1,721,006 research outputs found
N=p-harmonic maps: Regularity for the sphere case
Full text linkWe introduce n/pα-harmonic maps as critical points of the energy En;pα (v)= Rn δ α /2 v pα where pointwise v W D n → SN-1, for the N-sphere SN-1 RN and pα D n/α . This energy combines the non-local behaviour of the fractional harmonic maps introduced by Rivière and the first author with the degenerate arguments of the n-Laplacian. In this setting, we will prove Hölder continuity. © 2014 de Gruyter
Partial Regularity for Stationary Solutions to Liouville-Type Equation in dimension 3
No full textIn dimension , we prove that the singular set of any stationary solution to the Liouville equation , which belongs to , has Hausdorff dimension at most
On the Bellman equation for infinite horizon problems with unbounded cost functional
No full textWe study a class of infinite horizon control problems for nonlinear systems, which includes
the Linear Quadratic (LQ) problem, using the Dynamic Programming
approach. Sufficient conditions for the regularity of the value function are given. The value function is compared with sub- and supersolutions of the Bellman equation and a uniqueness theorem is proved for this equation among locally Lipschitz functions bounded below. As an application it is showed that an optimal control
for the LQ problem is nearly optimal for a large class of small unbounded nonlinear
and non-quadratic pertubations of the same problem
Symmetry properties of viscosity solutions to nonlinear uniformly elliptic equations
No full text\begin{abstract}We study uniformly elliptic fully nonlinear
equations
and prove results of Gidas-Ni-Nirenberg type for positive
viscosity solutions of such equations. We show that symmetries of
the equation and the domain are reflected by the solution, both in
bounded and unbounded domains
On the Bellman equation for some unbounded control problems.
No full textWe consider a class of finite horizon optimal control problems with unbounded data
for nonlinear systems, which includes the Linear-Quadratic (LQ)
problem. We give comparison results between the value function and
viscosity sub- and supersolutions of the Bellman equation, and prove
uniqueness for this equation among locally Lipschitz functions bounded
below. As an application we show that an optimal control for the LQ
problem is nearly optimal for a large class of small unbounded nonlinear
and non-quadratic perturbations of the same problem
On the Generalized Dirichlet Problem for Viscous Hamilton-Jacobi Equations,
No full textAbstractWe study the Dirichlet problem for viscous Hamilton–Jacobi equations. Despite this type of equations seems to be uniformly elliptic, loss of boundary conditions may occur because of the strong nonlinearity of the first-order part and therefore the Dirichlet boundary condition has to be understood in the sense of viscosity solutions theory. Under natural assumptions on the initial and boundary data, we prove a Strong Comparison Result which allows us to obtain the existence and the uniqueness of a continuous solution which is defined globally in time
Finite Time Horizon Risk Sensitive Control and the RobustLimit under a Quadratic Growth Assumption
No full textThe finite time--horizon risk sensitive limit problem for
continuous, nonlinear systems is considered. Previous results are
extended to cover more typical examples. In particular, the cost
may grow quadratically, and the diffusion coefficient may depend
on the state. It is shown that the risk sensitive value function
is the solution of the corresponding dynamic programming equation.
It is also shown that this value converges to the value of the
Robust control problem as the cost becomes infinitely risk averse
with corresponding scaling of the diffusion coefficient
Free boundary minimal surfaces: a nonlocal approach
No full textGiven a smooth closed embedded manifold N and a compact connected smooth Riemannian surface (S, g) with boundary, we consider half-harmonic maps from the boundary of S to N. These maps are critical points of the nonlocal energy given by the Dirichlet energy of the harmonic extension of u in S. We express this energy as a sum of the half-energies at each boundary component, plus a quadratic term which is continuous in the smooth topology. We show the regularity of half-harmonic maps. We also establish a connection between free boundary minimal surfaces and critical points of E with respect to variations of the pair (map, metric), in terms of the Teichmüller space of S
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