Geodesic
Barriers for a class of geometric evolution problems
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 8 (1997) no. 2, pp. 119-128.

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We present some general results on minimal barriers in the sense of De Giorgi for geometric evolution problems. We also compare minimal barriers with viscosity solutions for fully nonlinear geometric problems of the form \( u_{t} + F(t, x, \nabla u, \nabla^{2} u) = 0 \). If \( F \) is not degenerate elliptic, it turns out that we obtain the same minimal barriers if we replace \( F \) with \( F^{+} \), which is defined as the smallest degenerate elliptic function above \( F \).
Vengono presentati alcuni risultati di Carattere generale sulle minime barriere nel senso di De Giorgi per evoluzioni geometriche di insiemi. Vengono anche confrontate le minime barriere con le evoluzioni ottenute usando le soluzioni nel senso della viscosità, per problemi geometrici completamente non lineari della forma \( u_{t} + F(t, x, \nabla u, \nabla^{2} u) = 0 \). Se \( F \) non è ellittica degenere, si osserva che si ottengono le stesse minime barriere se, al posto di \( F \), si considera la funzione \( F^{+} \), definita come la più piccola funzione ellittica degenere maggiore o uguale a \( F \). 
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     title = {Barriers for a class of geometric evolution problems},
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Bellettini, Giovanni; Novaga, Matteo. Barriers for a class of geometric evolution problems. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 8 (1997) no. 2, pp. 119-128. http://geodesic.mathdoc.fr/item/RLIN_1997_9_8_2_a4/

[1] F. Almgren - J. E. Taylor - L. Wang, Curvature-driven flows: a variational approach. SIAM J. Control Optim., 31, 1993, 387-437. | DOI | MR | Zbl

[2] G. Bellettini - M. Novaga, Minimal barriers for geometric evolutions. Preprint Univ. Pisa, gennaio 1997; J. Diff. Eq., to appear. | DOI | MR | Zbl

[3] G. Bellettini - M. Novaga, Comparison results between minimal barriers and viscosity solutions for geometric evolutions. Preprint Univ. Pisa n. 2.252.998, ottobre 1996. | fulltext mini-dml | MR | Zbl

[4] G. Bellettini - M. Paolini, Some results on minimal barriers in the sense of De Giorgi applied to driven motion by mean curvature. Rend. Acc. Naz. Sci., XL Mem. Mat. (5), 19, 1995, 43-67. | MR | Zbl

[5] Y. G. Chen - Y. Giga - S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equation. J. Differential Geom., 33, 1991, 749-786. | fulltext mini-dml | MR | Zbl

[6] M. G. Crandall - H. Ishii - P.-L. Lions, Users guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.), 27, 1992, 1-67. | fulltext mini-dml | DOI | MR | Zbl

[7] E. De Giorgi, New problems on minimizing movements. In: J.-L. LIONS - C. BAIOCCHI (eds.), Boundary Value Problems for Partial Differential Equations and Applications. Vol. 29, Masson, Paris 1993. | MR | Zbl

[8] E. De Giorgi, Barriers, boundaries, motion of manifolds. Conference held at Dipartimento di Matematica of Pavia, March 18, 1994.

[9] E. De Giorgi, Congetture riguardanti alcuni problemi di evoluzione. Duke Math. J., 81, 1996, 255-268. | fulltext mini-dml | DOI | MR | Zbl

[10] L. C. Evans - J. Spruck, Motion of level sets by mean curvature. I. J. Differential Geom., 33, 1991, 635-681. | fulltext mini-dml | MR | Zbl

[11] Y. Giga - S. Goto - H. Ishii - M. H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ. Math. J., 40, 1991, 443-470. | DOI | MR | Zbl

[12] S. Goto, Generalized motions of hypersurfaces whose growth speed depends superlinearly on the curvature tensor. Differential Integral Equations, 7, 1994, 323-343. | MR | Zbl

[13] T. Ilmanen, The level-set flow on a manifold. In: R. GREENE - S. T. YAU (eds.), Proc. of the 1990 Summer Inst. in Diff. Geom. Amer. Math. Soc., 1992. | MR | Zbl

[14] T. Ilmanen, Elliptic Regularization and Partial Regularity for Motion by Mean Curvature. Memoirs of the Amer. Math. Soc., 250, 1994, 1-90. | MR | Zbl

[15] H. Ishii - P. E. Souganidis, Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor. Tohoku Math. J., 47, 1995, 227-250. | fulltext mini-dml | DOI | MR | Zbl

[16] H.-M. Soner, Motion of a set by the curvature of its boudnary. J. Differential Equations, 101, 1993, 313-372. | DOI | MR | Zbl