The $\Gamma$-limit and the related gradient flow for singular perturbation functionals of Perona-Malik type
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- by G. Bellettini and G. Fusco PDF
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Abstract:
We consider a class of nonconvex functionals of the gradient in one dimension, which we regularize with a second order derivative term. After a proper rescaling, suggested by the associated dynamical problems, we show that the sequence $\{F_\nu \}$ of regularized functionals $\Gamma$-converges, as $\nu \to 0^+$, to a particular class of free-discontinuity functionals $\mathcal {F}$, concentrated on $SBV$ functions with finite energy and having only the jump part in the derivative. We study the singular dynamic associated with $\mathcal {F}$, using the minimizing movements method. We show that the minimizing movement starting from an initial datum with a finite number of discontinuities has jump positions fixed in space and whose number is nonincreasing with time. Moreover, there are a finite number of singular times at which there is a dropping of the number of discontinuities. In the interval between two subsequent singular times, the vector of the survived jumps is determined by the system of ODEs which expresses the $L^2$-gradient of the $\Gamma$-limit. Furthermore the minimizing movement turns out to be continuous with respect to the initial datum. Some properties of a minimizing movement starting from a function with an infinite number of discontinuities are also derived.References
- Giovanni Alberti and Stefan Müller, A new approach to variational problems with multiple scales, Comm. Pure Appl. Math. 54 (2001), no. 7, 761–825. MR 1823420, DOI 10.1002/cpa.1013
- Roberto Alicandro, Andrea Braides, and Maria Stella Gelli, Free-discontinuity problems generated by singular perturbation, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), no. 6, 1115–1129. MR 1664085, DOI 10.1017/S0308210500027256
- Luigi Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 19 (1995), 191–246 (Italian, with English and Italian summaries). MR 1387558
- Luigi Ambrosio, Camillo De Lellis, and Carlo Mantegazza, Line energies for gradient vector fields in the plane, Calc. Var. Partial Differential Equations 9 (1999), no. 4, 327–255. MR 1731470, DOI 10.1007/s005260050144
- Luigi Ambrosio, Nicola Fusco, and Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. MR 1857292
- L. Ambrosio, N. Gigli, G. Savaré: Gradient Flows in Metric Spaces and in the Wesserstein Space of Probability Measures, ETH Lectures in Mathematics, Birkhäuser, 2004.
- G. I. Barenblatt, M. Bertsch, R. Dal Passo, and M. Ughi, A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow, SIAM J. Math. Anal. 24 (1993), no. 6, 1414–1439. MR 1241152, DOI 10.1137/0524082
- G. I. Barenblatt, M. Bertsch, R. Dal Passo, V. M. Prostokishin, and M. Ughi, A mathematical model of turbulent heat and mass transfer in stably stratified shear flow, J. Fluid Mech. 253 (1993), 341–358. MR 1233902, DOI 10.1017/S002211209300182X
- G. Bellettini: On gradient flows of some non-convex functionals of Perona-Malik type, Oberwolfach Report 26/2004.
- G. Bellettini and G. Fusco, A regularized Perona-Malik functional: some aspects of the gradient dynamics, EQUADIFF 2003, World Sci. Publ., Hackensack, NJ, 2005, pp. 639–644. MR 2185104, DOI 10.1142/9789812702067_{0}106
- Giovanni Bellettini, Giorgio Fusco, and Nicola Guglielmi, A concept of solution and numerical experiments for forward-backward diffusion equations, Discrete Contin. Dyn. Syst. 16 (2006), no. 4, 783–842. MR 2257160, DOI 10.3934/dcds.2006.16.783
- G. Bellettini, M. Novaga, and E. Paolini, Global solutions to the gradient flow equation of a nonconvex functional, SIAM J. Math. Anal. 37 (2006), no. 5, 1657–1687. MR 2215602, DOI 10.1137/050625333
- Guy Bouchitté, Andrea Braides, and Giuseppe Buttazzo, Relaxation results for some free discontinuity problems, J. Reine Angew. Math. 458 (1995), 1–18. MR 1310950, DOI 10.1515/crll.1995.458.1
- G. Bouchitté, C. Dubs, and P. Seppecher, Regular approximation of free-discontinuity problems, Math. Models Methods Appl. Sci. 10 (2000), no. 7, 1073–1097. MR 1780150, DOI 10.1142/S0218202500000549
- Andrea Braides, Approximation of free-discontinuity problems, Lecture Notes in Mathematics, vol. 1694, Springer-Verlag, Berlin, 1998. MR 1651773, DOI 10.1007/BFb0097344
- Andrea Braides, $\Gamma$-convergence for beginners, Oxford Lecture Series in Mathematics and its Applications, vol. 22, Oxford University Press, Oxford, 2002. MR 1968440, DOI 10.1093/acprof:oso/9780198507840.001.0001
- R.J. Braun, J.W. Cahn, G.B. McFadden, A.A. Wheeler: Anisotropy of interfaces in an ordered alloy: a multiple order parameter model. Trans. Royal Soc. A 355 (1997), 1787–1832.
