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A characterization of convex calibrable sets in $$\mathbb R^N$$. (English) Zbl 1108.35073
The authors extend a known characterization of calibrable sets in $${\mathbb R}^2$$ in term of the mean curvature of the boundary to every dimension. Precisely they prove that a convex and bounded set $$C \subset {\mathbb R}^n$$ of class $$C^{1,1}$$ is calibrable, i.e. once defined $$\lambda_C:= P(C)/| C|$$ where $$P(A)$$ denotes the perimeter of $$A$$ and $$| A|$$ the $$n$$-dimensional volume of $$A$$, there is a vector field $$z \in L^{\infty}(C, {\mathbb R}^n)$$, $$\| z\| _{\infty} \leqslant 1$$, such that $- \text{div} \, z = \lambda_C \text{ in } \;C\, , z \cdot \nu^C = -1 \text{ in } \;\partial C\, .$ if and only if ($${\mathbf H}_C$$ denotes the mean curvature) $(n-1) \text{ ess sup}_{x \in \partial C} {\mathbf H}_C (x) \leq \lambda_C \, .$
Moreover $$C$$ is a solution of the minimum problem $\min_{X \subset C} P(X) - \lambda_C | X| \, .$ As a by-product they obtain that any set $$C$$ as above contain a convex calibrate set $$K$$ and for every volume $$V \in [| K| , | C| ]$$ the solution of $\min_{X \subset C\, , | X| =V} P(X)$ is a convex set. The evolution of sets $$C = C_1 \cup \dots \cup C_k$$, with $$C_j$$ of class $$C^{1,1}$$ convex sets satisfying $${\overline C}_i \cap {\overline C}_j = \emptyset$$, by the minimizing total variation flow is also described.

##### MSC:
 35J70 Degenerate elliptic equations 49J40 Variational inequalities 35K65 Degenerate parabolic equations
##### Keywords:
calibrable sets; mean curvature
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##### References:
 [1] Alter, F., Caselles, V., Chambolle, A.: Evolution of Convex Sets in the Plane by the Minimizing Total Variation Flow. Interfaces Free Boundaries April 2005. · Zbl 1084.49003 [2] Ambrosio, L.: Corso introduttivo alla teoria geometrica della misura ed alle supefici minime. Scuola Normale Superiore, Pisa, 1997 · Zbl 0977.49028 [3] Ambrosio, L.: Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 19, 191–246 (1995) · Zbl 0957.49029 [4] Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs, 2000 · Zbl 0957.49001 [5] Andreu, F., Ballester, C., Caselles, V., Mazón, J.M.: Minimizing total variational flow. Differential Integral Equations 4(3), 321–360 (2001) · Zbl 1020.35037 [6] Andreu, F., Ballester, C., Caselles, V., Mazón, J.M.: The Dirichlet Problem for the Total Variation Flow. J. Funct. Anal. 180, 347–403 (2001) · Zbl 0973.35109 [7] Andreu, F., Caselles, V., Mazón, J.M.: A parabolic quasilinear problem for linear growth functionals. Rev. Mat. Iberoamericana 18, 135–185 (2002) · Zbl 1010.35063 [8] Anzellotti, G.: Pairings between measures and bounded functions and compensated compactness. Ann. Mat. Pura Appl. 135, 293–318 (1983) · Zbl 0572.46023 [9] Atkinson, F.V., Peletier, L.A., Bounds for vertical points of solutions of prescribed mean curvature type equations. Proc. Roy. Soc. Edinburgh Sect. A 112, 15–32 (1989) · Zbl 0685.35022 [10] Barozzi, E.: The curvature of a boundary with finite area. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 5, 149–159 (1994) · Zbl 0809.49038 [11] Bellettini, G., Caselles, V., Novaga, M.: The Total Variation Flow in . J. Differential Equations 184, 475–525 (2002) · Zbl 1036.35099 [12] Bellettini, G., Caselles, V., Novaga, M.: Explicit solutions of the eigenvalue problem Siam J. Mathematical Analysis, 2005. · Zbl 1162.35379 [13] Bellettini, G., Novaga, M., Paolini, M.: On a crystalline variational problem. II. BV regularity and structure of minimizers on facets. Arch. Ration. Mech. Anal. 157(3), 193–217 (2001) · Zbl 0976.58017 [14] Bellettini, G., Novaga, M., Paolini, M.: Characterization of facet–breaking for nonsmooth mean curvature flow in the convex case. Interfaces Free Bound. 3, 415–446 (2001) · Zbl 0989.35127 [15] Brezis, H.: Operateurs Maximaux Monotones. North Holland, 1973 [16] Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vision 20, 89–97 (2004) · Zbl 1366.94056 [17] Chambolle, A.: An algorithm for mean curvature motion. Interfaces Free Bound. 