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Some isoperimetric inequalities with respect to monomial weights

ESAIM - Control, Optimisation and Calculus of Variations, ISSN: 1262-3377, Vol: 27
2021
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  • Citations
    6
    • Citation Indexes
      6

Article Description

We solve a class of isoperimetric problems on R2+ with respect to monomial weights. Let a and β be real numbers such that 0 ≤ α < β + 1, β ≤ 2α. We show that, among all smooth sets ω in R2+ with fixed weighted measure RRω yβdxdy, the weighted perimeter Rαω yα 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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