Narrow escape problem with a mixed trap and the effect of orientation

We consider the mean first passage time (MFPT) of a two-dimensional diffusing particle to a small trap with a distribution of absorbing and reflecting sections. High-order asymptotic formulas for the MFPT and the fundamental eigenvalue of the Laplacian are derived which extend previously obtained results and show how the orientation of the trap affects the mean time to capture. We obtain a simple geometric condition which gives the optimal trap alignment in terms of the gradient of the regular part of a regular part of a Green's function and a certain alignment vector. We find that subdividing the absorbing portions of the trap reduces the mean first passage time of the diffusing particle. In the scenario where the trap undergoes prescribed motion in the domain, the MFPT is seen to be particularly sensitive to the orientation of the trap.