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 and 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.

[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.

[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. 

[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.

[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. 

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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. 

[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. 

[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.

[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.

[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.

[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.

[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.

[19]

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

[20]

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

[21]

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

[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.

[23]

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

[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.
[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.

[26]

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

[27]

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

[28]

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

[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.

[30]

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

[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.

[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.

[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.

[34]

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

[35]

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

[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.

[37]

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

[38]

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

[39]

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

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.

[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.

[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. 

[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.

[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. 

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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. 

[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. 

[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.

[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.

[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.

[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.

[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.

[19]

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

[20]

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

[21]

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

[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.

[23]

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

[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.
[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.

[26]

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

[27]

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

[28]

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

[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.

[30]

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

[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.

[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.

[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.

[34]

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

[35]

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

[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.

[37]

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

[38]

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

[39]

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

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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