We present a new method for studying the numerical radius of bounded operators on Hilbert $C^{*}$-modules. Our method enables us to obtain some new results and generalize some known theorems for bounded operators on Hilbert spaces to bounded adjointable operators on Hilbert $C^{*}$-module spaces.

Given an open subset $\mathrm{\Omega }$ of ${ℝ}^n$ and a Borel function $f:\mathrm{\Omega }\times ℝ\times {ℝ}^n\to [0,+\mathrm{\infty }[$, conditions on $f$ are given which assure the lower semicontinuity of the functional ${\int }_{\mathrm{\Omega }}f(x,u,Du)dx$ with respect to different topologies.