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.
- 12.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)Google Scholar
- 17.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. 439–455Google Scholar
- 20.H. Brezis, How to recognize constant functions. Usp. Mat. Nauk 57 (2002) (in Russian). English translation in Russian Math. Surveys 57, 693–708 (2002)Google Scholar
- 39.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)Google Scholar
- 42.H. Federer, Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 (Springer, New York 1969)Google Scholar
- 51.E. Giusti, Minimal Surface and Functions of Bounded Variation. Monographs in Mathematics, vol. 80 (Birkhäuser, Basel, 1984)Google Scholar
- 72.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. 135–152Google Scholar