Debye Layer in Poisson–Boltzmann Model with Isolated Singularities

  • Chia-Yu HsiehEmail author
  • Yong Yu


We show the existence of solutions to the charge conserved Poisson–Boltzmann equation with a Dirichlet boundary condition on \({\partial }\Omega \). Here \(\Omega \) is a smooth simply connected bounded domain in \({\mathbb {R}}^n\) with \(n \geqslant 2\). When \(n = 2\), the solutions can have isolated singularities at prescribed points in \(\Omega \), in which case they are essentially weak solutions of the charge conserved Poisson–Boltzmann equations with Dirac measures as source terms. By contrast, for higher dimensional cases \(n \geqslant 3\), all the isolated singularities are removable. As a small parameter \(\epsilon \) tends to zero, and the solutions develop a Debye boundary layer near the boundary \({\partial }\Omega \). In the interior of \(\Omega \), the solutions converge to a unique constant. The limiting constant is explicitly calculated in terms of a novel formula which depends only on the supplied Dirichlet data on \({\partial }\Omega \). In addition we also give a quantitative description on the asymptotic behaviour of the solutions as \(\epsilon \rightarrow 0\).



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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsThe Chinese University of Hong KongSha Tin, New TerritoriesHong Kong

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