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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.

##### MSC:
 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