Orlicz equi-integrability for scaled gradients
Piotr Antoni Kozarzewski, Elvira Zappale · Journal of Elliptic and Parabolic Equations · 2017
In the realm of 3D–2D dimensional reduction problems, we prove that, up to an extraction, it is possible to decompose a sequence $$(u_n)$$ , whose scaled gradients $$\left( abla _\alpha u_n, \frac{1}{\varepsilon _n} abla _3 u_n \right) $$ are bounded in $$L^\Phi (\omega \times (-1,1),\mathbb R^{3 \times 3})$$ for a suitable Orlicz function $$\Phi $$ , as $$u_n = v_n+z_n$$ , such that $$v_n$$ describes the oscillations, $$\left( \Phi \left( \left| abla _\alpha v_n,\frac{1}{\varepsilon _n} abla _3 v_n\right| \right) \right) $$ , is equi-integrable and the remainder $$z_n$$ , accounting for concentration effects, converges to zero in measure. In particular, we extend to the Orlicz–Sobolev setting the results contained in Bocea and Fonseca, (ESAIM: COCV 7:443–470, 2002) and Braides and Zeppieri (Calc Var 29:231–238, 2007).