  We investigate restricted diffusion in a bounded domain towards a small
partially reactive target in three- and higher-dimensional spaces. We propose a
simple explicit approximation for the principal eigenvalue of the Laplace
operator with mixed Robin-Neumann boundary conditions. This approximation
involves the harmonic capacity and the surface area of the target, the volume
of the confining domain, the diffusion coefficient and the reactivity. The
accuracy of the approximation is checked by using a finite-elements method. The
proposed approximation determines also the mean first-reaction time, the
long-time decay of the survival probability, and the overall reaction rate on
that target. We identify the relevant length scale of the target, which
determines its trapping capacity, and investigate its relation to the target
shape. In particular, we study the effect of target anisotropy on the principal
eigenvalue by computing the harmonic capacity of prolate and oblate spheroids
in various space dimensions. Some implications of these results in chemical
physics and biophysics are briefly discussed.
