A short proof of the C0,αC^{0,\alpha}--regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications

Recently I. Capuzzo Dolcetta, F. Leoni and A. Porretta obtain a very surprising regularity result for fully nonlinear, superquadratic, elliptic equations by showing that viscosity subsolutions of such equations are locally H\"older continuous, and even globally if the boundary of the domain is regular enough. The aim of this paper is to provide a simplified proof of their results, together with an interpretation of the regularity phenomena, some extensions and various applications

Uniqueness for unbounded solutions to stationary viscous Hamilton--Jacobi equations

We consider a class of stationary viscous Hamilton--Jacobi equations as \left\{\begin{array}{l} \la u-{\rm div}(A(x) \nabla u)=H(x,\nabla u)\mbox{in }\Omega, u=0{on}\partial\Omega\end{array} \right. where \la\geq 0, $A(x)$ is a bounded and uniformly elliptic matrix and $H(x,\xi)$ is convex in $\xi$ and grows at most like $|\xi|^q+f(x)$, with $1 < q < 2$ and f \in \elle {\frac N{q'}}. Under such growth conditions solutions are in general unbounded, and there is not uniqueness of usual weak solutions. We prove that uniqueness holds in the restricted class of solutions satisfying a suitable energy--type estimate, i.e. (1+|u|)^{\bar q-1} u\in \acca, for a certain (optimal) exponent $\bar q$. This completes the recent results in \cite{GMP}, where the existence of at least one solution in this class has been proved

On the regularizing effect for unbounded solutions of first-order Hamilton-Jacobi equations

We give a simplified proof of regularizing effects for first-order Hamilton-Jacobi Equations of the form $u\_t+H(x,t,Du)=0$ in $\R^N\times(0,+\infty)$ in the case where the idea is to first estimate $u\_t$. As a consequence, we have a Lipschitz regularity in space and time for coercive Hamiltonians and, for hypo-elliptic Hamiltonians, we also have an H\''older regularizing effect in space following a result of L. C. Evans and M. R. James

A PDE approach to large-time asymptotics for boundary-value problems for nonconvex Hamilton-Jacobi Equations

We investigate the large-time behavior of three types of initial-boundary value problems for Hamilton-Jacobi Equations with nonconvex Hamiltonians. We consider the Neumann or oblique boundary condition, the state constraint boundary condition and Dirichlet boundary condition. We establish general convergence results for viscosity solutions to asymptotic solutions as time goes to infinity via an approach based on PDE techniques. These results are obtained not only under general conditions on the Hamiltonians but also under weak conditions on the domain and the oblique direction of reflection in the Neumann case

Ergodic type problems and large time behaviour of unbounded solutions of Hamilton-Jacobi Equations

We study the large time behavior of Lipschitz continuous, possibly unbounded, viscosity solutions of Hamilton-Jacobi Equations in the whole space $\R^N$. The associated ergodic problem has Lipschitz continuous solutions if the analogue of the ergodic constant is larger than a minimal value $\lambda_{min}$. We obtain various large-time convergence and Liouville type theorems, some of them being of completely new type. We also provide examples showing that, in this unbounded framework, the ergodic behavior may fail, and that the asymptotic behavior may also be unstable with respect to the initial data

Some Homogenization Results for Non-Coercive Hamilton-Jacobi Equations

Recently, C. Imbert & R. Monneau study the homogenization of coercive Hamilton-Jacobi Equations with a $u/e$-dependence : this unusual dependence leads to a non-standard cell problem and, in order to solve it, they introduce new ideas to obtain the estimates on the oscillations of the solutions. In this article, we use their ideas to provide new homogenization results for ``standard'' Hamilton-Jacobi Equations (i.e. without a $u/e$-dependence) but in the case of {\it non-coercive Hamiltonians}. As a by-product, we obtain a simpler and more natural proof of the results of C. Imbert & R. Monneau, but under slightly more restrictive assumptions on the Hamiltonians

Local C0,αC^{0,\alpha} Estimates for Viscosity Solutions of Neumann-type Boundary Value Problems

In this article, we prove the local $C^{0,\alpha}$ regularity and provide $C^{0,\alpha}$ estimates for viscosity solutions of fully nonlinear, possibly degenerate, elliptic equations associated to linear or nonlinear Neumann type boundary conditions. The interest of these results comes from the fact that they are indeed regularity results (and not only a priori estimates), from the generality of the equations and boundary conditions we are able to handle and the possible degeneracy of the equations we are able to take in account in the case of linear boundary conditions

Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations

We obtain non-symmetric upper and lower bounds on the rate of convergence of general monotone approximation/numerical schemes for parabolic Hamilton Jacobi Bellman Equations by introducing a new notion of consistency. We apply our general results to various schemes including finite difference schemes, splitting methods and the classical approximation by piecewise constant controls

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