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

Bucur, Dorin; Giacomini, Alessandro; Trebeschi, Paola

doi:10.1016/j.anihpc.2019.07.007

Bucur, Dorin ¹ ; Giacomini, Alessandro ² ; Trebeschi, Paola ²

¹ Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA 73000 Chambéry, France
² DICATAM, Sezione di Matematica, Università degli Studi di Brescia, Via Branze 43, 25123 Brescia, Italy

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 7, pp. 1959-1986.

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Résumé

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.

MR Zbl

DOI : 10.1016/j.anihpc.2019.07.007

Classification : 49Q10, 26A45, 35R35, 35J65, 35J91, 49K20
Mots-clés : Free discontinuity problems, Functions of bounded variation, Sets of finite perimeter, Robin boundary conditions

Affiliations des auteurs :

Bucur, Dorin ¹ ; Giacomini, Alessandro ² ; Trebeschi, Paola ²

¹ Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA 73000 Chambéry, France
² DICATAM, Sezione di Matematica, Università degli Studi di Brescia, Via Branze 43, 25123 Brescia, Italy

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     author = {Bucur, Dorin and Giacomini, Alessandro and Trebeschi, Paola},
     title = {Best constant in {Poincar\'e} inequalities with traces: {A} free discontinuity approach},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1959--1986},
     publisher = {Elsevier},
     volume = {36},
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     year = {2019},
     doi = {10.1016/j.anihpc.2019.07.007},
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     zbl = {1428.49043},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2019.07.007/}
}

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Bucur, Dorin; Giacomini, Alessandro; Trebeschi, Paola. Best constant in Poincaré inequalities with traces: A free discontinuity approach. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 7, pp. 1959-1986. doi : 10.1016/j.anihpc.2019.07.007. http://www.numdam.org/articles/10.1016/j.anihpc.2019.07.007/

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