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

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

Cp(Ω)uLp(Ω)uLp(Ω;RN)+uLp(Ω)C_{p}(\Omega )\|u\|_{L^{p}(\Omega )} \leq \|\mathrm{∇}u\|_{L^{p}(\Omega ;\mathbb{R}^{N})} + \|u\|_{L^{p}(\partial \Omega )}

on the Sobolev space W1,p(Ω)W^{1,p}(\Omega ). 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