Uniform Equicontinuity for a family of Zero Order operators approaching the fractional Laplacian

In this paper we consider a smooth bounded domain $\Omega \subset \R^N$ and a parametric family of radially symmetric kernels $K_\epsilon: \R^N \to \R_+$ such that, for each $\epsilon \in (0,1)$, its $L^1-$norm is finite but it blows up as $\epsilon \to 0$. Our aim is to establish an $\epsilon$ independent modulus of continuity in ${\Omega}$, for the solution $u_\epsilon$ 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 $f \in C(\bar{\Omega})$ 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 $\epsilon\to 0$. 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 $u_\epsilon$ may have on $\partial \Omega$. 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 $\Omega$, 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 $C(\bar{\Omega} \times [0,+\infty))$ 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 $t \to +\infty$ under suitable assumptions on the data

Nonlocal ergodic control problem in Rd\mathbb{R}^d

We study the existence-uniqueness of solution $(u, \lambda)$ to the ergodic Hamilton-Jacobi equation $$(-\Delta)^s u + H(x, \nabla u) = f-\lambda\quad \text{in}\; \mathbb{R}^d,$$ and $u\geq 0$, where $s\in (\frac{1}{2}, 1)$. We show that the critical $\lambda=\lambda^*$, defined as the infimum of all $\lambda$ attaining a non-negative supersolution, attains a nonnegative solution $u$. Under suitable conditions, it is also shown that $\lambda^*$ is the supremum of all $\lambda$ for which a non-positive subsolution is possible. Moreover, uniqueness of the solution $u$, corresponding to $\lambda^*$, is also established. Furthermore, we provide a probabilistic characterization that determines the uniqueness of the pair $(u, \lambda^*)$ in the class of all solution pair $(u, \lambda)$ with $u\geq 0$. Our proof technique involves both analytic and probabilistic methods in combination with a new local Lipschitz estimate obtained in this article

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