July  2017, 16(4): 1427-1454. doi: 10.3934/cpaa.2017068

Minimizers of anisotropic perimeters with cylindrical norms

1. 

Dipartimento di Ingegneria dell'Informazione e Scienze Matematiche, Università degli studi di Siena, via Roma 56,53100 Siena, Italy

2. 

International Centre for Theoretical Physics (ICTP), Strada Costiera 11,34151 Trieste, Italy

3. 

Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5,56127 Pisa, Italy

4. 

Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Bonomea 265,34136 Trieste, Italy

Received  April 2016 Revised  February 2017 Published  April 2017

We study various regularity properties of minimizers of the $\Phi$–perimeter, where $\Phi$ is a norm. Under suitable assumptions on $\Phi$ and on the dimension of the ambient space, we prove that the boundary of a cartesian minimizer is locally a Lipschitz graph out of a closed singular set of small Hausdorff dimension. Moreover, we show the following anisotropic Bernstein-type result: any entire cartesian minimizer is the subgraph of a monotone function depending only on one variable.

Citation: Giovanni Bellettini, Matteo Novaga, Shokhrukh Yusufovich Kholmatov. Minimizers of anisotropic perimeters with cylindrical norms. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1427-1454. doi: 10.3934/cpaa.2017068
References:
[1]

G. AlbertiG. Bouchitté and G. Dal Maso, The calibration method for the Mumford-Shah functional and free-discontinuity problems, Calc. Var. Partial Differential Equations, 16 (2003), 299-333. doi: 10.1007/s005260100152. Google Scholar

[2]

F. Almgren Jr.R. Schoen and L. Simon, Regularity and singularity estimates on hypersurfaces minimizing elliptic variational integrals, Acta Math., 139 (1977), 217-265. doi: 10.1007/BF02392238. Google Scholar

[3]

F. Almgren and J. Taylor, Flat flow is motion by crystalline curvature for curves with crystalline energies, J. Differential Geom., 42 (1995), 1-22. Google Scholar

[4]

F. AlmgrenJ. Taylor and L. Wang, Curvature-driven flows: a variational approach, SIAM J. Control Optim., 31 (1993), 387-438. doi: 10.1137/0331020. Google Scholar

[5]

M. Amar and G. Bellettini, A notion of total variation depending on a metric with discontinuous coefficients, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 91-133. Google Scholar

[6]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, The Clarendon Press, Oxford University Press, New York, 2000. Google Scholar

[7]

G. BellettiniM. Novaga and M. Paolini, On a crystalline variational problem, part Ⅰ: first variation and global L-regularity, Arch. Ration. Mech. Anal., 157 (2001), 165-191. doi: 10.1007/s002050010127. Google Scholar

[8]

G. BellettiniV. CasellesA. Chambolle and M. Novaga, Crystalline mean curvature flow of convex sets, Arch. Ration. Mech. Anal., 179 (2006), 109-152. doi: 10.1007/s00205-005-0387-0. Google Scholar

[9]

G. BellettiniM. Paolini and S. Venturini, Some results on surface measures in calculus of variations, Ann. Mat. Pura Appl., 170 (1996), 329-357. doi: 10.1007/BF01758994. Google Scholar

[10]

E. Bombieri, Regularity theory for almost minimal currents, Arch. Ration Mech. Anal., 78 (1982), 99-130. doi: 10.1007/BF00250836. Google Scholar

[11]

E. BombieriE. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math., 7 (1969), 243-268. doi: 10.1007/BF01404309. Google Scholar

[12]

J. W. Cahn and D. W. Hoffman, A vector thermodynamics for anisotropic surfaces. Ⅰ. Fundamentals and applications to plane surface junctions, Surface Sci., 31 (1972), 368-388. Google Scholar

[13]

J. W. Cahn and D. W. Hoffman, A vector thermodynamics for anisotropic surfaces. Ⅱ. Curved and facetted surfaces, Acta Metall., 22 (1974), 1205-1214. Google Scholar

[14]

V. CasellesA. Chambolle and M. Novaga, Regularity for solutions of the total variation denoising problem, Rev. Mat. Iber., 27 (2011), 233-252. doi: 10.4171/RMI/634. Google Scholar

[15]

V. CasellesR. KimmelG. Sapiro and C. Sbert, Minimal surfaces based object segmentation, IEEE Trans. Pattern Anal. Mach. Intell., 19 (1997), 394-398. doi: 10.1007/s002110050294. Google Scholar

