An isoperimetric inequality related to a Bernoulli problem

Daniel Daners and Bernd Kawohl
Preprint, January 2010
Calculus of Variations and PDE's 39 (2010), 547-555.
Original article at doi:10.1007/s00526-010-0324-4
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Given a bounded domain Ω we look at the minimal parameter Λ(Ω) for which a Bernoulli free boundary value problem for the p-Laplacian has a solution minimising an energy functional. We show that amongst all domains of equal volume Λ(Ω) is minimal for the ball. Moreover, we show that the inequality is sharp with essentially only the ball minimising Λ(Ω). This resolves a problem related to a question asked in [Flucher et al., J. Reine Angew. Math. 486(1997), 165–204].

AMS Subject Classification (2000): 35R35, 49R05

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