- Francine Catté, Pierre-Louis Lions, Jean-Michel Morel, and Tomeu Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal. 29 (1992), no. 1, 182–193. MR 1149092, DOI 10.1137/0729012
- Guido Cortesani, Sequences of non-local functionals which approximate free-discontinuity problems, Arch. Rational Mech. Anal. 144 (1998), no. 4, 357–402. MR 1656480, DOI 10.1007/s002050050121
- Guido Cortesani and Rodica Toader, A density result in SBV with respect to non-isotropic energies, Nonlinear Anal. 38 (1999), no. 5, Ser. B: Real World Appl., 585–604. MR 1709990, DOI 10.1016/S0362-546X(98)00132-1
- Gianni Dal Maso, An introduction to $\Gamma$-convergence, Progress in Nonlinear Differential Equations and their Applications, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1201152, DOI 10.1007/978-1-4612-0327-8
- Ennio De Giorgi, New problems on minimizing movements, Boundary value problems for partial differential equations and applications, RMA Res. Notes Appl. Math., vol. 29, Masson, Paris, 1993, pp. 81–98. MR 1260440
- E. De Giorgi: Congetture riguardanti alcuni problemi di evoluzione, Duke Math. J. 81 (1995), 255-268
- E. De Giorgi: Su alcuni problemi instabili legati alla teoria della visione, Atti del Convegno in onore di Carlo Ciliberto (Napoli, 1995), T. Bruno, P. Buonocore, L. Carbone, V. Esposito, eds., 91–98, La Città del Sole, Napoli, 1997.
- Ennio De Giorgi, Antonio Marino, and Mario Tosques, Problems of evolution in metric spaces and maximal decreasing curve, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 68 (1980), no. 3, 180–187 (Italian, with English summary). MR 636814
- Sophia Demoulini, Young measure solutions for a nonlinear parabolic equation of forward-backward type, SIAM J. Math. Anal. 27 (1996), no. 2, 376–403. MR 1377480, DOI 10.1137/S0036141094261847
- Selim Esedoḡlu, An analysis of the Perona-Malik scheme, Comm. Pure Appl. Math. 54 (2001), no. 12, 1442–1487. MR 1852979, DOI 10.1002/cpa.3008
- Francesca Fierro, Roberta Goglione, and Maurizio Paolini, Numerical simulations of mean curvature flow in the presence of a nonconvex anisotropy, Math. Models Methods Appl. Sci. 8 (1998), no. 4, 573–601. MR 1634826, DOI 10.1142/S0218202598000263
- Massimo Gobbino, Gradient flow for the one-dimensional Mumford-Shah functional, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 (1998), no. 1, 145–193 (1999). MR 1658873
- Massimo Gobbino, Minimizing movements and evolution problems in Euclidean spaces, Ann. Mat. Pura Appl. (4) 176 (1999), 29–48. MR 1746533, DOI 10.1007/BF02505987
- M. Gobbino: Entire solutions of the one-dimensional Perona-Malik equation, Comm. Partial Differential Equations 32 (2007), 719–743.
