Abstract
We solve a class of isoperimetric problems on ℝ+2with respect to monomial weights. Letαandβbe real numbers such that 0 ≤α<β+ 1,β≤ 2α. We show that, among all smooth sets Ω in ℝ+2with fixed weighted measure ∬Ωyβdxdy, the weighted perimeter ∫∂Ωyαdsachieves its minimum for a smooth set which is symmetric w.r.t. to they-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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