Maximal monotonicity of operators with sufficiently large domain and application to the Hartree problem

Abstract

Suppose that L is a linear, closed operator and A is a hemi-continuous, (cyclically) monotone operator with D(A) ⊃ D(L). Both operators are defined in a Hilbert space. Then, it can be shown that A+L*L is maximal (cyclically) monotone, though A is not necessarily maximal. However, if A has the property (Au-Av, u-v)≥ge2‖L(u-v)‖2 then even A is maximal. Using this abstract result and the theory of sesquilinear forms, it requires only some technical calculation to show that the generalized Hartree operator $$F(u) = - a^2 \Delta u + (q(x) + b)u(x) + u(x)\int\limits_{\mathbb{R}^3 } {\frac{{u^2 }}{{|x - y|}}d^3 y} $$ is maximal (cyclical) monotone in the space L2 (ℝ3).