Fisherman killed a brown bear on the Kenai a few years ago with a 9mm. Not my choice at all, but if a 9 will do it, a 357 properly loaded will as well.
A strong 357 will push a 180 gr 1200 fps from a 4" barrel, and that'll penetrate the skull of any bear. As with any gun, you have to make the shot count.
Just remember, 95% of the people who give advice on bear guns have never even seen a grizzly or brown in the wild, under 1% of those giving advice have been uncomfortably close to one in the wild, and 0.00001% have had a DLP event.
