doi: 10.3934/dcds.2019242

Remarks on some minimization problems associated with bv norms

1. 

Rutgers University, Dept. of Math., Hill Center, Busch Campus, 110 Frelinghuysen RD, Piscataway, NJ 08854, USA

2. 

Dept. of Math. and Dept. of Computer SC., Technion, 32.000 Haifa, Israel

To Luis Caffarelli, a master of regularity, with esteem and affection

Received  November 2018 Revised  November 2019 Published  June 2019

Fund Project: This research was partially supported by NSF

The purpose of this paper is twofold. Firstly I present an optimal regularity result for minimizers of a $ 1D $ convex functional involving the BV-norm, under Neumann boundary condition. This functional is a simplified version of models occuring in Image Processing. Secondly I investigate the existence of minimizers for the same functional under Dirichlet boundary condition. Surprisingly, this turns out to be a delicate issue, which is still widely open.

Citation: Haïm Brezis. Remarks on some minimization problems associated with bv norms. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2019242
References:
[1] L. AmbrosioN. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, 2000.
[2]

M. Bonforte and A. Figalli, Total variation flow and sign fast diffusion in one dimension, J. Differential Equation, 252 (2012), 4455-4480. doi: 10.1016/j.jde.2012.01.003.

[3]

H. Brezis, Problèmes unilatéraux, J. Math. Pures Appl., 51 (1972), 1-168.

[4]

H. Brezis, Functional Analysis, Sobolev Spaces and PDEs, Springer, 2011.

[5]

H. Brezis, New approximations of the total variation and filters in imaging, Rend. Accad. Lincei, 26 (2015), 223-240. doi: 10.4171/RLM/704.

[6]

H. Brezis, Regularized interpolation involving the BV norm, to appear.

[7]

H. Brezis and D. Kinderlehrer, The smoothness of solutions to nonlinear variational inequalities, Indiana Univ. Math. J., 23 (1974), 831-844. doi: 10.1512/iumj.1974.23.23069.

[8]

H. Brezis and P. Mironescu, Sobolev Maps with Values into the Circle— from the Perspective of Analysis, Geometry and Topology, Birkhäuser, (in preparation).

[9]

H. Brezis and S. Serfaty, Variational formulation for the two-sided obstacle problem with measure data, Comm. Contemp. Math., 4 (2002), 357-374. doi: 10.1142/S0219199702000671.

[10]

H. Brezis and G. Stampacchia, Sur la régularité de la solution d'inéquations elliptiques, Bull. Soc. Math. Fr., 96 (1968), 153-180.

[11]

A. BrianiA. ChambolleM. Novaga and G. Orlandi, On the gradient flow of a one-homo-geneous functional, Confluentes Math., 3 (2011), 617-635. doi: 10.1142/S1793744211000461.

[12]

L. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl., 4 (1998), 383-402. doi: 10.1007/BF02498216.

[13]

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

[14]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, New York-London, 1980.

[15]

H. Lewy and G. Stampacchia, On the regularity of the solution of a variational inequality, Comm. Pure Appl. Math., 22 (1969), 153-188. doi: 10.1002/cpa.3160220203.

[16]

P. Mucha and P. Rybka, Well posedness of sudden directional diffusion equations, Math. Methods Appl. Sci., 36 (2013), 2359-2370. doi: 10.1002/mma.2759.

[17]

L. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise-removal algorithms, Phys. D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.

[18]

P. Sternberg and W. Ziemer, The Dirichlet problem for functions of least gradient, in: Degenerate Diffusions (Minneapolis, MN, 1991), IMA Vol. Math. Appl., Springer, 47 (1993), 197–214. doi: 10.1007/978-1-4612-0885-3_14.

[19]

P. SternbergG. Williams and W. Ziemer, Existence, uniqueness, and regularity for functions of least gradient, J. Reine Angew. Math., 430 (1992), 35-60.

