Boundary behaviour of positive harmonic functions on Lipschitz domains.
Carroll, Tom (2002)
Annales Academiae Scientiarum Fennicae. Mathematica
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Carroll, Tom (2002)
Annales Academiae Scientiarum Fennicae. Mathematica
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John L. Lewis, Andrew Vogel (1991)
Revista Matemática Iberoamericana
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Wen Sheng Wang (1995)
Revista Matemática Iberoamericana
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In any C domain, there is nonzero harmonic function C continuous up to the boundary such that the function and its gradient on the boundary vanish on a set of positive measure.
Martin Silverstein, Richard Wheeden (1971)
Studia Mathematica
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Vilhelm Adolfsson, Luis Escauriaza, Carlos Kenig (1995)
Revista Matemática Iberoamericana
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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.
I. Miyamoto, Minoru Yanagishita, H. Yoshida (2005)
Czechoslovak Mathematical Journal
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This paper shows that some characterizations of the harmonic majorization of the Martin function for domains having smooth boundaries also hold for cones.
Peter Lindqvist, Juan Manfredi (1997)
Revista Matemática de la Universidad Complutense de Madrid
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Frank Beatrous (1991)
Studia Mathematica
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Alano Ancona (1998)
Publicacions Matemàtiques
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Let L be a symmetric second order uniformly elliptic operator in divergence form acting in a bounded Lipschitz domain Ω of R and having Lipschitz coefficients in Ω. It is shown that the Rellich formula with respect to Ω and L extends to all functions in the domain D = {u ∈ H (Ω); L(u) ∈ L(Ω)} of L. This answers a question of A. Chaïra and G. Lebeau.