153 research outputs found
Lyapunov stabilizability of controlled diffusions via a superoptimality principle for viscosity solutions
We prove optimality principles for semicontinuous bounded viscosity solutions
of Hamilton-Jacobi-Bellman equations. In particular we provide a representation
formula for viscosity supersolutions as value functions of suitable obstacle
control problems. This result is applied to extend the Lyapunov direct method
for stability to controlled Ito stochastic differential equations. We define
the appropriate concept of Lyapunov function to study the stochastic open loop
stabilizability in probability and the local and global asymptotic
stabilizability (or asymptotic controllability). Finally we illustrate the
theory with some examples.Comment: 22 page
Convergence of nonlocal geometric flows to anisotropic mean curvature motion
We consider nonlocal curvature functionals associated with positive
interaction kernels, and we show that local anisotropic mean curvature
functionals can be retrieved in a blow-up limit from them. As a consequence, we
prove that the viscosity solutions to the rescaled nonlocal geometric flows
locally uniformly converge to the viscosity solution to the anisotropic mean
curvature motion. The result is achieved by combining a compactness argument
and a set-theoretic approach related to the theory of De Giorgi's barriers for
evolution equations.Comment: 19 page
The isoperimetric problem for nonlocal perimeters
We consider a class of nonlocal generalized perimeters which includes
fractional perimeters and Riesz type potentials. We prove a general
isoperimetric inequality for such functionals, and we discuss some
applications. In particular we prove existence of an isoperimetric profile,
under suitable assumptions on the interaction kernel.Comment: 17 p
Volume constrained minimizers of the fractional perimeter with a potential energy
We consider volume-constrained minimizers of the fractional perimeter with
the addition of a potential energy in the form of a volume inte- gral. Such
minimizers are solutions of the prescribed fractional curvature problem. We
prove existence and regularity of minimizers under suitable assumptions on the
potential energy, which cover the periodic case. In the small volume regime we
show that minimizers are close to balls, with a quantitative estimate
Liouville properties and critical value of fully nonlinear elliptic operators
We prove some Liouville properties for sub- and supersolutions of fully
nonlinear degenerate elliptic equations in the whole space. Our assumptions
allow the coefficients of the first order terms to be large at infinity,
provided they have an appropriate sign, as in Ornstein- Uhlenbeck operators. We
give two applications. The first is a stabilization property for large times of
solutions to fully nonlinear parabolic equations. The second is the solvability
of an ergodic Hamilton-Jacobi-Bellman equation that identifies a unique
critical value of the operator.Comment: 18 pp, to appear in J. Differential Equation
Symmetric self-shrinkers for the fractional mean curvature flow
We show existence of homothetically shrinking solutions of the fractional
mean curvature flow, whose boundary consists in a prescribed numbers of
concentric spheres. We prove that all these solutions, except from the ball,
are dynamically unstable.Comment: 12 page
Isoperimetric problems for a nonlocal perimeter of Minkowski type
We prove a quantitative version of the isoperimetric inequality for a non
local perimeter of Minkowski type. We also apply this result to study
isoperimetric problems with repulsive interaction terms, under convexity
constraints. We show existence of minimizers, and we describe the shape of
minimizers in certain parameter regimes
On a parabolic Hamilton-Jacobi-Bellman equation degenerating at the boundary
We derive the long time asymptotic of solutions to an evolutive
Hamilton-Jacobi-Bellman equation in a bounded smooth domain, in connection with
ergodic problems recently studied in \cite{bcr}. Our main assumption is an
appropriate degeneracy condition on the operator at the boundary. This
condition is related to the characteristic boundary points for linear operators
as well as to the irrelevant points for the generalized Dirichlet problem, and
implies in particular that no boundary datum has to be imposed. We prove that
there exists a constant such that the solutions of the evolutive problem
converge uniformly, in the reference frame moving with constant velocity ,
to a unique steady state solving a suitable ergodic problem.Comment: 12p
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