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Equi-integrability results for 3D-2D dimension reduction problems. (English) Zbl 1044.49010

Authors’ abstract: “3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients \((\nabla_\alpha u_\varepsilon\mid{1\over\varepsilon}\nabla_3 u_\varepsilon)\) bounded in \(L^p(\Omega; \mathbb{R}^9)\), \(1< p<+\infty\). Here it is shown that, up to a subsequence, \(u_\varepsilon\) may be decomposed as \(w_\varepsilon+ z_\varepsilon\), where \(z_\varepsilon\) carries all the concentration effects, i.e., \(\{|(\nabla_\alpha w_\varepsilon|{1\over\varepsilon} \nabla_3 w_\varepsilon)|^p\}\) is equi-integrable, and \(w_\varepsilon\) captures the oscillatory behavior, i.e., \(z_\varepsilon\to 0\) in measure. In addition, if \(\{u_\varepsilon\}\) is a recovering sequence then \(z_\varepsilon= z_\varepsilon(x_\alpha)\) nearby \(\partial\Omega\).”

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
74B20 Nonlinear elasticity
74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics
74K15 Membranes
74K35 Thin films
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