We prove a C 1,α interior regularity theorem for fully nonlinear uniformly elliptic integro-differential equations without assuming any regularity of the kernel. We then give some applications to linear theory and higher regularity of a special class of nonlinear operators.These procedures give a bijection between the I and the J. All of the constructions for nonlocal operators carry over to J in the obvious ways (e.g uniform ellipticity, J µ,λ ). We use the notation N + for the J corresponding to the extremal operator M + . Let φ be a smooth radial cutoff function ≡ 1 on B 1/2 and supported on B 1 , and set φ ǫ (x) = φ(x/ǫ). Let η ǫ (x) be a standard mollifier; in other words, η ǫ integrates to 1, is smooth, positive, radial, and supported on a ball of radius ǫ. Given a nonlocal operator in form J, recall that J z is the operator frozen at z, i.e.
Abstract. We study a free boundary problem arising from the theory of thermal insulation. The outstanding feature of this set optimization problem is that the boundary of the set being optimized is not a level surface of a harmonic function, but rather a hypersurface along which a harmonic function satisfies a Robin condition. We show that minimal sets exist, satisfy uniform density estimates, and, under some geometric conditions, have "locally flat" boundaries.
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