Displaying similar documents to “On the Martin compactification of a bounded Lipschitz domain in a riemannian manifold”

On the boundary limits of harmonic functions with gradient in L p

Yoshihiro Mizuta (1984)

Annales de l'institut Fourier

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This paper deals with tangential boundary behaviors of harmonic functions with gradient in Lebesgue classes. Our aim is to extend a recent result of Cruzeiro (C.R.A.S., Paris, 294 (1982), 71–74), concerning tangential boundary limits of harmonic functions with gradient in L n ( R + n ) , R + n denoting the upper half space of the n -dimensional euclidean space R n . Our method used here is different from that of Nagel, Rudin and Shapiro (Ann. of Math., 116 (1982), 331–360); in fact, we use the integral representation...

p -harmonic measure is not additive on null sets

José G. Llorente, Juan J. Manfredi, Jang-Mei Wu (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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When 1 < p < ∞ and p ≠ 2 the p -harmonic measure on the boundary of the half plane ℝ + 2 is not additive on null sets. In fact, there are finitely many sets E 1 , E 2 ,..., E κ in ℝ , of p -harmonic measure zero, such that E 1 ∪ E 2 ∪ . . . ∪ E κ = ℝ .

Hörmander systems and harmonic morphisms

Elisabetta Barletta (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Given a Hörmander system X = { X 1 , ⋯ , X m } on a domain Ω ⊆ 𝐑 n we show that any subelliptic harmonic morphism φ from Ω into a ν -dimensional riemannian manifold N is a (smooth) subelliptic harmonic map (in the sense of J. Jost & C-J. Xu, [9]). Also φ is a submersion provided that ν ≤ m and X has rank m . If Ω = 𝐇 n (the Heisenberg group) and X = 1 2 L α + L α ¯ , 1 2 i L α - L α ¯ , where L α ¯ = ∂ / ∂ z ¯ α - i z α ∂ / ∂ t is the Lewy operator, then a smooth map φ : Ω → N is a subelliptic harmonic morphism if and only if φ ∘ π : ( C ( 𝐇 n ) , F θ 0 ) → N is a harmonic morphism, where S 1 → C ( 𝐇 n ) → π → 𝐇 n is the canonical circle bundle and F θ 0 ...

Comparisons of kernel functions boundary Harnack principle and relative Fatou theorem on Lipschitz domains

Jang-Mei G. Wu (1978)

Annales de l'institut Fourier

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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...

Behavior of biharmonic functions on Wiener's and Royden's compactifications

Y. K. Kwon, Leo Sario, Bertram Walsh (1971)

Annales de l'institut Fourier

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Let R be a smooth Riemannian manifold of finite volume, Δ its Laplace (-Beltrami) operator. Canonical direct-sum decompositions of certain subspaces of the Wiener and Royden algebras of R are found, and for biharmonic functions (those for which Δ Δ u = 0 ) the decompositions are related to the values of the functions and their Laplacians on appropriate ideal boundaries.

Embedding of open riemannian manifolds by harmonic functions

Robert E. Greene, H. Wu (1975)

Annales de l'institut Fourier

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Let M be a noncompact Riemannian manifold of dimension n . Then there exists a proper embedding of M into R 2 n + 1 by harmonic functions on M . It is easy to find 2 n + 1 harmonic functions which give an embedding. However, it is more difficult to achieve properness. The proof depends on the theorems of Lax-Malgrange and Aronszajn-Cordes in the theory of elliptic equations.