Displaying similar documents to “A counterexample to the L p -Hodge decomposition”

Dirichlet problem with L p -boundary data in contractible domains of Carnot groups

Andrea Bonfiglioli, Ermanno Lanconelli (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

Let be a sub-laplacian on a stratified Lie group G . In this paper we study the Dirichlet problem for with L p -boundary data, on domains Ω which are contractible with respect to the natural dilations of G . One of the main difficulties we face is the presence of non-regular boundary points for the usual Dirichlet problem for . A potential theory approach is followed. The main results are applied to study a suitable notion of Hardy spaces.

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

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

Annales de l'institut Fourier

Similarity:

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.

p Harmonic Measure in Simply Connected Domains

John L. Lewis, Kaj Nyström, Pietro Poggi-Corradini (2011)

Annales de l’institut Fourier

Similarity:

Let Ω be a bounded simply connected domain in the complex plane, . Let N be a neighborhood of Ω , let p be fixed, 1 < p < , and let u ^ be a positive weak solution to the p Laplace equation in Ω N . Assume that u ^ has zero boundary values on Ω in the Sobolev sense and extend u ^ to N Ω by putting u ^ 0 on N Ω . Then there exists a positive finite Borel measure μ ^ on with support contained in Ω and such that | u ^ | p - 2 u ^ , φ d A = - φ d μ ^ whenever φ C 0 ( N ) . If p = 2 and if u ^ is the Green function for Ω with pole at x Ω N ¯ then the...

Uniform bounds for quotients of Green functions on C 1 , 1 -domains

H. Hueber, M. Sieveking (1982)

Annales de l'institut Fourier

Similarity:

Let Δ u = Σ i 2 x i 2 , L u = Σ i , j a i j 2 x i x j u + Σ i b i x i u + c u be elliptic operators with Hölder continuous coefficients on a bounded domain Ω R n of class C 1 , 1 . There is a constant c > 0 depending only on the Hölder norms of the coefficients of L and its constant of ellipticity such that c - 1 G Δ Ω G L Ω c G Δ Ω on Ω × Ω , where γ Δ Ω (resp. G L Ω ) are the Green functions of Δ (resp. L ) on Ω .

Discrete Green's function and maximum principles

Vejchodský, Tomáš, Šolín, Pavel

Similarity:

In this paper the discrete Green’s function (DGF) is introduced and its fundamental properties are proven. Further it is indicated how to use these results to prove the discrete maximum principle for 1D Poisson equation discretized by the h p -FEM with pure Dirichlet or with mixed Dirichlet-Neumann boundary conditions and with piecewise constant coefficient.

Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents

Fanghua Lin, Tristan Rivière (1999)

Journal of the European Mathematical Society

Similarity:

There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a S 1 -valued function defined on the boundary of a bounded regular domain of R n . When such extensions do not exist, we use the Ginzburg-Landau relaxation procedure. We prove that, up to a subsequence, a sequence of Ginzburg-Landau minimizers, as the coupling parameter tends to infinity, converges to a unimodular harmonic map away from a codimension-2 minimal current minimizing the area within...

Landau's theorem for p-harmonic mappings in several variables

Sh. Chen, S. Ponnusamy, X. Wang (2012)

Annales Polonici Mathematici

Similarity:

A 2p-times continuously differentiable complex-valued function f = u + iv in a domain D ⊆ ℂ is p-harmonic if f satisfies the p-harmonic equation Δ p f = 0 , where p (≥ 1) is a positive integer and Δ represents the complex Laplacian operator. If Ω ⊂ ℂⁿ is a domain, then a function f : Ω m is said to be p-harmonic in Ω if each component function f i (i∈ 1,...,m) of f = ( f , . . . , f m ) is p-harmonic with respect to each variable separately. In this paper, we prove Landau and Bloch’s theorem for a class of p-harmonic mappings...

On the order of starlikeness and convexity of complex harmonic functions with a two-parameter coefficient condition

Agnieszka Sibelska (2010)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

The article of J. Clunie and T. Sheil-Small [3], published in 1984, intensified the investigations of complex functions harmonic in the unit disc Δ . In particular, many papers about some classes of complex mappings with the coefficient conditions have been published. Consideration of this type was undertaken in the period 1998–2004 by Y. Avci and E. Złotkiewicz [2], A. Ganczar [5], Z. J. Jakubowski, G. Adamczyk, A. Łazinska and A. Sibelska [1], [8], [7], H. Silverman [12] and J. M. Jahangiri...

Duality on vector-valued weighted harmonic Bergman spaces

Salvador Pérez-Esteva (1996)

Studia Mathematica

Similarity:

We study the duals of the spaces A p α ( X ) of harmonic functions in the unit ball of n with values in a Banach space X, belonging to the Bochner L p space with weight ( 1 - | x | ) α , denoted by L p α ( X ) . For 0 < α < p-1 we construct continuous projections onto A p α ( X ) providing a decomposition L p α ( X ) = A p α ( X ) + M p α ( X ) . We discuss the conditions on p, α and X for which A p α ( X ) * = A q α ( X * ) and M p α ( X ) * = M q α ( X * ) , 1/p+1/q = 1. The last equality is equivalent to the Radon-Nikodým property of X*.

A class of functions containing polyharmonic functions in ℝⁿ

V. Anandam, M. Damlakhi (2003)

Annales Polonici Mathematici

Similarity:

Some properties of the functions of the form v ( x ) = i = 0 m | x | i h i ( x ) in ℝⁿ, n ≥ 2, where each h i is a harmonic function defined outside a compact set, are obtained using the harmonic measures.