• Previous Article
    The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals
  • DCDS Home
  • This Issue
  • Next Article
    A nondegeneracy condition for a semilinear elliptic system and the existence of 1- bump solutions
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

[1]

H. Beirão da Veiga. Vorticity and regularity for flows under the Navier boundary condition. Communications on Pure & Applied Analysis, 2006, 5 (4) : 907-918. doi: 10.3934/cpaa.2006.5.907

[2]

M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473

[3]

Jaeyoung Byeon, Sangdon Jin. The Hénon equation with a critical exponent under the Neumann boundary condition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4353-4390. doi: 10.3934/dcds.2018190

[4]

Dorina Mitrea, Marius Mitrea, Sylvie Monniaux. The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1295-1333. doi: 10.3934/cpaa.2008.7.1295

[5]

Khadijah Sharaf. A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1691-1706. doi: 10.3934/dcds.2017070

[6]

Yueding Yuan, Zhiming Guo, Moxun Tang. A nonlocal diffusion population model with age structure and Dirichlet boundary condition. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2095-2115. doi: 10.3934/cpaa.2015.14.2095

[7]

Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations & Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325

[8]

Franco Obersnel, Pierpaolo Omari. Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 305-320. doi: 10.3934/dcds.2013.33.305

[9]

Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101

[10]

Shouming Zhou, Chunlai Mu, Yongsheng Mi, Fuchen Zhang. Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935

[11]

Larissa V. Fardigola. Transformation operators in controllability problems for the wave equations with variable coefficients on a half-axis controlled by the Dirichlet boundary condition. Mathematical Control & Related Fields, 2015, 5 (1) : 31-53. doi: 10.3934/mcrf.2015.5.31

[12]

Jian Zhai, Zhihui Cai. $\Gamma$-convergence with Dirichlet boundary condition and Landau-Lifshitz functional for thin film. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1071-1085. doi: 10.3934/dcdsb.2009.11.1071

[13]

Lan Zeng, Guoxi Ni, Yingying Li. Low Mach number limit of strong solutions for 3-D full compressible MHD equations with Dirichlet boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5503-5522. doi: 10.3934/dcdsb.2019068

[14]

Kenji Nakanishi, Hideo Takaoka, Yoshio Tsutsumi. Local well-posedness in low regularity of the MKDV equation with periodic boundary condition. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1635-1654. doi: 10.3934/dcds.2010.28.1635

[15]

Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194

[16]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[17]

R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497

[18]

Samia Challal, Abdeslem Lyaghfouri. The heterogeneous dam problem with leaky boundary condition. Communications on Pure & Applied Analysis, 2011, 10 (1) : 93-125. doi: 10.3934/cpaa.2011.10.93

[19]

Nicolas Van Goethem. The Frank tensor as a boundary condition in intrinsic linearized elasticity. Journal of Geometric Mechanics, 2016, 8 (4) : 391-411. doi: 10.3934/jgm.2016013

[20]

Wenzhen Gan, Peng Zhou. A revisit to the diffusive logistic model with free boundary condition. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 837-847. doi: 10.3934/dcdsb.2016.21.837

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (187)
  • HTML views (273)
  • Cited by (0)

Other articles
by authors

[Back to Top]