Viscosity solutions to second order partial differential equations on Riemannian manifolds

Azagra, Daniel; Ferrera, Juan; Sanz, Beatriz

doi:10.1016/j.jde.2008.03.030

Cited by 40 publications

(88 citation statements)

References 11 publications

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“…The last equality in the above expression is proved in [5,Section 3]. Therefore, if M has sectional curvature bounded below by some constant −K 0 ≤ 0, we obtain from equation (5.5) and Proposition 3.2 that…”

Section: Comparisonmentioning

confidence: 73%

“…The result is proved in [9] for M i = R ni , i = 1, ..., k. As in the stationary case [5], we can reduce the problem to this situation by an adecuate composition with the exponential mappings. Let us give some details for completeness.…”

Section: Then We Have Thatmentioning

confidence: 82%

“…Another important ingredient of the proof of our main comparison result is the following Proposition, established in [5,Proposition 3.3].…”

Section: Some Technical Toolsmentioning

confidence: 99%

“…For instance, he proves that if noncompact initial data Γ 0 are allowed then one loses uniqueness of the generalized geometric evolution. In recent years, an interest has grown in the use of viscosity solutions of (first order) Hamilton-Jacobi equations defined on Riemannian manifolds (in relation to dynamical systems, to geometric problems, or from a theoretical point of view), see [32,11,12,4,31,10,23], but no second order theory, apart from Ilmanen's paper, has apparently been developed for parabolic equations (in the case of stationary, degenerate elliptic equations, such a study was recently started in [5]). …”

Section: Introductionmentioning

confidence: 99%

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Generalized motion of level sets by functions of their curvatures on Riemannian manifolds

2008

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Abstract. We consider the generalized evolution of compact level sets by functions of their normal vectors and second fundamental forms on a Riemannian manifold M . The level sets of a function u : M → R evolve in such a way whenever u solves an equation ut + F (Du, D 2 u) = 0, for some real function F satisfying a geometric condition. We show existence and uniqueness of viscosity solutions to this equation under the assumptions that M has nonnegative curvature, F is continuous off {Du = 0}, (degenerate) elliptic, and locally invariant by parallel translation. We then prove that this approach is geometrically consistent, hence it allows to define a generalized evolution of level sets by very general, singular functions of their curvatures. For instance, these assumptions on F are satisfied when F is given by the evolutions of level sets by their mean curvature (even in arbitrary codimension) or by their positive Gaussian curvature. We also prove that the generalized evolution is consistent with the classical motion by the corresponding function of the curvature, whenever the latter exists. When M is not of nonnegative curvature, the same results hold if one additionally requires that F is uniformly continuous with respect to D 2 u. Finally we give some counterexamples showing that several well known properties of the evolutions in R n are no longer true when M has negative sectional curvature.

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Section: Comparisonmentioning

confidence: 73%

Section: Then We Have Thatmentioning

confidence: 82%

“…Another important ingredient of the proof of our main comparison result is the following Proposition, established in [5,Proposition 3.3].…”

Section: Some Technical Toolsmentioning

confidence: 99%