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J. Differential Equations 245 (2008) 307336 www.elsevier.com/locate/jde

Summary: J. Differential Equations 245 (2008) 307­336
Viscosity solutions to second order partial differential
equations on Riemannian manifolds
Daniel Azagra
, Juan Ferrera, Beatriz Sanz
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense, 28040 Madrid, Spain
Received 19 February 2007; revised 28 March 2008
Available online 7 May 2008
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,d2u) = 0 defined on a finite-dimensional
Riemannian manifold M. Finest results (with hypothesis that require the function F to be degenerate ellip-
tic, 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 d2u in a uniformly continuous manner, then comparison results are
established with no restrictive assumptions on curvature.
© 2008 Elsevier Inc. All rights reserved.
MSC: 58J32; 49J52; 49L25; 35D05; 35J70
Keywords: Degenerate elliptic second order PDEs; Hamilton­Jacobi equations; Viscosity solution; Riemannian


Source: Azagra Rueda, Daniel - Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid


Collections: Mathematics