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Some corrector results for composites with imperfect interface. (English) Zbl 1129.35008

Summary: We give some corrector results for a problem modelling the stationary heat diffusion in a conductor with two components, a connected one \(\Omega_1^\varepsilon\) and a disconnected one \(\Omega_2^\varepsilon\), consisting of \(\varepsilon\)-periodic connected components of size \(\varepsilon\). The flow of heat is proportional, by mean of a function of order \(\varepsilon^\gamma\), \(\gamma>-1\), to the jump of the temperature field, due to a contact resistance on the interface. We prove a corrector result for the temperature in the component \(\Omega_1^\varepsilon\) . Moreover, for \(-1< \gamma\leq 1\) we prove the strong convergence to zero of the gradient of the temperature in the component \(\Omega_2^\varepsilon\). Due to different a priori estimates, the case \(\gamma>1\) needs to be treated separately.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
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