Highly oscillating thin obstacles (Q1946793)

The authors take a compact subset \(T\) of the unit ball \(B_{1}\) of \(\mathbb{R }^{n}\), \(n\geq 3\), and build the \(\varepsilon \)-periodic distribution of \(a_{\varepsilon }\)-copies of \(T\) defined through \(\mathcal{T} _{\varepsilon }=\bigcup _{k\in \mathbb{Z}^{n}}(a_{\varepsilon }T+k\varepsilon )\). Let \(\Omega \) be a domain of \(\mathbb{R}^{n}\) and \(\Gamma \) a hyperplane defined as \(\Gamma _{\nu }=\{x\in \mathbb{R}^{n}:x\cdot \nu =x^{0}\cdot \nu \}\) for given \(\nu \in S^{n-1}\) and \(x^{0}\in \mathbb{R}^{n}\). The authors consider the oscillating boundary \(\Gamma _{\varepsilon }=\Gamma _{\nu }\cap \mathcal{T}_{\varepsilon }\). They choose a function \(\psi \in L^{\infty }(\Omega )\cap H^{1}(\Omega )\) and introduce the obstacle \(\psi _{\varepsilon }=\psi \chi _{\Gamma _{\varepsilon }}\), where \(\chi _{\Gamma _{\varepsilon }}\) is the characteristic function of \(\Gamma _{\varepsilon }\) and the admissible set \(\mathcal{K}_{\Gamma _{\varepsilon }}=\{v\in H_{0}^{1}(\Omega )\mid v\geq \psi _{\varepsilon }\}\). The obstacle problem under consideration is written as: Find \(u_{\varepsilon }\in \mathcal{K} _{\Gamma _{\varepsilon }}\) such that \(\int_{\Omega }\nabla u_{\varepsilon }\cdot \nabla (v-u_{\varepsilon })dx\geq \int_{\Omega }(v-u_{\varepsilon })fdx\) for every \(v\in \mathcal{K}_{\Gamma _{\varepsilon }}\). It is known that there exists a unique solution \(u_{\varepsilon }\) of this problem which is the unique minimizer of the strictly convex and coercive functional \(J\) defined on \(\mathcal{K}_{\Gamma _{\varepsilon }}\) through \(J(v)=\int_{\Omega }(\frac{1}{2}\left| \nabla u_{\varepsilon }\right| ^{2}-fv)dx\). The main result of the paper proves that \(u_{\varepsilon }\) converges in the weak topology of \(H_{0}^{1}(\Omega )\) to the unique minimizer \(u\) of the functional \(J_{\nu }(v)=\int_{\Omega }(\frac{1}{2}\left| \nabla u_{\varepsilon }\right| ^{2}-fv)dx+\frac{1}{2}cap_{\nu }(T)\int_{\Gamma }((\psi -v)^{+})^{2}d\sigma \), \(v\geq 0\), where \(cap_{\nu }(T)=\int_{-\infty }^{+\infty }cap(T\cap \Gamma _{\nu }(s))ds\), the so-called averaged capacity of \(T\) with respect to \(\nu \). Here the authors assume that \(a_{\varepsilon }=\varepsilon ^{\frac{n}{n-1}}\), and the convergence result occurs for almost every \(\nu \in S^{n-1}\). For the proof, the authors first evaluate the cardinal of \(\Gamma _{\nu }\cap \mathcal{T}_{\varepsilon }\) using a uniform distribution result which is presented in an Appendix. They then introduce the solution \(w_{\varepsilon }^{k}\) of a local harmonic problem, from which they build a so-called corrector \(w_{\varepsilon }\). The main part of the proof consists to derive the properties of this corrector \(w_{\varepsilon }\). The paper ends with considerations on the same problem but with more general hypersurfaces.