Best constant in Poincaré inequalities with traces: A free discontinuity approach

Dorin Bucur; Alessandro Giacomini; Paola Trebeschi

doi:10.1016/j.anihpc.2019.07.007

JournalsaihpcVol. 36, No. 7pp. 1959–1986

Best constant in Poincaré inequalities with traces: A free discontinuity approach

Dorin Bucur
Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA 73000 Chambéry, France
Alessandro Giacomini
DICATAM, Sezione di Matematica, Università degli Studi di Brescia, Via Branze 43, 25123 Brescia, Italy
Paola Trebeschi
DICATAM, Sezione di Matematica, Università degli Studi di Brescia, Via Branze 43, 25123 Brescia, Italy

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Abstract

For $Ω \subset R^{N}$ open bounded and with a Lipschitz boundary, and $1 \leq p < + \infty$ , we consider the Poincaré inequality with trace term

C_{p} (Ω) ∥ u ∥_{L^{p} (Ω)} \leq ∥ \nabla u ∥_{L^{p} (Ω; R^{N})} + ∥ u ∥_{L^{p} (\partial Ω)}

on the Sobolev space $W^{1, p} (Ω)$ . We show that among all domains $Ω$ with prescribed volume, the constant is minimal on balls. The proof is based on the analysis of a free discontinuity problem.

Cite this article

Dorin Bucur, Alessandro Giacomini, Paola Trebeschi, Best constant in Poincaré inequalities with traces: A free discontinuity approach. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 7, pp. 1959–1986

DOI 10.1016/J.ANIHPC.2019.07.007