Viscosity solutions to second order partial differential equations on Riemannian manifolds

Details Viscosity solutions to second order partial differential equations on Riemannian manifolds Viscosity solutions to second order partial differential equations on Riemannian manifolds

2006-12-23

arxiv.org/abs/math/0612742v3

Mathematics

Analysis of PDEs

Final version: the domain of F in the equation F=0 has been changed in order to get more generality and simplicity in the defi

Scientific paper

We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations $F(x, u, du, d^{2}u)=0$ defined on a finite-dimensional Riemannian manifold $M$. Finest results (with hypothesis that require the function $F$ to be degenerate elliptic, that is nonincreasing in the second order derivative variable, and uniformly continuous with respect to the variable $x$) are obtained under the assumption that $M$ has nonnegative sectional curvature, while, if one additionally requires $F$ to depend on $d^{2}u$ in a uniformly continuous manner, then comparison results are established with no restrictive assumptions on curvature.

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