Lipschitz regularity for censored subdiffusive integro-differential equations with superfractional gradient terms

Abstract

Abstract. 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.

A priori Lipschitz estimates for solutions of local and nonlocal Hamilton–Jacobi equations with Ornstein–Uhlenbeck operator

“…where Φ(x, y) = C 1 (φ(x) + φ(y) + A), φ = φ µ defined by (12). Set ϕ(x, y, t) = (ψ(|x − y|) + δ)Φ(x, y) − T −t .…”

“…Recall that φ µ is the growth function defined in (12). Throughout this paper we work with solutions which belong to these classes and for simplicity, we will write indifferently φ µ or φ, µ > 0 being fixed.…”

“…The restriction of the growth (12) when comparing with ( 11) is due to "bad" firstorder nonlinearities coming from the dependence of H with respect to x and the possible anistropy of the higher-order operators. These terms are delicate to treat in this unbounded setting.…”

“…These terms are delicate to treat in this unbounded setting. We do not know if the growth (12) is optimal.…”

“…In the nonlocal setting, an important work is Barles et al [7] where Ishii-Lions' method is extended for bounded solutions to strictly elliptic (in a suitable sense) integro-differential equations in the whole space. See also [8,12,10], and [11] for an extension of the Bernstein method in the nonlocal case with coercive Hamiltonian.…”