| Given a planar domain O and p ∈ (1, infinity), let Hf denote the p-harmonic extension to O of a boundary function f : ∂O → R (as defined via Perron's method). We show that if f is a piecewise continuous function, O is piecewise smooth near where f is discontinuous, and g is a bounded function on ∂O such that g = f on ∂O E where E ⊂ R2 is a countable set with op( E, O) = 0, then Hf = H g. In particular, this solves a problem posed by Baernstein in 1998, who asked the question for 0-1 valued F and F˜ on the unit circle; it extends work of Bjorn, Bjorn, and Shanmugalingam, who answered the question for 1 < p ≤ 2. A key step is to show that p-harmonic extensions approximately agree with harmonic extensions in a neighborhood of a jump discontinuity.;For a general bounded domain O ⊂ R2 with n ≥ 2, we will also give several invariance results under perturbations on countable subsets of ∂O. Most of results are new for p > n. The main tool is tug-of-war with noise which was introduced by Yuval Peres and Scott Sheffield in [30]. In particular, when f is a continuous function on ∂O and g is a function on ∂O such that g = f except a point, we provide a necessary and sufficient condition for Hf = Hg where Hf and Hg denote the Perron solutions of f and g, respectively. It turns out that the point of f ≠ g should be of p-harmonic measure zero with respect to O. As a consequence, we can show that E ⊂ ∂O is a countable set of p-harmonic measure zero if and only if every point of E is of p-harmonic measure zero. Therefore, the p-harmonic measure is subadditive on {E ⊂ O : o p(E, O) = 0 and E is countable}. |
