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

Comparison principle for unbounded viscosity solutions of degenerate elliptic PDEs with gradient superlinear terms

International audienceWe are concerned with fully nonlinear possibly degenerate elliptic partial differential equations (PDEs) with superlinear terms with respect to $Du$. We prove several comparison principles among viscosity solutions which may be unbounded under some polynomial-type growth conditions. Our main result applies to PDEs with convex superlinear terms but we also obtain some results in nonconvex cases. Applications to monotone systems of PDEs are given