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On nonhomogeneous reinforcements of varying shape and different exponents. (English. Abridged French version) Zbl 0894.49007

The paper deals with the so-called reinforcement problem for a domain \(\Omega\) surrounded by a thin layer \(\Gamma\) whose thickness goes to zero together with the ellipticity coefficient of the surrounding material. In other words, the authors consider the \(\Gamma\)-limit of the sequence of functionals \[ J_\varepsilon(u)= F(u\mathop{\vbox{=.75em\vrule height.75em width1pt\hskip-1pt\vrule height1pt width.75em}}\Omega)- \int_{\Omega_\varepsilon} f_\varepsilon udx+ \int_{\Sigma_\varepsilon} {a_\varepsilon\circ\sigma\over h_\varepsilon\circ\sigma} G(h_\varepsilon\circ \sigma| Du|)dx, \] where \(\Omega_\varepsilon= \overline\Omega\cup \Sigma_\varepsilon\) and \[ \Sigma_\varepsilon= \{s+ t\nu(s): s\in\Gamma\subset\partial\Omega,\;0<t< h_\varepsilon(s)\}. \] Reinforcement problems have been considered by several authors; we recall here the papers by E. Sanchez-Palencia [J. Math. Pures Appl., IX. Sér. 53, 251-269 (1974; Zbl 0273.35007)], H. Brezis, L. A. Caffarelli and A. Friedman [Ann. Mat. Pura Appl, IV. Ser. 123, 219-246 (1980; Zbl 0434.35079)], and E. Acerbi and G. Buttazzo [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 273-284 (1986; Zbl 0607.73018)].
In the paper under review, the main difference consists in the fact that the growth exponents of the energy, \(p\) on \(\Omega\) and \(q\) on \(\Sigma_\varepsilon\), can be different, and the coefficients \(a_\varepsilon\) and \(h_\varepsilon\) are more general (in the paper by Acerbi and Buttazzo there is \(a_\varepsilon\equiv 1\) and \(h_\varepsilon=\varepsilon h(x)\)). The consequence is that in the \(\Gamma\)-limit we have \[ J(u)= F(u)- \int_\Omega fu dx+ \int_{\Gamma_2} aG(| u|)ds, \] where the boundary integral can be extended only to a portion \(\Gamma_2\) of \(\Gamma\).
Reviewer: G.Buttazzo (Pisa)

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
74K20 Plates
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