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We investigate restricted diffusion in a bounded domain towards a smallpartially reactive target in three- and higher-dimensional spaces. We propose asimple explicit approximation for the principal eigenvalue of the Laplaceoperator with mixed Robin-Neumann boundary conditions. This approximationinvolves the harmonic capacity and the surface area of the target, the volumeof the confining domain, the diffusion coefficient and the reactivity. Theaccuracy of the approximation is checked by using a finite-elements method. Theproposed approximation determines also the mean first-reaction time, thelong-time decay of the survival probability, and the overall reaction rate onthat target. We identify the relevant length scale of the target, whichdetermines its trapping capacity, and investigate its relation to the targetshape. In particular, we study the effect of target anisotropy on the principaleigenvalue by computing the harmonic capacity of prolate and oblate spheroidsin various space dimensions. Some implications of these results in chemicalphysics and biophysics are briefly discussed.

First-passage times to anisotropic partially reactive targets