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Asymptotics of solutions of the Neumann problem in a domain with closely posed components of the boundary. (English) Zbl 1175.35040
The authors studied he Neumann problem for the Poisson equation
\[ \begin{alignedat}{2} -\Delta_x u_\varepsilon(x)&= f(\varepsilon,x), &\quad x&\in \Omega_\varepsilon\subset{\mathbb R^n},\\ \partial_\nu u_\varepsilon (x)&= g(\varepsilon, y, z-\varepsilon), &\quad x&\in \Gamma_\varepsilon,\\ \partial_\nu u_\varepsilon (x)&=0, &\quad x&\in\Gamma. \end{alignedat} \]
The boundary components are posed at a small distance \(\varepsilon>0\) so that in the limit, as \(\varepsilon\to 0^+\) the components touch each other at the point \({\mathcal O}\) with the tangency exponent \(2m\geq 2.\) Asymptotics of the solution \(u_\varepsilon\) and the Dirichlet integral \(\|\nabla_x u_\varepsilon; L^2(\Omega_\varepsilon) \|^2\) are evaluated and it is shown that main asymptotic term of \(u_\varepsilon\) and the existence of the finite limit of the integral depend on the relation between the spatial dimension \(n\) and the exponent \(2m.\) Some generalizations are discussed and certain unsolved problems are formulated.

35J25 Boundary value problems for second-order elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
35B25 Singular perturbations in context of PDEs
35J61 Semilinear elliptic equations