On Consequences of the Strong Convergence in Lebesgue–Orlich Spaces

Abstract

We study the continuity in the sense of the strong topology for the flux function υ → l(υ) = |υ|^{p(⋅) − 2}υ acting from the Lebesgue–Orlicz space L^p(⋅)(Ω, ℝ^m) to the dual L^{p ′ (⋅)}(Ω, ℝ^m), where p^′(⋅) is the Hölder-conjugate exponent, under the assumption that p(·) is an L^∞(Ω)-function such that 1 < α ≤ p(·) ≤ β < ∞. We obtain estimates for the convergence \( {\left\Vert l\left({\upsilon}_n\right)-l\left(\upsilon \right)\right\Vert}_{p^{\prime}\left(\cdot \right)}\to 0 \) with respect to the smallness order as ‖υ_n − υ‖_p(⋅) → 0. The strong continuity of the energy functional \( \underset{\varOmega }{\int }{\left|\upsilon \right|}^{p\left(\cdot \right)} dx \) is a consequence of the strong continuity of the flux function.