Existence of infinitely many solutions for the one-dimensional Perona-Malik model

Calculus of Variations and Partial Differential Equations

Volumn 26, Issue 2, 2006, Pages 171-199

  Zhang, Kewei ^a  

^a UNIVERSITY OF SUSSEX   (United Kingdom)

Author keywords

Differential inclusion; Infinitely many solutions; One dimensional; Perona Malik model; Relaxation property

Indexed keywords

EID: 33645657795     PISSN: 09442669     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00526-005-0363-4     Document Type: Article

References (32)

1. Adams, R.A.: Sobolev spaces. Academic Press (1975)
2. Aubin, J.P., Cellina, A.: Differential inclusions: set-valued maps and viability theory. Springer-Verlag (1984)
3. Aubert, G., Kornprobst, P.: Mathematical problems in image processing. Partial differential equations and the calculus of variations. Applied Mathematical Sciences, vol. 147. New York, Springer (2002)
4. Alvarez, L., Guichard, F., Lions, P.L., Morel, S.M.: Axioms and fundamental equations of image processing. Arch. Rational Mech. Anal 123, 199-257 (1993)
5. Bellettini, G., Fusco, G.: A regularized Perona-Malik functional: some aspects of the gradient dynamics. Preprint (1993)
6. Ball, J.M., James, R.D.: Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100, 13-52 (1987)
7. Ball, J.M., James, R.D.: Proposed experimental tests of a theory of fine microstructures and the two-well problem. Phil. Royal Soc. Lon. 338A, 389-450 (1992)
8. Caselles, V., Morel, J. (eds): Special issue on partial differential equations and geometry-driven diffusion in image processing and analysis. IEEE Trans. Image Processing 7(3) (1998)
9. Dacorogna, B., Marcellini, P.: General existence theorems for Hamilton-Jacobi Equations in the scalar and vectorial cases. Acta Mathematica 178, 1-37 (1997)
10. Dacorogna, B., Marcellini, P.: Implicit partial differential equations. Progress in Nonlinear Differential Equations and their Applications, vol. 37. Birkhäuser (1999)
11. Dacorogna, B., Pisante, G.: A general existence theorem for differential inclusions in the vector valued case. Preprint (2004)
12. Esedoglu, S.: An analysis of the Perona-Malik scheme. Comm. Pure Appl. Math. 54, 1442-1487 (2001)
13. Friedman, A.: Partial differential equations of parabolic type. Prentice-Hall (1964)
14. Gromov, M.: Partial differential relations. Springer-Verlag (1986)
15. Höllig, K.: Existence of infinitely many solutions for a forward backward heat equation. Trans. Amer. Math. Soc. 278, 299-316 (1983)
16. Horstmann, D., Painter, K.J., Othmer, H.G.: Aggregation under local reinforcement, from lattice to continuum. Eur. J. Appl. Math. 15, 545-576 (2004)
17. Kichenassamy, S.: The Pernoa-Malik paradox. SLAM J. Appl. Math 57, 1328-1342 (1997)
18. Kirchheim, B.: Rigidity and Geometry of Microstructures. MPI for Mathematics in the Sciences Leipzig, Lecture notes (http://www.mis.mpg.de/ preprints/ln/lecturenote-1603.pdf) (2003)
19. Kawohl, B., Kutev, N.: Maximum and comparision principle for one-dimensional anisotropic diffusion. Math. Ann 311, 107-123 (1998)
20. Lieberman, G.M.: Second order parabolic differential equations. Singapore, London: World Scientific (1996)
21. Ladyzenskaya, O.A., Solonnikov, V.A., Uralceva, N.N.: Linear and quasilinear equations of parabolic type. Nauka, Moscow (1967)
22. Morini, M., Negri, M.: Mumford-Shah functional as Γ-limit of discrete Perona-Malik energies. Math. Models Methods Appl. Sci 13, 785-805 (2003)
23. Müller, S., Šverák, V.: Attainment results for the two-well problem by convex integration. In: Jost, J. (ed.) Geometric analysis and the calculus of variations, pp. 239-251. International Press (1996)
24. Müller, S., Šverák, V: Unexpected solutions of first and second order partial differential equations. Doc. Math. J. DMV, Extra Vol. ICM 98, pp. 691-702
25. Müller, S., Šverák, V.: Convex integration with constraints and applications to phase transitions and partial differential equations. J. Eur. Math. Soc. 1, 393-422 (1999)
26. Müller, S., Šverák, V.: Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. Math 157, 715-742 (2003)
27. Müller, S., Sychev, M.A.: Optimal existence theorems for nonhomogeneous differential inclusions. J. Funct. Anal. 181, 447-475 (2001)
28. Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell 12, 629-639 (1990)
29. Sapiro, G.: Geometric partial differential equations and image analysis. Cambridge: Cambridge University Press (2001)
30. Sychev, M.A.: Comparing two methods of resolving homogeneous differential inclusions. Calc. Var. PDEs 13, 213-229 (2001)
31. Taheri, S., Tang, Q., Zhang, K.: Young measure solutions and instability of the one-dimensional Perona-Malik equation. J. Math. Anal. Appl. 308, 467-490 (2005)
32. Weickert, J.: Anisotropic diffusion in image processing. ECMI Series, Teubner, Stuttgart (1998)