Homogenization of variational inequality for the Laplace operator with nonlinear constraint on the flow in a domain perforated by arbitrary shaped sets. Critical case
In the paper under review, the authors study the homogenization of a variational inequality for the Laplace equation with nonlinear constraint in a perforated domain by small holes of critical size. More precisely, they consider the asymptotic behavior of the solutions \(u_\epsilon\) of a variational inequality for the Laplacian in a set \(\Omega_\epsilon \subseteq \mathbb{R}^n\) (\(n \geq 3\)), perforated by arbitrary shaped sets \(G_\epsilon^j\) of radius equal to \(C_0 \epsilon^{\frac{n}{n-2}}\) with \(C_0>0\). On the boundary of the sets \(G_\epsilon^j\) they impose constraints of the type \[ u_\epsilon \geq 0\, , \quad \partial_\nu u_\epsilon\geq -\epsilon^{\frac{n}{2-n}}\sigma(u_\epsilon)\, , \quad u_\epsilon \big(\partial_\nu u_\epsilon+\epsilon^{\frac{n}{2-n}}\sigma(u_\epsilon)\big)=0\, , \] whereas they impose \( u_\epsilon=0 \) on the remaining part of the boundary of \(\Omega_\epsilon\). Then they construct special test functions and establish the asymptotics of the solutions to the variational inequality.
scientific article; zbMATH DE number 6919794