Existence and Homogenization for a Singular Problem Through Rough Surfaces
Patrizia Donato, Daniela Giachetti · SIAM Journal on Mathematical Analysis · 2016
The paper deals with existence and homogenization for elliptic problems with lower order terms singular in the u-variable (u is the solution) in a cylinder $Q$ in $\mathbb{R}^N$, so that the lower order term becomes infinite on the set $\{u=0\}$. A rapidly oscillating interface inside $Q$ separates the cylinder in two composite connected components. The interface has a periodic microstructure and it is situated in a small neighborhood of a hyperplane which separates the two components of $Q$. At the interface we suppose the following transmission conditions: (i) the flux is continuous, (ii) the jump of a solution at the interface is proportional to the flux through the interface. This is a steady state model for the heat conduction in two heterogeneous electrically conducting materials with an imperfect contact between them. On the exterior boundary Dirichlet boundary conditions are prescribed. We also derive a corrector result for every value of the two parameters $\gamma$ and $\kappa$ which are related, respectively, to the microstructure period and to the amplitude of the interface oscillations.