Inserted: 4 dec 1996
Last Updated: 10 nov 2018
Journal: Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei, serie IX
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.