Abstract
The word perimeter comes from the Greek peri (around) and meter (measure). A perimeter is usually used with two senses: it is the boundary that surrounds an N-dimensional set, and it is the measure of such boundary. We will see in these two first sections that these two concepts must be well precise.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs (Clarendon, Oxford, 2000)
F. Andreu-Vaillo, J.M. Mazón, J.D. Rossi, J. Toledo, Nonlocal Diffusion Problems. Mathematical Surveys and Monographs, vol. 165 (American Mathematical Society, Providence, 2010)
J. Bourgain, H. Brezis, P. Mironescu, Another Look at Sobolev Spaces, ed. by J.L. Menaldi et al. Optimal Control and Partial Differential Equations. A volume in honour of A. Bensoussan’s 60th birthday (IOS Press, Amsterdam, 2001), pp. 439–455
H. Brezis, How to recognize constant functions. Usp. Mat. Nauk 57 (2002) (in Russian). English translation in Russian Math. Surveys 57, 693–708 (2002)
L. Caffarelli, J.M. Roquejoffre, O. Savin, Nonlocal minimal surfaces. Commun. Pure Appl. Math. 63, 1111–1144 (2010)
J. Dávila, On an open question about functions of bounded variation. Calc. Var. Partial Differ. Equ. 15, 519–527 (2002)
E. De Giorgi, Definizione ed espressione analitica del perimetro di un insieme. Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. 14, 390–393 (1953)
E. De Giorgi, Su una teoria generale della misura (r − 1)-dimensionale in uno spazio ad r dimensioni. (Italian) Ann. Mat. Pura Appl. 36, 191–213 (1954)
H. Federer, Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 (Springer, New York 1969)
H. Federer, W. Fleming, Normal and integral currents. Ann. Math. 72, 458–520 (1960)
H.W. Fleming, R. Rishel, An integral formula for total gradient variation. Arch. Math. 11, 218–222 (1960)
E. Giusti, Minimal Surface and Functions of Bounded Variation. Monographs in Mathematics, vol. 80 (Birkhäuser, Basel, 1984)
M. Miranda, Distribuzioni aventi derivate misure insiemi di perimetro localmente finito. Ann. Scuola Norm. Sup. Pisa 18, 27–56 (1964)
A. Ponce, A new approach to Sobolev spaces and connections to Γ-convergence. Calc. Var. Partial Differ. Equ. 19, 229–255 (2004)
A. Ponce, An estimate in the spirit of Poincare’s inequality. J. Eur. Math. Soc. 6, 1–15 (2004)
J. Van Schaftingen, M. Willem, Set Transformations, Symmetrizations and Isoperimetric Inequalities. (English summary) Nonlinear Analysis and Applications to Physical Sciences (Springer Italia, Milan, 2004), pp. 135–152
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Mazón, J.M., Rossi, J.D., Toledo, J.J. (2019). Nonlocal Perimeter. In: Nonlocal Perimeter, Curvature and Minimal Surfaces for Measurable Sets. Frontiers in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-06243-9_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-06243-9_1
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-06242-2
Online ISBN: 978-3-030-06243-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)