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We show that p-harmonic functions in the plane satisfy a nonlinear asymptotic mean value property for p > 1. This extends previous results of Manfredi and Lindqvist for certain range of p's. div(|∇u| p−2 ∇u) = 0

Let (X, d, µ) be a proper metric measure space and let Ω ⊂ X be a bounded domain. For each x ∈ Ω, we choose a radius 0 < ̺(x) ≤ dist(x, ∂Ω) and let Bx be the closed ball centered at x with radius ̺(x). If α ∈ R, consider the following operator in C(Ω),Under appropriate assumptions on α, X, µ and the radius function ̺ we show that solutions u ∈ C(Ω) of the functional equation Tαu = u satisfy a local Hölder or Lipschitz condition in Ω. The motivation comes from the so called p-harmonious functions in euclidean domains.

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We consider abstract inverse problems between infinite-dimensional Banach spaces. These inverse problems are typically nonlinear and ill-posed, making the inversion with limited and noisy measurements a delicate process. In this work, we assume that the unknown belongs to a finite-dimensional manifold: this assumption arises in many real-world scenarios where natural objects have a low intrinsic dimension and belong to a certain submanifold of a much larger ambient space. We prove uniqueness and Hölder and Lipschitz stability results in this general setting, also in the case when only a finite discretization of the measurements is available. Then, a Landweber-type reconstruction algorithm from a finite number of measurements is proposed, for which we prove global convergence, thanks to a new criterion for finding a suitable initial guess. These general results are then applied to several examples, including two classical nonlinear ill-posed inverse boundary value problems. The first is Calderón’s inverse conductivity problem, for which we prove a Lipschitz stability estimate from a finite number of measurements for piece-wise constant conductivities with discontinuities on an unknown triangle. A similar stability result is then obtained for Gel’fand-Calderón’s problem for the Schrödinger equation, in the case of piece-wise constant potentials with discontinuities on a finite number of non-intersecting balls.