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).

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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We characterise regular boundary points of the parabolic $p$-Laplacian in terms of a family of barriers, both when $p>2$ and $1

Abstract. We show to a general class of parabolic equations that every bounded superparabolic function is a weak supersolution and, in particular, has derivatives in a Sobolev sense. To this end, we establish various comparison principles between super-and subparabolic functions, and show that a pointwise limit of uniformly bounded weak supersolutions is a weak supersolution.