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Minimal barriers for geometric evolutions. (English) Zbl 0882.35028
Minimal barriers have been introduced by De Giorgi in order to provide a notion of weak solution for partial differential equations as, for example, the mean curvature flow, which is suitable to describe the evolution even past singularities. The authors study general properties of minimal barriers for the evolution equation $$\partial u/\partial t+F(\nabla u, \nabla^2 u)=0$$. They prove that for lower semicontinuous $$F$$ local and global barriers are the same. Further, they show that in this case the minimal barrier coincides with that one where $$F$$ is replaced by $$F^+$$, the smallest degenerate elliptic function above $$F$$. One section is devoted to the joint and disjoint set property in terms of the function $$F$$.

##### MSC:
 35D10 Regularity of generalized solutions of PDE (MSC2000) 35K55 Nonlinear parabolic equations 35K40 Second-order parabolic systems
##### Keywords:
evolution past singularities; mean curvature flow
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##### References:
 [1] Ambrosio, L; Soner, H.-M, A level set approach to the evolution of surfaces of any codimension, J. differential geom., 43, 693-737, (1996) · Zbl 0868.35046 [2] Barles, G; Soner, H.-M; Souganidis, PE, Front propagation and phase field theory, SIAM J. control optim., 31, 439-469, (1993) · Zbl 0785.35049 [3] G. Bellettini, M. Novaga, Comparison results between minimal barriers and viscosity solutions for geometric evolutions · Zbl 0904.35041 [4] G. Bellettini, M. Novaga, Barriers for a class of geometric evolution problems, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei · Zbl 0990.35070 [5] Bellettini, G; Paolini, M, Two examples of fattening for the curvature flow with a driving force, Atti accad. naz. lincei cl. sci. fis. mat. natur. rend. lincei (9) mat. appl., 5, 229-236, (1994) · Zbl 0826.35051 [6] Bellettini, G; Paolini, M, Some results on minimal barriers in the sense of De Giorgi applied to driven motion by Mean curvature, Rend. accad. naz. sci. XL mem. mat. (5), 19, 43-67, (1995) · Zbl 0944.53039 [7] Chen, YG; Giga, Y; Goto, S, Uniqueness and existence of viscosity solutions of generalized Mean curvature flow equation, J. differential geom., 33, 749-786, (1991) · Zbl 0696.35087 [8] E. De Giorgi, New ideas in calculus of variations and geometric meaasure theory, Motion by Mean Curvature and Related Topics, Proc. of the Conference held in Trento, July 20-24, 1992, 63, 69, Walter de Gruyter, Berlin [9] E. De Giorgi, Barriers, boundaries, motion of manifolds, Conference held at Department of Mathematics of Pavia, March 18, 1994. [10] E. De Giorgi, November 4, 1993, Congetture riguardanti barriere, superfici minime, movimenti secondo la curvatura media, Lecce [11] Evans, LC; Spruck, J, Motion of level sets by Mean curvature. I, J. differential geom., 33, 635-681, (1991) · Zbl 0726.53029 [12] Giga, Y; Goto, S; Ishii, H; Sato, MH, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana univ. math. J., 40, 443-470, (1991) · Zbl 0836.35009 [13] T. Ilmanen, 1992, The level-set flow on a manifold, Proceedings of the 1990 Summer Inst. in Diff. Geom. Amer. Math. Soc. Providence, RI · Zbl 0759.53035 [14] Ilmanen, T, Generalized flow of sets by Mean curvature on a manifold, Indiana univ. math. J., 41, 671-705, (1992) · Zbl 0759.53035 [15] Ishii, H; Souganidis, PE, Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor, Tohoku math. J., 47, 227-250, (1995) · Zbl 0837.35066 [16] Soner, H.-M; Souganidis, P.E, Singularities and uniqueness of cylindrically symmetric surfaces moving by Mean curvature, Comm. partial differential equations, 18, 859-894, (1993) · Zbl 0804.53006 [17] White, B, The topology of hypersurfaces moving by Mean curvature, Comm. anal. geom., 3, 317-333, (1995) · Zbl 0858.58047
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