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\(p\)-harmonic measure is not additive on null sets. (English) Zbl 1105.31002

Let \(\Omega\) be some domain in \(\mathbb R^n\), \(n\geq 2\). Define for \(p\in (1,\infty)\), \(p\neq 2\), the \(p\)-Laplace operator \(L_p(u) := \text{div}(|\nabla u|^{p-2} \nabla u)\). A function is said to be \(p\)-harmonic if it is a weak solution of \(L_p(u)=0\). It is well-known that \(p\)-harmonic functions are locally of class \(C^{1,\alpha}\) with \(\alpha\) only depending on \(p\) and \(n\). Consider the corresponding \(p\)-harmonic measure of the set \(E\subset\partial\Omega\) at a point \(X\in\Omega\) which is given by \[ \omega_p(X,E,\Omega):=\inf\{v(X): v\in\mathcal C(E,\Omega)\} \] where \(\mathcal C(E,\Omega)\) denote all nonnegative \(p\)-superharmonic functions \(v\) on \(\Omega\) such that \[ \liminf_{\Omega\ni X\to \zeta} v(X)\geq \mathbf{1}_E(\zeta), \quad \zeta\in\partial \Omega. \] The main result of the paper states that, even in the case \(\Omega=\mathbb R_+^2\), \(\partial \Omega = \mathbb R \supset E\), the set-function \(E\mapsto \omega(X,E,\Omega)\) is not additive. In fact, there exist finitely many disjoint \(\omega_p\)-null sets \(E_j\subset\mathbb R\) such that \(\mathbb R = E_1\cup\cdots\cup E_k\). Consequently \(E\mapsto \omega(X,E,\Omega)\) cannot be a Choquet capacity. The proof of these facts relies on a (unpublished) result of T.Wolff (1984, Gap series constructions for the \(p\)-Laplacian), see also J. L.Lewis [Holomorphic functions and moduli I, Proc. Workshop, Berkeley/Calif. 1986, Publ., Math. Sci. Res. Inst. 10, 93–100 (1988; Zbl 0667.35016)] where a basic \(p\)-harmonic function was constructed which shows that the mean-value property fails in the nonlinear (i.e.\(p\neq 2\)) case. The last section of the paper contains a number of interesting open problems for the \(p\)-harmonic measure.

MSC:

31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
35J70 Degenerate elliptic equations
60G46 Martingales and classical analysis

Citations:

Zbl 0667.35016
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References:

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