July  2009, 8(4): 1203-1229. doi: 10.3934/cpaa.2009.8.1203

Weak solutions of linear degenerate parabolic equations and an application in image processing

1. 

Center for Industrial Mathematics Fachbereich 3, University of Bremen, Postfach 33 04 40, D-28334 Bremen, Germany

Received  July 2008 Revised  December 2008 Published  March 2009

In this paper, linear degenerate parabolic diffusion equations of second order with discontinuous coefficients are studied with respect to existence and uniqueness of weak solutions. We consider the full degenerate case where the diffusion is given by a tensor field which is only positive semi-definite and essentially bounded in the whole domain. Existence of solutions in Hilbert spaces incorporating the diffusion tensor is proven and uniqueness in a certain sense is established. Moreover, we examine replacements for the missing compactness by the Lions-Aubin lemma, proving that the set of solutions associated with bounded data and bounded semi-definite coefficients is weakly relatively compact in a space of weakly continuous functions. Finally, an application to the image-processing problem of edge-preserving denoising is presented. A method based on the considered equations is introduced and numerical examples are given.
Citation: Kristian Bredies. Weak solutions of linear degenerate parabolic equations and an application in image processing. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1203-1229. doi: 10.3934/cpaa.2009.8.1203
[1]

Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781

[2]

Hiroshi Watanabe. Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 177-189. doi: 10.3934/dcdss.2014.7.177

[3]

Pierpaolo Soravia. Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients. Communications on Pure & Applied Analysis, 2006, 5 (1) : 213-240. doi: 10.3934/cpaa.2006.5.213

[4]

Kenneth Hvistendahl Karlsen, Nils Henrik Risebro. On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1081-1104. doi: 10.3934/dcds.2003.9.1081

[5]

Gui-Qiang Chen, Kenneth Hvistendahl Karlsen. Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients. Communications on Pure & Applied Analysis, 2005, 4 (2) : 241-266. doi: 10.3934/cpaa.2005.4.241

[6]

Zhigang Wang, Lei Wang, Yachun Li. Renormalized entropy solutions for degenerate parabolic-hyperbolic equations with time-space dependent coefficients. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1163-1182. doi: 10.3934/cpaa.2013.12.1163

[7]

Renhai Wang, Yangrong Li. Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4145-4167. doi: 10.3934/dcdsb.2019054

[8]

Weihong Guo, Jing Qin. A geometry guided image denoising scheme. Inverse Problems & Imaging, 2013, 7 (2) : 499-521. doi: 10.3934/ipi.2013.7.499

[9]

M. Sango. Weak solutions for a doubly degenerate quasilinear parabolic equation with random forcing. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 885-905. doi: 10.3934/dcdsb.2007.7.885

[10]

Luisa Moschini, Guillermo Reyes, Alberto Tesei. Nonuniqueness of solutions to semilinear parabolic equations with singular coefficients. Communications on Pure & Applied Analysis, 2006, 5 (1) : 155-179. doi: 10.3934/cpaa.2006.5.155

[11]

Takesi Fukao, Masahiro Kubo. Nonlinear degenerate parabolic equations for a thermohydraulic model. Conference Publications, 2007, 2007 (Special) : 399-408. doi: 10.3934/proc.2007.2007.399

[12]

Young-Sam Kwon. Strong traces for degenerate parabolic-hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1275-1286. doi: 10.3934/dcds.2009.25.1275

[13]

Jiebao Sun, Boying Wu, Jing Li, Dazhi Zhang. A class of doubly degenerate parabolic equations with periodic sources. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1199-1210. doi: 10.3934/dcdsb.2010.14.1199

[14]

Wenjun Wang, Lei Yao. Spherically symmetric Navier-Stokes equations with degenerate viscosity coefficients and vacuum. Communications on Pure & Applied Analysis, 2010, 9 (2) : 459-481. doi: 10.3934/cpaa.2010.9.459

[15]

Takahiro Hashimoto. Nonexistence of weak solutions of quasilinear elliptic equations with variable coefficients. Conference Publications, 2009, 2009 (Special) : 349-358. doi: 10.3934/proc.2009.2009.349

[16]

Fangfang Dong, Yunmei Chen. A fractional-order derivative based variational framework for image denoising. Inverse Problems & Imaging, 2016, 10 (1) : 27-50. doi: 10.3934/ipi.2016.10.27

[17]

Wei Zhu, Xue-Cheng Tai, Tony Chan. Augmented Lagrangian method for a mean curvature based image denoising model. Inverse Problems & Imaging, 2013, 7 (4) : 1409-1432. doi: 10.3934/ipi.2013.7.1409

[18]

Qiang Liu, Zhichang Guo, Chunpeng Wang. Renormalized solutions to a reaction-diffusion system applied to image denoising. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1839-1858. doi: 10.3934/dcdsb.2016025

[19]

Feishe Chen, Lixin Shen, Yuesheng Xu, Xueying Zeng. The Moreau envelope approach for the L1/TV image denoising model. Inverse Problems & Imaging, 2014, 8 (1) : 53-77. doi: 10.3934/ipi.2014.8.53

[20]

Serena Dipierro, Aram Karakhanyan, Enrico Valdinoci. Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6073-6090. doi: 10.3934/dcds.2018262

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]