Abstract We study the mean first passage time (MFPT) to an absorbing target of a one-dimensional Brownian particle subject to an external potential v ( x ) in a finite domain. We focus on the cases in which the external potential is confining, of the form v ( x ) = k | x − x 0 | n / n , and where the particle’s initial position coincides with x 0 . We first consider a particle between an absorbing target at x = 0 and a reflective wall at x = c . At fixed x 0 , we show that when the target distance c exceeds a critical value, there exists a nonzero optimal stiffness k opt that minimizes the MFPT to the target. However, when c lies below the critical value, the optimal stiffness k opt vanishes. Hence, for any value of n , the optimal potential stiffness undergoes a continuous ‘freezing’ transition as the domain size is varied. On the other hand, when the reflective wall is replaced by a second absorbing target, the freezing transition in k opt becomes discontinuous. The phase diagram in the ( x 0 , n )-plane then exhibits three dynamical phases and metastability, with a ‘triple’ point at ( x 0 / c ≃ 0.171 85, n ≃ 0.395 39). For harmonic or higher order potentials ( n ⩾ 2), the MFPT always increases with k at small k , for any x 0 or domain size. These results are contrasted with problems of diffusion under optimal resetting in bounded domains.