A sharp trace inequality for functions of bounded variation in the ball

Andrea Cianchi

doi:10.1017/S0308210511000758

Abstract

The best constant in a mean-value trace inequality for functions of bounded variation on admissible domains Ω ⊂ ℝn is shown to agree with an isoperimetric constant associated with Ω. The existence and form of extremals is also discussed. This result is exploited to compute the best constant in the relevant trace inequality when Ω is a ball. The existence and the form of extremals in this special case turn out to depend on the dimension n. In particular, the best constant is not achieved when Ω is a disc in ℝ2.

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Ferone, Vincenzo Nitsch, Carlo and Trombetti, Cristina 2014. On a conjectured reverse Faber-Krahn inequality for a Steklov--type Laplacian eigenvalue. Communications on Pure and Applied Analysis, Vol. 14, Issue. 1, p. 63.

Obersnel, Franco Omari, Pierpaolo and Rivetti, Sabrina 2014. Asymmetric Poincaré inequalities and solvability of capillarity problems. Journal of Functional Analysis, Vol. 267, Issue. 3, p. 842.

Steinerberger, Stefan 2015. Sharp L 1-Poincaré inequalities correspond to optimal hypersurface cuts. Archiv der Mathematik, Vol. 105, Issue. 2, p. 179.

Kuznetsov, Nikolay and Nazarov, Alexander 2015. SHARP CONSTANTS IN THE POINCARÉ, STEKLOV AND RELATED INEQUALITIES (A SURVEY). Mathematika, Vol. 61, Issue. 2, p. 328.

Brandolini, Barbara Della Pietra, Francesco Nitsch, Carlo and Trombetti, Cristina 2015. Symmetry breaking in a constrained Cheeger type isoperimetric inequality. ESAIM: Control, Optimisation and Calculus of Variations, Vol. 21, Issue. 2, p. 359.

Cheng, Wanli and Olshanskii, Maxim A. 2017. Finite stopping times for freely oscillating drop of a yield stress fluid. Journal of Non-Newtonian Fluid Mechanics, Vol. 239, Issue. , p. 73.

Cianchi, Andrea Ferone, Vincenzo Nitsch, Carlo and Trombetti, Cristina 2017. Balls minimize trace constants in BV. Journal für die reine und angewandte Mathematik (Crelles Journal), Vol. 2017, Issue. 725, p. 41.

De Nápoli, P. L. Haddad, J. Jiménez, C. H. and Montenegro, M. 2018. The sharp affine $$L^2$$ L 2 Sobolev trace inequality and variants. Mathematische Annalen, Vol. 370, Issue. 1-2, p. 287.

Cianchi, Andrea Ferone, Vincenzo Nitsch, Carlo and Trombetti, Cristina 2018. Poincaré Trace Inequalities in $$\textit{BV}({\mathbb {B}}^n)$$ BV ( B n ) with Non-standard Normalization. The Journal of Geometric Analysis, Vol. 28, Issue. 4, p. 3522.

Dan, Su and Yang, Qiaohua 2022. Trace Hardy-Sobolev-Maz'ya inequalities on half space and sharp constant in dimension two. Journal of Mathematical Analysis and Applications, Vol. 516, Issue. 1, p. 126488.

Leite, Edir Junior Ferreira and Montenegro, Marcos 2022. Minimization to the Zhang's energy on BV(Ω) and sharp affine Poincaré-Sobolev inequalities. Journal of Functional Analysis, Vol. 283, Issue. 10, p. 109646.

Horta, Raul Fernandes and Montenegro, Marcos 2025. Asymmetric affine Poincaré–Sobolev–Wirtinger inequalities on BV(Ω) and characterization of extremizers in one-dimension. Nonlinear Analysis, Vol. 251, Issue. , p. 113673.

Article contents

A sharp trace inequality for functions of bounded variation in the ball

Abstract

Access options

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Article contents

A sharp trace inequality for functions of bounded variation in the ball

Abstract

Access options

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests