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**11 - 15**of**15**### EXISTENCE OF ISOPERIMETRIC SETS WITH DENSITIES “CONVERGING FROM BELOW ” ON RN

"... Abstract. In this paper, we consider the isoperimetric problem in the space RN with density. Our result states that, if the density f is l.s.c. and converges to a limit a> 0 at infinity, being f ≤ a far from the origin, then isoperimetric sets exist for all volumes. Several known results or count ..."

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Abstract. In this paper, we consider the isoperimetric problem in the space RN with density. Our result states that, if the density f is l.s.c. and converges to a limit a> 0 at infinity, being f ≤ a far from the origin, then isoperimetric sets exist for all volumes. Several known results or counterexamples show that the present result is essentially sharp. The special case of our result for radial and increasing densities posively answers a conjecture made in [10]. 1.

### An Isoperimetric Problem With Density and the Hardy Sobolev Inequality in R2.

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