37 research outputs found
Uniform Equicontinuity for a family of Zero Order operators approaching the fractional Laplacian
In this paper we consider a smooth bounded domain and a
parametric family of radially symmetric kernels
such that, for each , its norm is finite but it blows
up as . Our aim is to establish an independent
modulus of continuity in , for the solution of the
homogeneous Dirichlet problem \begin{equation*} \left \{ \begin{array}{rcll} -
\I_\epsilon [u] \&=\& f \& \mbox{in} \ \Omega. \\ u \&=\& 0 \& \mbox{in} \
\Omega^c, \end{array} \right . \end{equation*} where
and the operator \I_\epsilon has the form \begin{equation*} \I_\epsilon[u](x)
= \frac12\int \limits_{\R^N} [u(x + z) + u(x - z) - 2u(x)]K_\epsilon(z)dz
\end{equation*} and it approaches the fractional Laplacian as .
The modulus of continuity is obtained combining the comparison principle with
the translation invariance of \I_\epsilon, constructing suitable barriers
that allow to manage the discontinuities that the solution may
have on . Extensions of this result to fully non-linear
elliptic and parabolic operators are also discussed
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
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
Existence, Uniqueness and Asymptotic Behavior for Nonlocal Parabolic Problems with Dominating Gradient Terms
In this paper we deal with the well-posedness of Dirichlet problems associated to nonlocal Hamilton-Jacobi parabolic equations in a bounded, smooth domain , in the case when the classical boundary condition may be lost. We address the problem for both coercive and noncoercive Hamiltonians: for coercive Hamiltonians, our results rely more on the regularity properties of the solutions, while noncoercive case are related to optimal control problems and the arguments are based on a careful study of the dynamics near the boundary of the domain. Comparison principles for bounded sub and supersolutions are obtained in the context of viscosity solutions with generalized boundary conditions, and consequently we obtain the existence and uniqueness of solutions in by the application of Perron's method. Finally, we prove that the solution of these problems converges to the solutions of the associated stationary problem as under suitable assumptions on the data
Lipschitz Regularity for Censored Subdiffusive Integro-Differential Equations with Superfractional Gradient Terms
In this paper we are interested in integro-differential elliptic and
parabolic equations involving nonlocal operators with order less than one, and
a gradient term whose coercivity growth makes it the leading term in the
equation. We obtain Lipschitz regularity results for the associated stationary
Dirichlet problem in the case when the nonlocality of the operator is confined
to the domain, feature which is known in the literature as censored
nonlocality. As an application of this result, we obtain strong comparison
principles which allow us to prove the well-posedness of both the stationary
and evolution problems, and steady/ergodic large time behavior for the
associated evolution problem
Nonlocal ergodic control problem in
We study the existence-uniqueness of solution to the ergodic
Hamilton-Jacobi equation and , where . We
show that the critical , defined as the infimum of all
attaining a non-negative supersolution, attains a nonnegative
solution . Under suitable conditions, it is also shown that is
the supremum of all for which a non-positive subsolution is possible.
Moreover, uniqueness of the solution , corresponding to , is also
established. Furthermore, we provide a probabilistic characterization that
determines the uniqueness of the pair in the class of all
solution pair with . Our proof technique involves both
analytic and probabilistic methods in combination with a new local Lipschitz
estimate obtained in this article