[16]

A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vision, 20 (2004), 89-97. doi: 10.1023/B:JMIV.0000011320.81911.38. Google Scholar

[17]

A. ChambolleV. CasellesM. NovagaD. Cremers and T. Pock, An introduction to total variation for image analysis. Theoretical foundations and numerical methods for sparse recovery, Radon Ser. Comput. Appl. Math., 9 (2010), 263-340. doi: 10.1515/9783110226157.263. Google Scholar

[18]

G. Dal Maso, Integral representation on BV (Ω) of Γ-limits of variational integrals, Manuscripta Math., 30 (1980), 387-416. doi: 10.1007/BF01301259. Google Scholar

[19]

H. Federer, Geometric Measure Theory, Springer-Verlag New York Inc. , 1969. Google Scholar

[20]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9486-0. Google Scholar

[21]

M. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, The Clarendon Press, Oxford University Press, New York, 1993. Google Scholar

[22]

R. Jerrard, A. Moradifam and A. Nachman, Existence and uniqueness of minimizers of general least gradient problems, J. Reine Angew. Math. , to appear. doi: 10.1515/crelle-2014-0151. Google Scholar

[23]

H. Jenkins, On two-dimensional variational problems in parametric form, Arch. Ration. Mech. Anal., 8 (1961), 181-206. doi: 10.1007/BF00277437. Google Scholar

[24] F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems: an Introduction to Geometric Measure Theory, Cambridge Studies in Advanced Mathematics no. 135, Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139108133. Google Scholar
[25]

J. M. Mazón, The Euler-Lagrange equation for the anisotropic least gradient problem, Nonlinear Anal. Real World Appl., 31 (2016), 452-472. doi: 10.1016/j.nonrwa.2016.02.009. Google Scholar

[26]

G. Mercier, Curve-and-Surface Evolutions for Image Processing, PhD Thesis, École Polytechnique, 2015.Google Scholar

[27]

M. Miranda, Superfici cartesiane generalizzate ed insiemi di perimetro localmente finito sui prodotti cartesiani, Ann. Sc. Norm. Super. Pisa, 18 (1964), 515-542. Google Scholar

[28]

J. Moll, The anisotropic total variation flow, Math. Ann., 332 (2005), 177-218. doi: 10.1007/s00208-004-0624-0. Google Scholar

[29]

F. Morgan, The cone over the Clifford torus in $\mathbb{R}^4$ is Φ-minimizing, Math. Ann., 289 (1991), 341-354. doi: 10.1007/BF01446576. Google Scholar

[30]

R. Neumayer, A strong form of the quantitative Wulff inequality, SIAM J. Math. Anal., 48 (2016), 1727-1772. doi: 10.1137/15M1013675. Google Scholar

[31]

M. Novaga and E. Paolini, Regularity results for some 1-homogeneous functionals, Nonlinear Anal. Real World Appl., 3 (2002), 555-566. doi: 10.1016/S1468-1218(01)00048-7. Google Scholar

[32]

M. Novaga and E. Paolini, Regularity results for boundaries in $\mathbb{R}^2$ with prescribed anisotropic curvature, Ann. Mat. Pura Appl., 184 (2005), 239-261. doi: 10.1007/s10231-004-0112-x. Google Scholar

[33]

P. Overath and H. von der Mosel, On minimal immersions in Finsler space, Ann. Global Anal. Geom., 48 (2015), 397-422. doi: 10.1007/s10455-015-9476-y. Google Scholar

[34]

L. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268. Google Scholar

[35]

L. Simon, On some extensions of Bernstein's theorem, Math. Z., 154 (1977), 265-273. doi: 10.1007/BF01214329. Google Scholar

[36]

I. Tamanini, Boundaries of Caccioppoli sets with Hölder-continuous normal vector, J. Reine Angew. Math., 334 (1982), 27-39. doi: 10.1515/crll.1982.334.27. Google Scholar

[37]

J. Taylor, Crystalline variational problems, Bull. Amer. Math. Soc., 84 (1978), 568-588. doi: 10.1090/S0002-9904-1978-14499-1. Google Scholar

[38]

J. Taylor, Complete catalog of minimizing embedded crystalline cones, Proc. Sympos. Pure Math., 44 (1984), 379-403. doi: 10.1090/pspum/044/840288. Google Scholar

[39]

