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The nonlinear Poisson-Boltzmann equation (NPBE) is an elliptic partialdifferential equation used in applications such as protein interactions andbiophysical chemistry (among many others). It describes the nonlinearelectrostatic potential of charged bodies submerged in an ionic solution. Thekinetic presence of the solvent molecules introduces randomness to the shape ofa protein, and thus a more accurate model that incorporates these randomperturbations of the domain is analyzed to compute the statistics of quantitiesof interest of the solution. When the parameterization of the randomperturbations is high-dimensional, this calculation is intractable as it issubject to the curse of dimensionality. However, if the solution of the NPBEvaries analytically with respect to the random parameters, the problem becomesamenable to techniques such as sparse grids and deep neural networks. In thispaper, we show analyticity of the solution of the NPBE with respect to analyticperturbations of the domain by using the analytic implicit function theorem andthe domain mapping method. Previous works have shown analyticity of solutionsto linear elliptic equations but not for nonlinear problems. We further showhow to derive \emph{a priori} bounds on the size of the region of analyticity.This method is applied to the trypsin molecule to demonstrate that theconvergence rates of the quantity of interest are consistent with theanalyticity result. Furthermore, the approach developed here is sufficientlygeneral enough to be applied to other nonlinear problems in uncertaintyquantification.

Uncertainty quantification and complex analyticity of the nonlinear Poisson-Boltzmann equation for the interface problem with random domains