nucleus.Many problems in geometry and calculus of change are essentially functional problems in infinite dimensional spacesBased on the extensive application of Lojasiewicz-Simon inequality on the semilin-ear partial differential evolution equation and structural feateres of Hartree energy func-tional,We give sufficient condition that energy functional in the following form satisfies Lojasiewicz-Simon inequality:Where u ? V and defined on a Hilbert spaceIn this paper,the relation between the important inequality of Lojasiewicz-Simon gradient inequality and the Hartree equation is proposed for the first time.Our theory generates some conclusions from[36],[48]and others.We not only give a concrete exam-ple to show the sufficient condition of the Lojasiewicz-Simon inequality,but also calculate the specific form of the derivative of the energy functional of Hartree equation and prove the coerciveness of the bilinear form of Hartree equation.Finally,We give the difference estimation results of energy functional of Hartree equation.