Some isoperimetric inequalities with respect to monomial weights

We solve a class of isoperimetric problems on ℝ+2 with respect to monomial weights. Let α and β be real numbers such that 0 ≤ α < β + 1, β ≤ 2α. We show that, among all smooth sets Ω in ℝ+2 with fixed weighted measure ∬Ωyβdxdy, the weighted perimeter ∫∂Ωyα ds achieves its minimum for a smooth set which is symmetric w.r.t. to the y-axis, and is explicitly given. Our results also imply an estimate of a weighted Cheeger constant and a bound for eigenvalues of some nonlinear problems.

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