Reactive capacitance of flat patches of arbitrary shape

We investigate the capacity of a flat partially reactive patch of arbitrary shape to trap independent particles that undergo steady-state diffusion in the three-dimensional space. We focus on the total flux of particles onto the patch that determines its reactive capacitance. To disentangle the respective roles of the reactivity and the shape of the patch, we employ a spectral expansion of the reactive capacitance over a suitable Steklov eigenvalue problem. We derive several bounds on the reactive capacitance to reveal its monotonicity with respect to the reactivity and the shape. Two probabilistic interpretations are presented as well. An efficient numerical tool is developed for solving the associated Steklov spectral problem for patches of arbitrary shape. We propose and validate, both theoretically and numerically, a simple, fully explicit approximation for the reactive capacitance that depends only on the surface area and the electrostatic capacitance of the patch. This approximation opens promising ways to access various characteristics of diffusion-controlled reactions in general domains with multiple small well-separated patches. Direct applications of these results in statistical physics and physical chemistry are discussed.