This paper introduces a new branch of mathematics, multilayer interface mathematics.A mathematical model describing the energy of multilayer interfaces is constructed based on the variational method, which includes constructing the total energy functional from interface free energy, electrochemical energy, and diffusion transport energy. The orresponding Euler-Lagrange equations are derived. Eachstep of the mathematical derivation is explained in detail, clearly introducing howperturbation methods, the chain rule, and integration by parts are used to rigorouslyderive the target equations. The performance of fuel cells and lithium-ion batteries is largely limited by the rapid exchange efficiency of key ions in multilayer interfaces. Based on traditional models, we further introduce physical equations describing proton transport in hydrogen fuel cells and the rapid exchange of lithium ions in multilayer interfaces, providing quantitative design guidance for the fastestexchange of protons in fuel cell membrane electrodes and lithium ions in lithium-ionbatteries through specific numerical examples and parameter sensitivity analysis.