205 research outputs found
Regularity theory for fully nonlinear integro-differential equations
We consider nonlinear integro-differential equations, like the ones that
arise from stochastic control problems with purely jump L\`evy processes. We
obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior
regularity for general fully nonlinear integro-differential
equations. Our estimates remain uniform as the degree of the equation
approaches two, so they can be seen as a natural extension of the regularity
theory for elliptic partial differential equations.Comment: Minor typos corrected, and some extra comments adde
Regularity properties of nonlocal minimal surfaces via limiting arguments
We prove an improvement of flatness result for nonlocal minimal surfaces
which is independent of the fractional parameter when .
As a consequence, we obtain that all the nonlocal minimal cones are flat and
that all the nonlocal minimal surfaces are smooth when the dimension of the
ambient space is less or equal than 7 and is close to 1
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