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

Yoshihiro Mizuta (1984)

Annales de l'institut Fourier

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...

On the boundary values of harmonic functions.

Paul R. Garabedian (1985)

Revista Matemática Iberoamericana

Over the years many methods have been discovered to prove the existence of a solution of the Dirichlet problem for Laplace's equation. A fairly recent collection of proofs is based on representations of the Green's functions in terms of the Bergman kernel function or some equivalent linear operator [3]. Perhaps the most fundamental of these approaches involves the projection of an arbitrary function onto the class of harmonic functions in a Hilbert space whose norm is defined by the Dirichlet integral...

On the Choquet integrals associated to Bessel capacities

Keng Hao Ooi (2022)

Czechoslovak Mathematical Journal

We characterize the Choquet integrals associated to Bessel capacities in terms of the preduals of the Sobolev multiplier spaces. We make use of the boundedness of local Hardy-Littlewood maximal function on the preduals of the Sobolev multiplier spaces and the minimax theorem as the main tools for the characterizations.

On the continuity of degenerate n-harmonic functions

Flavia Giannetti, Antonia Passarelli di Napoli (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We study the regularity of finite energy solutions to degenerate n-harmonic equations. The function K(x), which measures the degeneracy, is assumed to be subexponentially integrable, i.e. it verifies the condition exp(P(K)) ∈ Lloc1. The function P(t) is increasing on  [0,∞[  and satisfies the divergence condition ∫ 1 ∞ P ( t ) t 2 d t = ∞ . ∫ 1 ∞ P ( t ) t 2   d t = ∞ .

On the continuity of degenerate n-harmonic functions

Flavia Giannetti, Antonia Passarelli di Napoli (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We study the regularity of finite energy solutions to degenerate n-harmonic equations. The function K(x), which measures the degeneracy, is assumed to be subexponentially integrable, i.e. it verifies the condition exp(P(K)) ∈ Lloc1. The function P(t) is increasing on  [0,∞[  and satisfies the divergence condition ∫ 1 ∞ P ( t ) t 2   d t = ∞ .

On the dimension of p -harmonic measure in space

John L. Lewis, Kaj Nyström, Andrew Vogel (2013)

Journal of the European Mathematical Society

Let Ω ⊂ ℝ n , n ≥ 3 , and let p , 1 < p < ∞ , p ≠ 2 , be given. In this paper we study the dimension of p -harmonic measures that arise from non-negative solutions to the p -Laplace equation, vanishing on a portion of ∂ Ω , in the setting of δ -Reifenberg flat domains. We prove, for p ≥ n , that there exists δ ˜ = δ ˜ ( p , n ) > 0 small such that if Ω is a δ -Reifenberg flat domain with δ < δ ˜ , then p -harmonic measure is concentrated on a set of σ -finite H n − 1 -measure. We prove, for p ≥ n , that for sufficiently flat Wolff snowflakes the Hausdorff dimension of p -harmonic measure...

On the Dirichlet problem associated with the Dunkl Laplacian

Mohamed Ben Chrouda (2016)

Annales Polonici Mathematici

This paper deals with the questions of the existence and uniqueness of a solution to the Dirichlet problem associated with the Dunkl Laplacian Δ k as well as the hypoellipticity of Δ k on noninvariant open sets.

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