p Harmonic Measure in Simply Connected Domains

• Let Omega be a bounded simply connected domain in the complex plane, C. Let N be a neighborhood of partial derivative Omega, let p be fixed, 1 < p < infinity, and let (u) over cap be a positive weak solution to the p Laplace equation in Omega boolean AND N. Assume that (u) over cap has zero boundary values on partial derivative Omega in the Sobolev sense and extend (u) over cap to N \ Omega by putting 11 E 0 on N Then there exists a positive finite Borel measure (mu) over cap on C with support contained in partial derivative Omega and such that integral vertical bar del(u) over cap vertical bar(p-2) dA = - integral phi d (mu) over cap whenever phi is an element of C(0)(infinity)(N). If p = 2 and if (u) over cap is the Green function for Omega with pole at x is an element of Omega\(N) over bar then the measure (mu) over cap coincides with harmonic measure at x, omega = omega(x), associated to the Laplace equation. In this paper we continue the studies initiated by the first author by establishing new results, in simply connected domains, concerning the Hausdorff dimension of the support of the measure (mu) over cap. In particular, we prove results, for 1 < p < infinity, p not equal 2, reminiscent of the famous result of Makarov concerning the Hausdorff dimension of the support of harmonic measure in simply connected domains.

Ämnesord

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Nyckelord

Harmonic function

harmonic measure

p harmonic measure

p harmonic function

simply connected domain

Hausdorff measure

Hausdorff dimension

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