The study of dynamics and thermodynamics of single biological molecules at confined liquid-solid interfaces is crucially important, especially in the case of low-copy number molecules in a single cell. Using a high-throughput single molecule imaging system and Lagrangian coordinates of single molecule images, we discovered that the local equilibrium isotherms of single lambdaDNA molecules at a confined liquid-solid interface varied from a stair type for the regions of single or double molecular DNA to a mild "S" type for the regions of triple molecular DNA spots, which does not agree with the conventional equilibrium isotherms in the literature. Single molecule images in time sequence for different lambdaDNA concentrations were statistically analyzed by measuring preferential partitioning from shearing effects, which were used to measure the local velocity of DNA molecules by directly observing the migration of DNA fluorescence spots for the 12 continuous images. The local linear velocity of hydrodynamic flow was calculated by the Hagen-Poiseuille equation in different microregions with a local Lagrangian approach. The local single molecule isotherms for the tracked molecules in the regions of single, double, or triple molecular DNA layers within the laminar flows were obtained according to the average local velocities of both the stochastic molecule events and the corresponding local Poiseuille flows. A millisecond and microvolume approach to directly determine local single molecule isotherms at confined liquid-solid interfaces was established, and the microspace scale effects on the types of isotherms were discovered. This study may have significant impact on preparations of low-copy number proteins in a single cell, membrane separations, and other bioseparation studies.