A p-laplacian approach and its analysis for the calculation of ensemble average solvation energy

Implicit solvent models are pivotal in predicting solvation energy for chemical and biological systems at the molecular level due to their lower computational cost compared to explicit solvent models and their satisfactory accuracy. However, traditional implicit solvent models often overlook the randomness inherent in solutesolvent interfaces caused by thermodynamic fluctuations. In our prior work [36], we addressed this limitation by introducing the concept of ensemble average solvation energy (EASE), which captures the average solvation energy across all microstates within an ensemble. We demonstrated that EASE can be calculated by using a total variation model with a two-sided obstacle. In this study, we employed a p-energy regularization method, previously developed in [11], to study this model. This approach allowed us to overcome the challenges associated with the variational analysis of the total variation model. We derived two computational models using the Euler-Lagrange equation and the L2-gradient flow of the p-energy functionals. We conducted a comprehensive analysis of these models, exploring properties such as the existence and uniqueness of weak solutions, energy dissipation, and convergence of weak solutions to steady-state solutions. Additionally, we investigated a time-discretized problem associated with the L2-gradient flow, establishing properties like unconditional stability, energy dissipation, and convergence. The analysis of these computational models is expected to establish a solid theoretical groundwork for the development of effective numerical schemes for the p-energy functionals.