Asymptotic analysis of charge conserving Poisson-Boltzmann equations with variable dielectric coefficients

Chiun-Chang  Lee

doi:10.3934/dcds.2016.36.3251

Article Contents

2016, Volume 36, Issue 6: 3251-3276. Doi: 10.3934/dcds.2016.36.3251

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Asymptotic analysis of charge conserving Poisson-Boltzmann equations with variable dielectric coefficients

Chiun-Chang Lee^1,

1.
Department of Applied Mathematics, National Hsinchu University of Education, Hsinchu City 30014

Received: August 31, 2015

Revised: September 30, 2015

Published: May 2017

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Abstract

Concerning the influence of the dielectric constant on the electrostatic potential in the bulk of electrolyte solutions, we investigate a charge conserving Poisson-Boltzmann (CCPB) equation [31,32] with a variable dielectric coefficient and a small parameter $\epsilon$ (related to the Debye screening length) in a bounded connected domain with smooth boundary. Under the Robin boundary condition with a given applied potential, the limiting behavior (as $\epsilon\downarrow0$) of the solution (the electrostatic potential) has been rigorously studied. In particular, under the charge neutrality constraint, our result exactly shows the effects of the dielectric coefficient and the applied potential on the limiting value of the solution in the interior domain. The main approach is the Pohozaev's identity of this model. On the other hand, under the charge non-neutrality constraint, we show that the maximum difference between the boundary and interior values of the solution has a lower bound $\log\frac{1}{\epsilon}$ as $\epsilon$ goes to zero. Such an asymptotic blow-up behavior describes an unstable phenomenon which is totally different from the behavior of the solution under the charge neutrality constraint.

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Mathematics Subject Classification: Primary: 35J25, 35J60, 35C20; Secondary: 35Q92, 45K05.

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