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We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction Ωε that is the union of a domain Ω0 and a large number 2N of thin rods with thickness of order ε = O(N −1 ). Depending on their lengths, the thin rods are divided into two levels. In addition, the rods from each level are ε-periodically alternated. Inhomogeneous Neumann boundary conditions are given on the vertical sides of the thin rods of the first level, and homogeneous Dirichlet boundary conditions are given on the vertical sides of the rods of the second level. We investigate the asymptotic behavior of a solution of this problem as ε → 0 and prove a convergence theorem and the convergence of the energy integral. Statement of the Problem and Main ResultAsymptotic methods for the investigation of boundary-value problems in domains with complex dependence on a small parameter (perforated domains, partially perforated domains, and skeleton structures) were considered in numerous papers (see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and the bibliography therein).Boundary-value problems in thick junctions (the number of components of these junctions increases infinitely if the perturbation parameter ε tends to zero) have specific difficulties and are of special interest. As shown in [15], boundary-value problems in thick junctions lose coercivity as ε → 0, which substantially complicates asymptotic studies. Note that the first investigations in this direction were carried out in [16][17][18], where the asymptotic behavior of the Green function of the Neumann problem for the Helmholtz equation in an unbounded thick junction was studied. In [19][20][21][22][23][24][25][26][27][28][29][30], thick junctions were classified, asymptotic methods for the investigation of main boundaryvalue problems of mathematical physics in thick singularly degenerating junctions of various types were developed, the first terms of asymptotic expansions were constructed, asymptotic estimates were proved, and the influence of boundary conditions given at the boundaries of thick junctions and the geometric configuration of thick junctions on the asymptotic behavior of solutions were investigated.In the present paper, we consider a new type of thick junctions, namely thick multilevel junctions. A thick multilevel junction is the union of a domain Ω 0 , which is called the junction body, and a large number N = O(ε −δ ) of thin domains with thickness of order O(ε). The thin domains are divided into finitely many levels, depending on their lengths. In addition, the thin domains from each level are ε-periodically alternated along a certain set at the boundary of the junction body. This set is called the junction zone. The junction body and the junction zone may also depend on the small parameter ε.For the first time, the problem in a plane two-level junction was considered in [31], where the asymptotic behavior of eigenvalues and eigenfunctions of a spectral problem was investigated in the case where the number of attached thin rods in ...

We consider a mixed boundary-value problem for a Poisson equation in a plane two-level junction Ωε that is the union of a domain Ω0 and a large number 3N of thin rods with thickness of order ε = O(N −1 ). The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are ε-periodically alternated. The homogeneous Dirichlet conditions and inhomogeneous Neumann conditions are given on the sides of the thin rods from the first level and the second level, respectively. Using the method of matched asymptotic expansions and special junction-layer solutions, we construct an asymptotic approximation for the solution and prove the corresponding estimates in the Sobolev space H 1 (Ωε) as ε → 0 (N → +∞).