J. Taylor and J. Cahn, Catalog of saddle shaped surfaces in crystals, Acta Metall., 34 (1986), 1-12. Google Scholar

show all references

References:
[1]

G. AlbertiG. Bouchitté and G. Dal Maso, The calibration method for the Mumford-Shah functional and free-discontinuity problems, Calc. Var. Partial Differential Equations, 16 (2003), 299-333. doi: 10.1007/s005260100152. Google Scholar

[2]

F. Almgren Jr.R. Schoen and L. Simon, Regularity and singularity estimates on hypersurfaces minimizing elliptic variational integrals, Acta Math., 139 (1977), 217-265. doi: 10.1007/BF02392238. Google Scholar

[3]

F. Almgren and J. Taylor, Flat flow is motion by crystalline curvature for curves with crystalline energies, J. Differential Geom., 42 (1995), 1-22. Google Scholar

[4]

F. AlmgrenJ. Taylor and L. Wang, Curvature-driven flows: a variational approach, SIAM J. Control Optim., 31 (1993), 387-438. doi: 10.1137/0331020. Google Scholar

[5]

M. Amar and G. Bellettini, A notion of total variation depending on a metric with discontinuous coefficients, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 91-133. Google Scholar

[6]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, The Clarendon Press, Oxford University Press, New York, 2000. Google Scholar

[7]

G. BellettiniM. Novaga and M. Paolini, On a crystalline variational problem, part Ⅰ: first variation and global L-regularity, Arch. Ration. Mech. Anal., 157 (2001), 165-191. doi: 10.1007/s002050010127. Google Scholar

[8]

G. BellettiniV. CasellesA. Chambolle and M. Novaga, Crystalline mean curvature flow of convex sets, Arch. Ration. Mech. Anal., 179 (2006), 109-152. doi: 10.1007/s00205-005-0387-0. Google Scholar

[9]

G. BellettiniM. Paolini and S. Venturini, Some results on surface measures in calculus of variations, Ann. Mat. Pura Appl., 170 (1996), 329-357. doi: 10.1007/BF01758994. Google Scholar

[10]

E. Bombieri, Regularity theory for almost minimal currents, Arch. Ration Mech. Anal., 78 (1982), 99-130. doi: 10.1007/BF00250836. Google Scholar

[11]

E. BombieriE. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math., 7 (1969), 243-268. doi: 10.1007/BF01404309. Google Scholar

[12]

J. W. Cahn and D. W. Hoffman, A vector thermodynamics for anisotropic surfaces. Ⅰ. Fundamentals and applications to plane surface junctions, Surface Sci., 31 (1972), 368-388. Google Scholar

[13]

J. W. Cahn and D. W. Hoffman, A vector thermodynamics for anisotropic surfaces. Ⅱ. Curved and facetted surfaces, Acta Metall., 22 (1974), 1205-1214. Google Scholar

[14]

V. CasellesA. Chambolle and M. Novaga, Regularity for solutions of the total variation denoising problem, Rev. Mat. Iber., 27 (2011), 233-252. doi: 10.4171/RMI/634. Google Scholar

[15]

V. CasellesR. KimmelG. Sapiro and C. Sbert, Minimal surfaces based object segmentation, IEEE Trans. Pattern Anal. Mach. Intell., 19 (1997), 394-398. doi: 10.1007/s002110050294. Google Scholar

[16]

A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vision, 20 (2004), 89-97. doi: 10.1023/B:JMIV.0000011320.81911.38. Google Scholar

[17]

A. ChambolleV. CasellesM. NovagaD. Cremers and T. Pock, An introduction to total variation for image analysis. Theoretical foundations and numerical methods for sparse recovery, Radon Ser. Comput. Appl. Math., 9 (2010), 263-340. doi: 10.1515/9783110226157.263. Google Scholar

[18]

G. Dal Maso, Integral representation on BV (Ω) of Γ-limits of variational integrals, Manuscripta Math., 30 (1980), 387-416. doi: 10.1007/BF01301259. Google Scholar

[19]

H. Federer, Geometric Measure Theory, Springer-Verlag New York Inc. , 1969. Google Scholar

[20]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9486-0. Google Scholar

[21]

M. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, The Clarendon Press, Oxford University Press, New York, 1993. Google Scholar

[22]

R. Jerrard, A. Moradifam and A. Nachman, Existence and uniqueness of minimizers of general least gradient problems, J. Reine Angew. Math. , to appear. doi: 10.1515/crelle-2014-0151. Google Scholar

