| Colloquium |
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| Time |
2026-10-29 16:20-17:10 |
| Venue |
31106, Department of Mathematics |
| Speaker |
Dr. Lyu, Jhih-Hong(Postdoctoral Research Fellow, National Center for Theoretical Sciences) |
| Title |
Boundary Layers, Spectral Asymptotics, and Potential Blow-Up in the Charge-Conserving Poisson--Boltzmann Equation |
| Abstract |
The charge-conserving Poisson--Boltzmann (CCPB) equation describes electrostatic equilibria with prescribed total amounts of ionic species. Under electroneutrality, solutions in smooth bounded domains with multiple boundary components exhibit a nearly constant bulk potential and boundary layers on the scale of the dimensionless Debye length (\sqrt{\epsilon}). In this talk, we begin with asymptotic expansions of these equilibria under Robin boundary conditions scaled with (\sqrt{\epsilon}), including mean-curvature corrections to the potential and electric field. Based on this analysis, we study the linearized CCPB operator and establish two distinct spectral scales. The first eigenvalue is simple and satisfies (\lambda_1(\epsilon)\sim\Lambda_1\sqrt{\epsilon}), with an explicit constant (\Lambda_1>0), while its eigenfunction is asymptotically constant in the bulk. Every higher eigenvalue of fixed index converges to a common positive limit determined by associated half-line Robin Schr"odinger operators. We then investigate the effect of adding a permanent charge while keeping the total charge of the mobile ions zero. For spatially constant permanent charges (\epsilon^\theta\bar\rho), with fixed (\bar\rho\ne0) and (\theta>0), the scale (\sqrt{\epsilon}) separates three regimes: the perturbed equilibrium approaches the unperturbed one when (\theta>1/2); an order-one bulk shift occurs at (\theta=1/2) under an explicit range condition on the scaled total permanent charge; and the boundary (L^\infty)-norm of the potential diverges when (0<\theta<1/2). This divergence follows from an exact Robin flux balance and occurs even though the added permanent charge tends to zero as (\epsilon\to0^+). |
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