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
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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 |
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.
Keywords:
Non parametric minimal surfaces,
anisotropy,
sets of finite perimeter,
minimal cones,
anisotropic Bernstein problem.
Mathematics Subject Classification:
Primary: 49Q05; Secondary: 53A10.
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. Alberti, G. 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. Almgren, J. 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. Bellettini, M. 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. Bellettini, V. Caselles, A. 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. Bellettini, M. 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. Bombieri, E. 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. Caselles, A. 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. Caselles, R. Kimmel, G. 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. Chambolle, V. Caselles, M. Novaga, D. 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. Rudin, S. 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. Alberti, G. 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. Almgren, J. 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. Bellettini, M. 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. Bellettini, V. Caselles, A. 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. Bellettini, M. 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. Bombieri, E. 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. Caselles, A. 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. Caselles, R. Kimmel, G. 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. Chambolle, V. Caselles, M. Novaga, D. 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. Rudin, S. 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 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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