Calculus of Variations and Geometric Measure Theory
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G. Bellettini - M. Novaga

Barriers for a Class of Geometric Evolutions Problems

created on 04 Dec 1996
modified by novaga on 10 Nov 2018


Published Paper

Inserted: 4 dec 1996
Last Updated: 10 nov 2018

Journal: Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei, serie IX
Volume: VIII
Number: 2
Pages: 119-128
Year: 1997


We prove that the minimal barriers in the sense of De Giorgi are equivalent to the viscosity solutions for fully nonlinear parabolic geometric problems of the form $u_t + F(t, x,\nabla u, \nabla^2 u) =0$, under the assumptions on $F$ made by Giga-Goto-Ishii-Sato in a recent paper. More generally, we prove that the minimal barrier is the maximal between all viscosity subsolutions assuming a given initial datum. All results can be extended to the case in which $F$ is not degenerate elliptic, provided that also $F^+$, which is defined as the smallest degenerate elliptic function above $F$, satisfies the assumptions of Giga-Goto-Ishii-Sato.

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