Boundary limits of Green's potentials along curves II. Lipschitz domains

Jang-Mei Wu

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Shiying Zhao (1994)

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The following results concerning boundary behavior of subharmonic functions in the unit ball of $ℝ^{n}$ are generalized to nontangential accessible domains in the sense of Jerison and Kenig [7]: (i) The classical theorem of Littlewood on the radial limits. (ii) Ziomek’s theorem on the $L^{p}$ -nontangential limits. (iii) The localized version of the above two results and nontangential limits of Green potentials under a certain nontangential condition.

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Jang-Mei G. Wu (1978)

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On a Lipschitz domain $D$ in $R^{n}$ , three theorems on harmonic functions are proved. The first (boundary Harnack principle) compares two positive harmonic functions at interior points near an open subset of the boundary where both functions vanish. The second extends some familiar geometric facts about the Poisson kernel on a sphere to the Poisson kernel on $D$ . The third theorem, on non-tangential limits of quotient of two positive harmonic functions in $D$ , generalizes Doob’s relative Fatou...

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We show that a harmonic function which vanishes continuously on an open set of the boundary of a convex domain cannot have a normal derivative which vanishes on a subset of positive surface measure. We also prove a similar result for caloric functions vanishing on the lateral boundary of a convex cylinder.