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Remarks on some minimization problems associated with BV norms
The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals
1. | SISSA, Via Bonomea 265, Trieste, 34136, Italy |
2. | Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY, 10012, USA |
3. | Mathematisches Institut, Universität Leipzig, Augustus Platz 10, Leipzig, D04109, Germany |
In this paper we investigate the "area blow-up" set of a sequence of smooth co-dimension one manifolds whose first variation with respect to an anisotropic integral is bounded. Following the ideas introduced by White in [
References:
[1] |
W. K. Allard, A characterization of the area integrand, in Symposia Mathematica, (Convegno di Teoria Geometrica dell'Integrazione e Varietà Minimali, INDAM, Rome, 1973), Academic Press, London, Volume XIV, 1974,429-444. Google Scholar |
[2] |
W. K. Allard,
An a priori estimate for the oscillation of the normal to a hypersurface whose first and second variation with respect to an elliptic integrand is controlled, Invent. Math., 73 (1983), 287-331.
doi: 10.1007/BF01394028. |
[3] |
W. K. Allard, An integrality theorem and a regularity theorem for surfaces whose first variation with respect to a parametric elliptic integrand is controlled, in Geometric Measure Theory and the Calculus of Variations, Proceedings of Symposia in Pure Mathematics, (eds. F. J. Allard ad W. K. Almgren Jr.), 44, 1986.
doi: 10.1090/pspum/044/840267. |
[4] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.
![]() |
[5] |
G. De Philippis, A. De Rosa and F. Ghiraldin,
Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies, Communications on Pure and Applied Mathematics, 71 (2018), 1123-1148.
doi: 10.1002/cpa.21713. |
[6] |
G. De Philippis and F. Maggi,
Dimensional estimates for singular sets in geometric variational problems with free boundaries, J. Reine Angew. Math., 725 (2017), 217-234.
doi: 10.1515/crelle-2014-0100. |
[7] |
L. Simon, Lectures on geometric measure theory, in Proceedings of the Centre for Mathematical Analysis, Australian National University, Centre for Mathematical Analysis, Canberra, 3, 1983. |
[8] |
L. Simon,
Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom., 1 (1993), 281-326.
doi: 10.4310/CAG.1993.v1.n2.a4. |
[9] |
B. Solomon and B. White,
A strong maximum principle for varifolds that are stationary with respect to even parametric elliptic functionals, Indiana Univ. Math. J., 38 (1989), 683-691.
doi: 10.1512/iumj.1989.38.38032. |
[10] |
B. White,
Curvature estimates and compactness theorems in {$3$}-manifolds for surfaces that are stationary for parametric elliptic functionals, Invent. Math., 88 (1987), 243-256.
doi: 10.1007/BF01388908. |
[11] |
B. White,
Existence of smooth embedded surfaces of prescribed genus that minimize parametric even elliptic functionals on $3$-manifolds, J. Differential Geom., 33 (1991), 413-443.
doi: 10.4310/jdg/1214446325. |
[12] |
B. White,
Controlling area blow-up in minimal or bounded mean curvature varieties, J. Differential Geom., 102 (2016), 501-535.
doi: 10.4310/jdg/1456754017. |
show all references
References:
[1] |
W. K. Allard, A characterization of the area integrand, in Symposia Mathematica, (Convegno di Teoria Geometrica dell'Integrazione e Varietà Minimali, INDAM, Rome, 1973), Academic Press, London, Volume XIV, 1974,429-444. Google Scholar |
[2] |
W. K. Allard,
An a priori estimate for the oscillation of the normal to a hypersurface whose first and second variation with respect to an elliptic integrand is controlled, Invent. Math., 73 (1983), 287-331.
doi: 10.1007/BF01394028. |
[3] |
W. K. Allard, An integrality theorem and a regularity theorem for surfaces whose first variation with respect to a parametric elliptic integrand is controlled, in Geometric Measure Theory and the Calculus of Variations, Proceedings of Symposia in Pure Mathematics, (eds. F. J. Allard ad W. K. Almgren Jr.), 44, 1986.
doi: 10.1090/pspum/044/840267. |
[4] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.
![]() |
[5] |
G. De Philippis, A. De Rosa and F. Ghiraldin,
Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies, Communications on Pure and Applied Mathematics, 71 (2018), 1123-1148.
doi: 10.1002/cpa.21713. |
[6] |
G. De Philippis and F. Maggi,
Dimensional estimates for singular sets in geometric variational problems with free boundaries, J. Reine Angew. Math., 725 (2017), 217-234.
doi: 10.1515/crelle-2014-0100. |
[7] |
L. Simon, Lectures on geometric measure theory, in Proceedings of the Centre for Mathematical Analysis, Australian National University, Centre for Mathematical Analysis, Canberra, 3, 1983. |
[8] |
L. Simon,
Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom., 1 (1993), 281-326.
doi: 10.4310/CAG.1993.v1.n2.a4. |
[9] |
B. Solomon and B. White,
A strong maximum principle for varifolds that are stationary with respect to even parametric elliptic functionals, Indiana Univ. Math. J., 38 (1989), 683-691.
doi: 10.1512/iumj.1989.38.38032. |
[10] |
B. White,
Curvature estimates and compactness theorems in {$3$}-manifolds for surfaces that are stationary for parametric elliptic functionals, Invent. Math., 88 (1987), 243-256.
doi: 10.1007/BF01388908. |
[11] |
B. White,
Existence of smooth embedded surfaces of prescribed genus that minimize parametric even elliptic functionals on $3$-manifolds, J. Differential Geom., 33 (1991), 413-443.
doi: 10.4310/jdg/1214446325. |
[12] |
B. White,
Controlling area blow-up in minimal or bounded mean curvature varieties, J. Differential Geom., 102 (2016), 501-535.
doi: 10.4310/jdg/1456754017. |
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