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Abstract. We discuss and compare various notions of weak solution for the p-Laplace equationand its parabolic counterpartIn 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 characterize p-harmonic functions in terms of an asymptotic mean value property. A p-harmonic function u is a viscosity solution to ∆ p u = div(|∇u| p−2 ∇u) = 0 with 1 < p ≤ ∞ in a domain Ω if and only if the expansionholds as ε → 0 for x ∈ Ω in a weak sense, which we call the viscosity sense. Here the coefficients α, β are determined by α + β = 1 and α/β = (p − 2)/(N + 2).

Abstract. Convex functions in Euclidean space can be characterized as universal viscosity subsolutions of all homogeneous fully nonlinear second order elliptic partial differential equations. This is the starting point we have chosen for a theory of convex functions on the Heisenberg group.

We consider functions that satisfy the identityHere ε > 0 and α, and β are suitable nonnegative coefficients such that α + β = 1. In particular, we show that these functions are uniquely determined by their boundary values, approximate p-harmonic functions for 2 ≤ p < ∞ (for a choice of p that depends on α and β), and satisfy the strong comparison principle. We also analyze their relation to the theory of tug-of-war games with noise.

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