236 research outputs found
Uniqueness results for convex Hamilton-Jacobi equations under growth conditions on data
Unbounded stochastic control problems may lead to Hamilton-Jacobi-Bellman
equations whose Hamiltonians are not always defined, especially when the
diffusion term is unbounded with respect to the control. We obtain existence
and uniqueness of viscosity solutions growing at most like at
infinity for such HJB equations and more generally for degenerate parabolic
equations with a superlinear convex gradient nonlinearity. If the corresponding
control problem has a bounded diffusion with respect to the control, then our
results apply to a larger class of solutions, namely those growing like
at infinity. This latter case encompasses some equations related
to backward stochastic differential equations
Gradient bounds for nonlinear degenerate parabolic equations and application to large time behavior of systems
We obtain new oscillation and gradient bounds for the viscosity solutions of
fully nonlinear degenerate elliptic equations where the Hamiltonian is a sum of
a sublinear and a superlinear part in the sense of Barles and Souganidis
(2001). We use these bounds to study the asymptotic behavior of weakly coupled
systems of fully nonlinear parabolic equations. Our results apply to some
"asymmetric systems" where some equations contain a sublinear Hamiltonian
whereas the others contain a superlinear one. Moreover, we can deal with some
particular case of systems containing some degenerate equations using a
generalization of the strong maximum principle for systems
Lipschitz regularity results for nonlinear strictly elliptic equations and applications
Most of lipschitz regularity results for nonlinear strictly elliptic
equations are obtained for a suitable growth power of the nonlinearity with
respect to the gradient variable (subquadratic for instance). For equations
with superquadratic growth power in gradient, one usually uses weak
Bernstein-type arguments which require regularity and/or convex-type
assumptions on the gradient nonlinearity. In this article, we obtain new
Lipschitz regularity results for a large class of nonlinear strictly elliptic
equations with possibly arbitrary growth power of the Hamiltonian with respect
to the gradient variable using some ideas coming from Ishii-Lions' method. We
use these bounds to solve an ergodic problem and to study the regularity and
the large time behavior of the solution of the evolution equation
Large time behavior for some nonlinear degenerate parabolic equations
We study the asymptotic behavior of Lipschitz continuous solutions of
nonlinear degenerate parabolic equations in the periodic setting. Our results
apply to a large class of Hamilton-Jacobi-Bellman equations. Defining S as the
set where the diffusion vanishes, i.e., where the equation is totally
degenerate, we obtain the convergence when the equation is uniformly parabolic
outside S and, on S, the Hamiltonian is either strictly convex or satisfies an
assumption similar of the one introduced by Barles-Souganidis (2000) for
first-order Hamilton-Jacobi equations. This latter assumption allows to deal
with equations with nonconvex Hamiltonians. We can also release the uniform
parabolic requirement outside S. As a consequence, we prove the convergence of
some everywhere degenerate second-order equations
Lipschitz regularity for integro-differential equations with coercive hamiltonians and application to large time behavior
In this paper, we provide suitable adaptations of the "weak version of
Bernstein method" introduced by the first author in 1991, in order to obtain
Lipschitz regularity results and Lipschitz estimates for nonlinear
integro-differential elliptic and parabolic equations set in the whole space.
Our interest is to obtain such Lipschitz results to possibly degenerate
equations, or to equations which are indeed "uniformly el-liptic" (maybe in the
nonlocal sense) but which do not satisfy the usual "growth condition" on the
gradient term allowing to use (for example) the Ishii-Lions' method. We treat
the case of a model equation with a superlinear coercivity on the gradient term
which has a leading role in the equation. This regularity result together with
comparison principle provided for the problem allow to obtain the ergodic large
time behavior of the evolution problem in the periodic setting
Uniqueness Results for Second Order Bellman-Isaacs Equations under Quadratic Growth Assumptions and Applications
In this paper, we prove a comparison result between semicontinuous viscosity
sub and supersolutions growing at most quadratically of second-order degenerate
parabolic Hamilton-Jacobi-Bellman and Isaacs equations. As an application, we
characterize the value function of a finite horizon stochastic control problem
with unbounded controls as the unique viscosity solution of the corresponding
dynamic programming equation
Regularity Results and Large Time Behavior for Integro-Differential Equations with Coercive Hamiltonians
In this paper we obtain regularity results for elliptic integro-differential
equations driven by the stronger effect of coercive gradient terms. This
feature allows us to construct suitable strict supersolutions from which we
conclude H\"older estimates for bounded subsolutions. In many interesting
situations, this gives way to a priori estimates for subsolutions. We apply
this regularity results to obtain the ergodic asymptotic behavior of the
associated evolution problem in the case of superlinear equations. One of the
surprising features in our proof is that it avoids the key ingredient which are
usually necessary to use the Strong Maximum Principle: linearization based on
the Lipschitz regularity of the solution of the ergodic problem. The proof
entirely relies on the H\"older regularity
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