6(2). 195–218 (2004). · Zbl 1061.35147 [18] Chen, J.T.: On the existence of capillary free surfaces in the absence of gravity. Pacific J. Math. 88, 323–361 (1980) · Zbl 0483.49036 [19] Concus, P., Finn, R.: On Capillary Free Surfaces in the Absence of Gravity. Acta Math. 132, 177–198 (1974) · Zbl 0382.76003 [20] Crandall, M.G., Liggett, T.M.: Generation of semigroups of nonlinear transformations on general Banach spaces. Am. J. Math. 93, 265–298 (1971) · Zbl 0226.47038 [21] Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Math. CRC Press, Ann Harbor, 1992 · Zbl 0804.28001 [22] Finn, R.: Existence and non existence of capillary surfaces. Manuscripta Math. 28, 1–11 (1979) · Zbl 0421.49043 [23] Finn, R.: A subsidiary variational problem and existence criteria for capillary surfaces. J. Reine Angew. Math. 353, 196–214 (1984) · Zbl 0542.49021 [24] Finn, R.: Equilibrium Capillary Surfaces. Springer Verlag, 1986 · Zbl 0583.35002 [25] Giga, M.H., Giga, Y., Kobayashi, R.: Very singular diffusion equations. Proc. of Taniguchi Conf. on Math. Advanced Studies in Pure Mathematics 31, 93–125 (2001) · Zbl 1002.35074 [26] Giga, Y.: Singular diffusivity-facets, shocks and more. Hokkaido University Preprint Series in Mathematics, Series 604, September 2003 [27] Giusti, E.: On the equation of surfaces of prescribed mean curvature. Existence and uniqueness without boundary conditions. Invent. Math. 46, 111–137 (1978) · Zbl 0381.35035 [28] Giusti, E.: Boundary Value Problems for Non-Parametric Surfaces of Prescribed Mean Curvature. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3, 501–548 (1976) · Zbl 0344.35036 [29] Gonzalez, E., Massari, U.: Variational Mean Curvatures. Rend. Sem. Mat. Univ. Politec. Torino 52, 1–28 (1994) · Zbl 0819.49025 [30] Korevaar, N.: Capillary surface convexity above convex domains. Indiana Univ. Math. J. 32, 73–82 (1983) · Zbl 0507.35010 [31] Korevaar, N., Simon, L.: Equations of mean curvature type with contact angle boundary conditions. Geometric analysis and the calculus of variations, Internat. Press, Cambridge, MA, 1996, pp. 175–201 · Zbl 0932.35091 [32] Lichnewski, A., Temam, R.: Pseudosolutions of the Time Dependent Minimal Surface Problem. J. Differential Equations 30, 340–364 (1978) · Zbl 0392.49020 [33] Massari, U.: Frontiere orientate di curvatura media assegnata in Lp. Rend. Sem. Mat. Univ. Padova 53, 37–52 (1975) · Zbl 0358.49019 [34] Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations. The fifteenth Dean Jacqueline B. Lewis memorial lectures. University Lecture Series, 22. American Mathematical Society, Providence, RI, 2001 · Zbl 0987.35003 [35] Miranda, M.: Un principio di massimo forte per le frontiere minimali e una sua applicazione alla risoluzione del problema al contorno per l’equazione delle superfici di area minima. Rend. Sem. Mat. Univ. Padova 45, 355–366 (1971) · Zbl 0266.49034 [36] Rosales, C.: Isoperimetric regions in rotationally symmetric convex bodies. Indiana Univ. Math. J. 52, 1201–1214 (2003) · Zbl 1088.53039 [37] Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992) · Zbl 0780.49028 [38] Santaló, L.A.: Integral geometry and geometric probability. Encyclopedia of Mathematics and its Applications, Vol. 1. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976 · Zbl 0342.53049 [39] Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, 1993 [40] Simon, L., Spruck, J.: Existence and Regularity of a Capillary Surface with Prescribed Contact Angle. Arch. Ration. Mech. Anal. 61, 19–34 (1976) · Zbl 0361.35014 [41] Stredulinsky, E., Ziemer, W.P.: Area Minimizing Sets Subject to a Volume Constraint in a Convex Set. J. Geom. Anal. 7, 653–677 (1997) · Zbl 0940.49025 [42] Ziemer, W.P.: Weakly Differentiable Functions. Springer Verlag, Ann Harbor, 1989 · Zbl 0692.46022
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