Some Weighted Isoperimetric Problems on R-+(N) with Stable Half Balls Have No Solutions

摘要

We show the counter-intuitive fact that some weighted isoperimetric problems on the half-space R-+(N), for which half-balls centered at the origin are stable, have no solutions. A particular case is the measure d mu=xN alpha dx, with alpha is an element of(-1,0). Some results on stability and nonexistence for weighted isoperimetric problems on R-N are also obtained.