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December  2019, 39(12): 7013-7029. 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, 2019, 39 (12) : 7013-7029. 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.   Google Scholar
[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.  Google Scholar

[3]

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

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[14]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Reprint of the 1980 original. Classics in Applied Mathematics, 31. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

[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.  Google Scholar

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.   Google Scholar
[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.  Google Scholar

[3]

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

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[14]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Reprint of the 1980 original. Classics in Applied Mathematics, 31. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

[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.  Google Scholar

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