- M. E. Gurtin, H. M. Soner, and P. E. Souganidis, Anisotropic motion of an interface relaxed by the formation of infinitesimal wrinkles, J. Differential Equations 119 (1995), no. 1, 54–108. MR 1334488, DOI 10.1006/jdeq.1995.1084
- Klaus Höllig, Existence of infinitely many solutions for a forward backward heat equation, Trans. Amer. Math. Soc. 278 (1983), no. 1, 299–316. MR 697076, DOI 10.1090/S0002-9947-1983-0697076-8
- Dirk Horstmann, Kevin J. Painter, and Hans G. Othmer, Aggregation under local reinforcement: from lattice to continuum, European J. Appl. Math. 15 (2004), no. 5, 546–576. MR 2128611, DOI 10.1017/S0956792504005571
- D. Horstmann, B. Schweizer: Comparison of two solution concepts for forward-backward diffusion, Preprint (2006).
- Bernd Kawohl and Nikolai Kutev, Maximum and comparison principle for one-dimensional anisotropic diffusion, Math. Ann. 311 (1998), no. 1, 107–123. MR 1624275, DOI 10.1007/s002080050179
- Satyanad Kichenassamy, The Perona-Malik paradox, SIAM J. Appl. Math. 57 (1997), no. 5, 1328–1342. MR 1470926, DOI 10.1137/S003613999529558X
- Antonio Marino, Claudio Saccon, and Mario Tosques, Curves of maximal slope and parabolic variational inequalities on nonconvex constraints, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 16 (1989), no. 2, 281–330. MR 1041899
- Massimiliano Morini, Sequences of singularly perturbed functionals generating free-discontinuity problems, SIAM J. Math. Anal. 35 (2003), no. 3, 759–805. MR 2048406, DOI 10.1137/S0036141001395388
- Stefan Müller, Variational models for microstructure and phase transitions, Calculus of variations and geometric evolution problems (Cetraro, 1996) Lecture Notes in Math., vol. 1713, Springer, Berlin, 1999, pp. 85–210. MR 1731640, DOI 10.1007/BFb0092670
- M. Nitzberg, D. Mumford, and T. Shiota, Filtering, segmentation and depth, Lecture Notes in Computer Science, vol. 662, Springer-Verlag, Berlin, 1993. MR 1226232, DOI 10.1007/3-540-56484-5
- K. J. Painter, D. Horstmann, and H. G. Othmer, Localization in lattice and continuum models of reinforced random walks, Appl. Math. Lett. 16 (2003), no. 3, 375–381. MR 1961428, DOI 10.1016/S0893-9659(03)80060-5
- P. Perona, J. Malik: Scale space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell. 12 (1990), 629–639.
- P. I. Plotnikov, Passage to the limit with respect to viscosity in an equation with a variable direction of parabolicity, Differentsial′nye Uravneniya 30 (1994), no. 4, 665–674, 734 (Russian, with Russian summary); English transl., Differential Equations 30 (1994), no. 4, 614–622. MR 1299852
- M. Slemrod, Dynamics of measure valued solutions to a backward-forward heat equation, J. Dynam. Differential Equations 3 (1991), no. 1, 1–28. MR 1094722, DOI 10.1007/BF01049487
Additional Information
- G. Bellettini
- Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica 00133 Roma, Italy – and – INFN Laboratori Nazionali di Frascati, Italy
- Email: belletti@mat.uniroma2.it
- G. Fusco
- Affiliation: Dipartimento di Matematica, Università di L’Aquila, via Vetoio, loc. Coppito, 67100 l’Aquila, Italy
- MR Author ID: 70195
- Email: fusco@univaq.it
- Received by editor(s): December 1, 2004
- Received by editor(s) in revised form: September 22, 2006
- Published electronically: April 25, 2008
- Additional Notes: The authors gratefully acknowledge the hospitality and the support of the Centro De Giorgi of the Scuola Normale Superiore di Pisa, where this paper was completed. The first author gratefully acknowledges also the hospitality and the support of the Max Planck Institute for Gravitational Physics in Golm. The authors are also grateful to the referee for some useful comments.
- © Copyright 2008 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 360 (2008), 4929-4987
- MSC (2000): Primary 49J45, 35B25, 74H40, 74G65
- DOI: https://doi.org/10.1090/S0002-9947-08-04495-4
- MathSciNet review: 2403710