[20]

T. Sznigir, Various minimization problems involving the total variation in one dimension, PhD Rutgers University, Sept., 2017.

[21]

T. Sznigir, A one-dimensional problem involving the total variation, to appear.

[22]

J. L. Vázquez, Two nonlinear diffusion equation with finite speed of propagation, in: Problems Involving Change of Type, Stuttgart, 1988, Lecture Notes in Phys., Springer, 359 (1990), 197–206. doi: 10.1007/3-540-52595-5_96.

show all references

References:
[1] L. AmbrosioN. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, 2000.
[2]

M. Bonforte and A. Figalli, Total variation flow and sign fast diffusion in one dimension, J. Differential Equation, 252 (2012), 4455-4480. doi: 10.1016/j.jde.2012.01.003.

[3]

H. Brezis, Problèmes unilatéraux, J. Math. Pures Appl., 51 (1972), 1-168.

[4]

H. Brezis, Functional Analysis, Sobolev Spaces and PDEs, Springer, 2011.

[5]

H. Brezis, New approximations of the total variation and filters in imaging, Rend. Accad. Lincei, 26 (2015), 223-240. doi: 10.4171/RLM/704.

[6]

H. Brezis, Regularized interpolation involving the BV norm, to appear.

[7]

H. Brezis and D. Kinderlehrer, The smoothness of solutions to nonlinear variational inequalities, Indiana Univ. Math. J., 23 (1974), 831-844. doi: 10.1512/iumj.1974.23.23069.

[8]

H. Brezis and P. Mironescu, Sobolev Maps with Values into the Circle— from the Perspective of Analysis, Geometry and Topology, Birkhäuser, (in preparation).

[9]

H. Brezis and S. Serfaty, Variational formulation for the two-sided obstacle problem with measure data, Comm. Contemp. Math., 4 (2002), 357-374. doi: 10.1142/S0219199702000671.

[10]

H. Brezis and G. Stampacchia, Sur la régularité de la solution d'inéquations elliptiques, Bull. Soc. Math. Fr., 96 (1968), 153-180.

[11]

A. BrianiA. ChambolleM. Novaga and G. Orlandi, On the gradient flow of a one-homo-geneous functional, Confluentes Math., 3 (2011), 617-635. doi: 10.1142/S1793744211000461.

[12]

L. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl., 4 (1998), 383-402. doi: 10.1007/BF02498216.

[13]

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

[14]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, New York-London, 1980.

[15]

H. Lewy and G. Stampacchia, On the regularity of the solution of a variational inequality, Comm. Pure Appl. Math., 22 (1969), 153-188. doi: 10.1002/cpa.3160220203.

[16]

P. Mucha and P. Rybka, Well posedness of sudden directional diffusion equations, Math. Methods Appl. Sci., 36 (2013), 2359-2370. doi: 10.1002/mma.2759.

[17]

L. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise-removal algorithms, Phys. D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.

[18]

P. Sternberg and W. Ziemer, The Dirichlet problem for functions of least gradient, in: Degenerate Diffusions (Minneapolis, MN, 1991), IMA Vol. Math. Appl., Springer, 47 (1993), 197–214. doi: 10.1007/978-1-4612-0885-3_14.

[19]

P. SternbergG. Williams and W. Ziemer, Existence, uniqueness, and regularity for functions of least gradient, J. Reine Angew. Math., 430 (1992), 35-60.

[20]

T. Sznigir, Various minimization problems involving the total variation in one dimension, PhD Rutgers University, Sept., 2017.

[21]

T. Sznigir, A one-dimensional problem involving the total variation, to appear.

[22]

J. L. Vázquez, Two nonlinear diffusion equation with finite speed of propagation, in: Problems Involving Change of Type, Stuttgart, 1988, Lecture Notes in Phys., Springer, 359 (1990), 197–206. doi: 10.1007/3-540-52595-5_96.

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