Geodesic
Best constant in Poincaré inequalities with traces: A free discontinuity approach
Bucur, Dorin1 ; Giacomini, Alessandro2 ; Trebeschi, Paola2
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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For Ω⊂RN open bounded and with a Lipschitz boundary, and 1≤p<+∞, we consider the Poincaré inequality with trace term

Cp(Ω)‖u‖Lp(Ω)≤‖∇u‖Lp(Ω;RN)+‖u‖Lp(∂Ω)
on the Sobolev space W1,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.

DOI : 10.1016/j.anihpc.2019.07.007
Classification : 49Q10, 26A45, 35R35, 35J65, 35J91, 49K20
Keywords: Free discontinuity problems, Functions of bounded variation, Sets of finite perimeter, Robin boundary conditions

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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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

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