Regularity for solutions of nonlocal parabolic equations II

Abstract

Abstract. We prove boundary regularity and a compactness result for parabolic nonlocal equations of the form ut − Iu = f , where the operator I is not necessarily translation invariant. As a consequence of this and the regularity results for translation invariant case, we obtain C 1,α interior estimates in space for non translation invariant operators under some hypothesis on the time regularity of the boundary data.

Further Time Regularity for Nonlocal, Fully Nonlinear Parabolic Equations

“…When P .0; 1/, the case of bounded data was treated by J. Serra in [22] and, as in our case, proves a Hölder estimate in time for every exponent less than 1. When P .1; 2/, as in our theorem, the best-known results assert that (for bounded data) u is Hölder-continuous in time for some exponent slightly bigger than 1= (see [8,22]), which is substantially weaker than our result when is away from 1.…”

“…We may therefore extract a subsequence u " 3 u locally uniformly, and u h g on ¢ f 1g. Applying (2) and a standard argument about viscosity solutions, we obtain that I u h f on ¢ 1; 0; the details may be found in [8].…”

“…This allowed them to recover uniform estimates as the order 3 2 , extending the second-order theory of fully nonlinear equations to nonlocal problems. In the parabolic setting, the first author of this paper in collaboration with G. Dávila considered in [6][7][8] the corresponding parabolic estimates by adapting the strategies from [3][4][5]. We discuss further developments of the theory in the following paragraphs, in particular those closely related with the present work.…”

Optimal existence and uniqueness theory for the fractional heat equation

“…For the case of fully nonlinear (i.e. in non-divergence form) nonlocal equations we refer to [15,16] for Hölder continuity results. In the case of (possibly degenerate/singular)nonlinear diffusion, it is a difficult problem to prove full regularity.…”

Integro-partial differential equations with singular terminal condition

“…Proof. Recall that u n is jointly continuous in (t, x) and from (13), u n is bounded on [0, T − ε] × R d uniformly in n. But if the function involved in C9 is bounded, then we can take δ = 0 in Lemma 6. Thus, the problem ∂ t v + Lv + I(t, x, v) + f n (t, x, v, ∇vσ(t, x), B(t, x, v)) = 0, with terminal condition φ = u n (T − ε, .)…”