Asymptotic Expansions of Solutions to the Poisson Equation with Alternating Boundary Conditions on an Open Arc

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Abstract
We consider a mixed boundary value problem for the Poisson equation in a bounded planar domain. Most of the boundary is endowed with the Neumann condition, while the Dirichlet one is imposed on a periodic family of short segments of length [Formula: see text]. Consequently, the limit problem, corresponding to [Formula: see text], has mixed boundary conditions with two collision points (where the type of boundary condition changes). The asymptotic representation of the solution to the original, singularly perturbed problem includes boundary layers of two types, namely the exponential one near the segments and the power-law one in the vicinity of the collision points. These boundary layers are solutions of certain problems in the half-strip and the half-plane, respectively. The power-law boundary layer causes some surprising phenomena, including the square-root singularities of the main correction asymptotic term at the collision points and their linear depending on the quantity [Formula: see text].
Funding Information
  • Research Council of Finland (342957)
  • Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni
  • Suomalainen Tiedeakatemia

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