The study of the optimal constant Kq(Ω) in the Sobolev inequality ∥u∥Lq(Ω) ≤ 1/Kq(Ω)∥Du∥(double-struck Rn), 1 ≤ q < 1∗, for BV functions which are zero outside Ω and with zero mean value inside Ω, leads to the definition of a Cheeger type constant. We are interested in finding the best possible embedding constant in terms of the measure of Ω alone. We set up an optimal shape problem and we completely characterize, on varying the exponent q, the behaviour of optimal domains. Among other things we establish the existence of a threshold value 1 ≤ q < 1∗ above which the symmetry of optimal domains is broken. Several differences between the cases n = 2 and n ≥ 3 are emphasized.

Brandolini B., Della Pietra F., Nitsch C., & Trombetti C. (2015). Symmetry breaking in a constrained cheeger type isoperimetric inequality. ESAIM. COCV, 21(2), 359-371 [10.1051/cocv/2014016].

Symmetry breaking in a constrained cheeger type isoperimetric inequality

Brandolini B.
;
2015

Abstract

The study of the optimal constant Kq(Ω) in the Sobolev inequality ∥u∥Lq(Ω) ≤ 1/Kq(Ω)∥Du∥(double-struck Rn), 1 ≤ q < 1∗, for BV functions which are zero outside Ω and with zero mean value inside Ω, leads to the definition of a Cheeger type constant. We are interested in finding the best possible embedding constant in terms of the measure of Ω alone. We set up an optimal shape problem and we completely characterize, on varying the exponent q, the behaviour of optimal domains. Among other things we establish the existence of a threshold value 1 ≤ q < 1∗ above which the symmetry of optimal domains is broken. Several differences between the cases n = 2 and n ≥ 3 are emphasized.
Settore MAT/05 - Analisi Matematica
Brandolini B., Della Pietra F., Nitsch C., & Trombetti C. (2015). Symmetry breaking in a constrained cheeger type isoperimetric inequality. ESAIM. COCV, 21(2), 359-371 [10.1051/cocv/2014016].
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10447/493970
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