A geometric interpretation of the heat equation with multivalued initial data

SIAM Journal on Mathematical Analysis

Volumn 27, Issue 4, 1996, Pages 932-958

  Evans, Lawrence C ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Heat equation; Level set method; Viscosity solutions

Indexed keywords

EID: 0030534516     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141094275439     Document Type: Article

References (25)

1. S. ANGENENT, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), pp. 79-96.
2. _, Nodal properties of solutions of parabolic equations, Rocky Mountain J. Math., 21 (1991), pp. 585-592.
3. -2(∂u/∂cursive Greek chi)] - (∂u/∂t) = 0, J. Math. Phys., 21 (1980), pp. 1019-1023.
4. _, Symmetries and Differential Equations, Springer-Verlag, Berlin, New York, Heidelberg, 1989.
5. A. M. BRUCKSTEIN AND R. KIMMEL, Shape from shading in level sets, Report CIS-9209, Computer Science Department, Technion, Haifa, Israel, 1992.
6. J. CAHN, C. HANDWERKER, AND J. TAYLOR, Geometric models of crystal growth, preprint.
7. C. CARATHEODORY, Calculus of Varieties of Partial Differential Equations of the First Order, Chelsea, New York, 1982.
8. Y. G. CHEN, Y. GIGA, AND S. GOTO, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), pp. 749-786.
9. L. C. EVANS AND J. SPRUCK, Motion of level sets by mean curvature I, J. Differential Geom., 33 (1991), pp. 635-681.
10. _, Motion of level sets by mean curvature II, Trans. Amer. Math. Soc., 330 (1992), pp. 321-332.
11. _, Motion of level sets by mean curvature III, J. Geom. Anal., 2 (1992), pp. 121-150.
12. _, Motion of level sets by mean curvature IV, J. Geom. Anal., 5 (1995), pp. 77-114.
13. H. FEDERER, Geometric Measure Theory, Springer-Verlag, Berlin, New York, Heidelberg, 1969.
14. -2Scursive Greek chi occurring in two-phase flow in porous media, SIAM J. Appl. Math., 42 (1982), pp. 318-332.
15. M. GAGE AND R. HAMILTON, The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), pp. 69-96.
16. M. GRAYSON, The heat equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), pp. 285-314.
17. H. ISHII, Degenerate parabolic PDE's with discontinuities and generalized evolutions of surfaces, Adv. Differential Equations , 1 (1996), pp. 51-72.
18. O. A. LADYZHENSKAYA, V. A. SOLONNIKOV, AND N. N. URAL'CEVA, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968.
19. J. OAKS, Singularities and self-intersections of curves evolving on surfaces, Indiana U. Math J., 43 (1994), pp. 959-981.
20. M. OHNUMA AND M.-H. SATO, Singular degenerate parabolic equations with applications to geometric evolutions, Differential Integral Equations, 6 (1993), pp. 1265-1280.
21. P. OLVER, Applications of Lie Groups to Differential Equations, Springer-Verlag, Berlin, New York, Heidelberg, 1986.
22. S. OSHER, A level set formulation for the solution of the Dirichlet problem for Hamilton-Jacobi equations, SIAM J. Math. Anal., 24 (1993), pp. 1145-1153.
23. S. OSHER AND J. SETHIAN, Fronts propagation with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), pp. 12-49.
24. M. SONER, Motion of a set by the curvature of its boundary, J. Differential Equations, 101 (1993), pp. 313-372.
25. G. B. WHITHAM, Linear and Nonlinear Waves, Wiley Interscience, New York, 1974.