We introduce function spaces for the treatment of non-linear parabolic partial differential equations with variable log–Hölder continuous exponents that only incorporate information symmetric part a gradient. As an analogue Korn's inequality these functions is not available, construction appropriate smoothing method proves to be difficult. Using point-wise Poincaré near boundary bounded Lipschi...

Abstract We provide necessary conditions on Euclidean domains for inclusions $$W^{1,p(\cdot )}(\Omega ) \hookrightarrow L^{q(\cdot $$ W 1 , p ( · ) Ω <mml:mo...