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(⋅) − 2υ acting from the Lebesgue–Orlicz space Lp(⋅)(Ω, ℝm) to the dual Lp ′ (⋅)(Ω, ℝm), where p′(⋅) is the Holder-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 ‖υn − 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.