  Hole spin relaxation in $p$-type (111) GaAs quantum wells is investigated in
the case with only the lowest hole subband, which is heavy-hole like in (111)
GaAs/AlAs and light-hole like in (111) GaAs/InP quantum wells, being relevant.
The subband L\"{o}wdin perturbation method is applied to obtain the effective
Hamiltonian including the Dresselhaus and Rashba spin-orbit couplings. Under a
proper gate voltage, the total in-plane effective magnetic field in (111)
GaAs/AlAs quantum wells can be strongly suppressed in the whole momentum space,
while the one in (111) GaAs/InP quantum wells can be suppressed only on a
special momentum circle. The hole spin relaxation due to the D'yakonov-Perel'
and Elliott-Yafet mechanisms is calculated by means of the fully microscopic
kinetic spin Bloch equation approach with all the relevant scatterings
explicitly included. For (111) GaAs/AlAs quantum wells, extremely long
heavy-hole spin relaxation time (upto hundreds of nanoseconds) is predicted. In
addition, we predict a pronounced peak in the gate-voltage dependence of the
heavy-hole spin relaxation time due to the D'yakonov-Perel' mechanism. This
peak origins from the suppression of the unique inhomogeneous broadening in
(111) GaAs/AlAs quantum wells. Moreover, the Elliott-Yafet mechanism influences
the spin relaxation only around the peak area due to the small spin mixing
between the heavy and light holes in quantum wells with small well width. We
also show the anisotropy of the spin relaxation. In (111) GaAs/InP quantum
wells, a mild peak, similar to the case for electrons in (111) GaAs quantum
wells, is also predicted in the gate-voltage dependence of the light-hole spin
relaxation time. The contribution of the Elliott-Yafet mechanism is always
negligible in this case.
