Viscosity solutions to second order partial differential equations on Riemannian manifolds

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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.
Final version: the domain of F in the equation F=0 has been changed in order to get more generality and simplicity in the definitions and assumptions, and several important misprints have been corrected

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Analysis of PDEs, Differential Geometry, 58J32, 49J52, 49L25, 35D05, 35J70

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http://salesiana.dossiersoluciones.com/handle/123456789/156760

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/0612742
 http://arxiv.org/abs/math/0612742

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arXiv