NON - LINEAR STOCHASTIC PDEs
Description and why it is important
Stochastic Non Linear Poisson Equation
Application - Statistics of potential fields for protein in solvents (Protein Interactions)
The nonlinear Poisson-Boltzmann equation (nPBE) is an elliptic partial differential equation used in applications such as protein interactions and biophysical chemistry (among many others). It describes the nonlinear electro- static potential of charged bodies submerged in an ionic solution. The kinetic presence of the solvent molecules introduces randomness to the shape of a protein, and thus a more accurate model that incorporates these random perturbations of the domain is analyzed to compute the statistics of quantities of interest of the solution.
Complexified Non-linear Poisson Boltzmann Equation
We prove the existence and uniqueness of the complexified Nonlinear Poisson-Boltzmann Equation (nPBE) in a bounded domain in ℝ3. The nPBE is a model equation in nonlinear electrostatics. The standard convex optimization argument to the complexified nPBE no longer applies, but instead, a contraction mapping argument is developed. Furthermore, we show that uniqueness can be lost if the hypotheses given are not satisfied. The complixified nPBE is highly relevant to regularity analysis of the solution of the real nPBE with respect to the dielectric (diffusion) and Debye-Hückel coefficients. This approach is also well-suited to investigate the existence and uniqueness problem for a wide class of semi-linear elliptic Partial Differential Equations (PDEs).
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Stochastic Machine Learning Group
- Department of Mathematics and Statistics Boston University, 665 Commonwealth Ave. Boston, MA 02215
- + (617) 353-9549
- jcandas@bu.edu
- mkon@math.bu.edu
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