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.
We consider the Bernoulli one-phase free boundary problem in a domain Ω and show that the free boundary F is C 1,1/2 regular in a neighborhood of the fixed boundary ∂Ω. We achieve this by relating the behavior of F near ∂Ω to a Signorini-type obstacle problem.
Abstract. In this paper we extend previous results on the regularity of solutions of integro-differential parabolic equations. The kernels are non necessarily symmetric which could be interpreted as a non-local drift with the same order as the diffusion. We provide an Oscillation Lemma and a Harnack Inequality which can be used to prove higher regularity estimates.
We establish Hölder estimates for the time derivative of solutions of non-local parabolic equations under mild assumptions for the boundary data. As a consequence we are able to extend the Evans-Krylov estimate for rough kernels to parabolic equations.L. Caffarelli and L. Silvestre studied in a series of papers [2,4,3] the interior regularity of the elliptic problem Iu = f (x). Their approach adapted the Krylov-Safanov and Evans-Krylov theory for fully non-linear equations to the non-local setting. This allowed them to recover uniform estimates as the order σ → 2 − , extending the second order theory of fully non-linear equations to non-local problems. In the parabolic setting, the first author of this 1991 Mathematics Subject Classification. 35B45, 35B65, 35K55, 35R09.
In this work we demonstrate that a class of some one and two phase free boundary problems can be recast as nonlocal parabolic equations on a submanifold. The canonical examples would be onephase Hele Shaw flow, as well as its two-phase analog. We also treat nonlinear versions of both one and two phase problems. In the special class of free boundaries that are graphs over R d , we give a precise characterization that shows their motion is equivalent to that of a solution of a nonlocal (fractional), nonlinear parabolic equation for functions on R d . Our main observation is that the free boundary condition defines a nonlocal operator having what we call the Global Comparison Property. A consequence of the connection with nonlocal parabolic equations is that for free boundary problems arising from translation invariant elliptic operators in the positive and negative phases, one obtains, in a uniform treatment for all of the problems (one and two phase), a propagation of modulus of continuity for viscosity solutions of the free boundary flow.
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