[23]

H. Jenkins, On two-dimensional variational problems in parametric form, Arch. Ration. Mech. Anal., 8 (1961), 181-206. doi: 10.1007/BF00277437. Google Scholar

[24] F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems: an Introduction to Geometric Measure Theory, Cambridge Studies in Advanced Mathematics no. 135, Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139108133. Google Scholar
[25]

J. M. Mazón, The Euler-Lagrange equation for the anisotropic least gradient problem, Nonlinear Anal. Real World Appl., 31 (2016), 452-472. doi: 10.1016/j.nonrwa.2016.02.009. Google Scholar

[26]

G. Mercier, Curve-and-Surface Evolutions for Image Processing, PhD Thesis, École Polytechnique, 2015.Google Scholar

[27]

M. Miranda, Superfici cartesiane generalizzate ed insiemi di perimetro localmente finito sui prodotti cartesiani, Ann. Sc. Norm. Super. Pisa, 18 (1964), 515-542. Google Scholar

[28]

J. Moll, The anisotropic total variation flow, Math. Ann., 332 (2005), 177-218. doi: 10.1007/s00208-004-0624-0. Google Scholar

[29]

F. Morgan, The cone over the Clifford torus in $\mathbb{R}^4$ is Φ-minimizing, Math. Ann., 289 (1991), 341-354. doi: 10.1007/BF01446576. Google Scholar

[30]

R. Neumayer, A strong form of the quantitative Wulff inequality, SIAM J. Math. Anal., 48 (2016), 1727-1772. doi: 10.1137/15M1013675. Google Scholar

[31]

M. Novaga and E. Paolini, Regularity results for some 1-homogeneous functionals, Nonlinear Anal. Real World Appl., 3 (2002), 555-566. doi: 10.1016/S1468-1218(01)00048-7. Google Scholar

[32]

M. Novaga and E. Paolini, Regularity results for boundaries in $\mathbb{R}^2$ with prescribed anisotropic curvature, Ann. Mat. Pura Appl., 184 (2005), 239-261. doi: 10.1007/s10231-004-0112-x. Google Scholar

[33]

P. Overath and H. von der Mosel, On minimal immersions in Finsler space, Ann. Global Anal. Geom., 48 (2015), 397-422. doi: 10.1007/s10455-015-9476-y. Google Scholar

[34]

L. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268. Google Scholar

[35]

L. Simon, On some extensions of Bernstein's theorem, Math. Z., 154 (1977), 265-273. doi: 10.1007/BF01214329. Google Scholar

[36]

I. Tamanini, Boundaries of Caccioppoli sets with Hölder-continuous normal vector, J. Reine Angew. Math., 334 (1982), 27-39. doi: 10.1515/crll.1982.334.27. Google Scholar

[37]

J. Taylor, Crystalline variational problems, Bull. Amer. Math. Soc., 84 (1978), 568-588. doi: 10.1090/S0002-9904-1978-14499-1. Google Scholar

[38]

J. Taylor, Complete catalog of minimizing embedded crystalline cones, Proc. Sympos. Pure Math., 44 (1984), 379-403. doi: 10.1090/pspum/044/840288. Google Scholar

[39]

J. Taylor and J. Cahn, Catalog of saddle shaped surfaces in crystals, Acta Metall., 34 (1986), 1-12. Google Scholar

Figure 1.  $C_1^{(2)}(0, 0)\cup C_2^{(2)}(l, 0)$ with $l>0$ in Example 2.7(a) and its boundary. The right picture is a slight rotation of the left picture
Figure 2.  Union $C$ of $C_1^{(2)}(0, 0)$ and the $(-\pi/2)$-rotation of $C_2^{(2)}(0, 0)$ in Example 2.7(b). Notice that $C_t$ for $t=0$ is not a minimizer of the Euclidean perimeter in $\mathbb{R}^2$; however, this does not affect the minimality of $C$
Figure 3.  In case $0 < \gamma\leq l, $ among all sets connecting two components of $E$ the strip parallel to $\xi_1$-axis has the "smallest" $\Phi$-perimeter
Figure 4.  "Roof" like cone (left) and its section (right) along $(\partial H_1\cap\partial H_2)^\perp$
Figure 5.  Sections of cones when $\lambda_1 < +\infty$ and $\lambda_1=+\infty$
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