124 research outputs found
Hamilton-Jacobi Equations and Distance Functions on Riemannian Manifolds
The paper is concerned with the properties of the distance function from a
closed subset of a Riemannian manifold, with particular attention to the set of
singularities
Linear PDEs and eigenvalue problems corresponding to ergodic stochastic optimization problems on compact manifolds
We consider long term average or `ergodic' optimal control poblems with a
special structure: Control is exerted in all directions and the control costs
are proportional to the square of the norm of the control field with respect to
the metric induced by the noise. The long term stochastic dynamics on the
manifold will be completely characterized by the long term density and
the long term current density . As such, control problems may be
reformulated as variational problems over and . We discuss several
optimization problems: the problem in which both and are varied
freely, the problem in which is fixed and the one in which is fixed.
These problems lead to different kinds of operator problems: linear PDEs in the
first two cases and a nonlinear PDE in the latter case. These results are
obtained through through variational principle using infinite dimensional
Lagrange multipliers. In the case where the initial dynamics are reversible we
obtain the result that the optimally controlled diffusion is also
symmetrizable. The particular case of constraining the dynamics to be
reversible of the optimally controlled process leads to a linear eigenvalue
problem for the square root of the density process
A Note on the Strong Maximum Principle for Fully Nonlinear Equations on Riemannian Manifolds
We investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci\u2019s extremal operators, some singular operators such as those modeled on the p- and 1e-Laplacian, and mean curvature-type problems. As a byproduct, we establish new strong comparison principles for some second-order uniformly elliptic problems when the manifold has nonnegative sectional curvature
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