Abstract. We discuss and compare various notions of weak solution for the p-Laplace equation −div(|∇u|p−2∇u) = 0 and its parabolic counterpart ut − div(|∇u|p−2∇u) = 0. In addition to the usual Sobolev weak solutions based on integration by parts, we consider the p-superharmonic (or p-superparabolic) functions from nonlinear potential theory and the viscosity solutions based on generalized pointwise derivatives (jets). Our main result states that in both the elliptic and the parabolic case, the viscosity supersolutions coincide with the potential-theoretic supersolutions.

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Abstract. We prove C1,γ regularity of Lipschitz free boundaries of two-phase problems for a class of homogeneous fully nonlinear elliptic operators F(D2u(x), x) with Hölder dependence on x, containing convex (concave) operators. 1. Introduction. In this paper

We study the comparison principle and interior Hölder continuity of viscosity solutions of F (x; u; Du;D 2 u) +H(x;Du) = f; where F satisfies the standard "structure condition" and H has a superlinear growth with respect to Du. Following Caffarelli, Crandall, Kocan and Swiech [3], we first present the comparison principle between L p-viscosity subsolutions and L p -strong supersolutions. We next show the interior Holder continuity for C-viscosity solutions of the above equation. For this purpose, modifying some arguments in [1] by Caffarelli, we obtain the Harnack inequality for them when the growth order of H with respect to Du is less than 2.