On the Dirichlet problem in the Cegrell classes

Rafał Czyż; Per Åhag

Displaying similar documents to “On the Dirichlet problem in the Cegrell classes”

The comparison principle and Dirichlet problem in the class $_{p} (f)$ , p > 0

Pham Hoang Hiep (2006)

Annales Polonici Mathematici

Similarity:

We establish the comparison principle in the class $_{p} (f)$ . The result obtained is applied to the Dirichlet problem in $_{p} (f)$ .

On a Monge-Ampère type equation in the Cegrell class $_{χ}$

Rafał Czyż (2010)

Annales Polonici Mathematici

Similarity:

Let Ω be a bounded hyperconvex domain in ℂn and let μ be a positive and finite measure which vanishes on all pluripolar subsets of Ω. We prove that for every continuous and strictly increasing function χ:(-∞,0) → (-∞,0) there exists a negative plurisubharmonic function u which solves the Monge-Ampère type equation $- χ (u) (d d^{c} u) ⁿ = d μ$ . Under some additional assumption the solution u is uniquely determined.

A regularity theorem for the complex Monge-Ampère equation in ℂPⁿ

Sławomir Kołodziej (2003)

Annales Polonici Mathematici

Similarity:

$C^{1, 1}$ regularity of the solutions of the complex Monge-Ampère equation in ℂPⁿ with the n-root of the right hand side in $C^{1, 1}$ is proved.

The Bohr inequality for ordinary Dirichlet series

R. Balasubramanian, B. Calado, H. Queffélec (2006)

Studia Mathematica

Similarity:

We extend to the setting of Dirichlet series previous results of H. Bohr for Taylor series in one variable, themselves generalized by V. I. Paulsen, G. Popescu and D. Singh or extended to several variables by L. Aizenberg, R. P. Boas and D. Khavinson. We show in particular that, if $f (s) = \sum_{n = 1}^{\infty} a ₙ n^{- s}$ with ${| | f | |}_{\infty} : = s u p_{ℜ s > 0} | f (s) | < \infty$ , then $\sum_{n = 1}^{\infty} | a ₙ | n^{- 2} {\leq | | f | |}_{\infty}$ and even slightly better, and $\sum_{n = 1}^{\infty} | a ₙ | n^{- 1 / 2} {\leq C | | f | |}_{\infty}$ , C being an absolute constant.

On Dirichlet type spaces on the unit ball of $ℂ^{n}$

Małgorzata Michalska (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

In this paper we discuss characterizations of Dirichlet type spaces on the unit ball of $ℂ^{n}$ obtained by P. Hu and W. Zhang [2], and S. Li [4].

On the Dirichlet problem associated with the Dunkl Laplacian

Mohamed Ben Chrouda (2016)

Annales Polonici Mathematici

Similarity:

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.

The Dirichlet-Bohr radius

Daniel Carando, Andreas Defant, Domingo A. Garcí, Manuel Maestre, Pablo Sevilla-Peris (2015)

Acta Arithmetica

Similarity:

Denote by Ω(n) the number of prime divisors of n ∈ ℕ (counted with multiplicities). For x∈ ℕ define the Dirichlet-Bohr radius L(x) to be the best r > 0 such that for every finite Dirichlet polynomial $\sum_{n \leq x} a_{n} n^{- s}$ we have $\sum_{n \leq x} | a_{n} | r^{Ω (n)} \leq s u p_{t \in ℝ} | \sum_{n \leq x} a_{n} n^{- i t} |$ . We prove that the asymptotically correct order of L(x) is ${(l o g x)}^{1 / 4} x^{- 1 / 8}$ . Following Bohr’s vision our proof links the estimation of L(x) with classical Bohr radii for holomorphic functions in several variables. Moreover, we suggest a general setting which allows translating various results...

On the Monge-Ampere differential equation $r t - s^{2} + λ^{2} (x, y) = 0$

B. N. Rachajsky (1969)

Matematički Vesnik

Similarity:

A priori estimates for weak solutions of complex Monge-Ampère equations

Slimane Benelkourchi, Vincent Guedj, Ahmed Zeriahi (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

Let $X$ be a compact Kähler manifold and $ω$ be a smooth closed form of bidegree $(1, 1)$ which is nonnegative and big. We study the classes $ℰ_{χ} (X, ω)$ of $ω$ -plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight $χ$ has fast growth at infinity, the corresponding functions are close to be bounded. We show that if a positive Radon measure is suitably dominated by the Monge-Ampère capacity, then it belongs to the range of the Monge-Ampère operator on some class $ℰ_{χ} (X, ω)$ . This is done by...

Hölder regularity for solutions to complex Monge-Ampère equations

Mohamad Charabati (2015)

Annales Polonici Mathematici

Similarity:

We consider the Dirichlet problem for the complex Monge-Ampère equation in a bounded strongly hyperconvex Lipschitz domain in ℂⁿ. We first give a sharp estimate on the modulus of continuity of the solution when the boundary data is continuous and the right hand side has a continuous density. Then we consider the case when the boundary value function is $^{1, 1}$ and the right hand side has a density in $L^{p} (Ω)$ for some p > 1, and prove the Hölder continuity of the solution.

Boundary values of functions in Cegrell’s class $_{ψ}$

Pham Hoang Hiep (2007)

Annales Polonici Mathematici

Similarity:

We study boundary values of functions in Cegrell’s class $_{ψ}$ .

Dirichlet forms on quotients of shift spaces

Manfred Denker, Atsushi Imai, Susanne Koch (2007)

Colloquium Mathematicae

Similarity:

We define thin equivalence relations ∼ on shift spaces $^{\infty}$ and derive Dirichlet forms on the quotient space $Σ =^{\infty} / \sim$ in terms of the nearest neighbour averaging operator. We identify the associated Laplace operator. The conditions are applied to some non-self-similar extensions of the Sierpiński gasket.

Isometric composition operators on weighted Dirichlet space

Shi-An Han, Ze-Hua Zhou (2016)

Czechoslovak Mathematical Journal

Similarity:

We investigate isometric composition operators on the weighted Dirichlet space $𝒟_{α}$ with standard weights ${(1 - | z |}^{2})^{α}$ , $α > - 1$ . The main technique used comes from Martín and Vukotić who completely characterized the isometric composition operators on the classical Dirichlet space $𝒟$ . We solve some of these but not in general. We also investigate the situation when $𝒟_{α}$ is equipped with another equivalent norm.

Hölder continuous solutions to Monge–Ampère equations

Jean-Pierre Demailly, Sławomir Dinew, Vincent Guedj, Pham Hoang Hiep, Sławomir Kołodziej, Ahmed Zeriahi (2014)

Journal of the European Mathematical Society

Similarity:

Let $(X, ω)$ be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on $X$ with $L^{p}$ right hand side, $p > 1$ . The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range $(X, ω)$ of the complex Monge-Ampère operator acting on $ω$ -plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Kołodziej’s result that measures with $L^{p}$ -density belong to $(X, ω)$ and proving that...

Single-point blow-up on the boundary where the zero Dirichlet boundary condition is imposed

Marek Fila, Michael Winkler (2008)

Journal of the European Mathematical Society

Similarity:

We consider a reaction-diffusion-convection equation on the halfline (0,1) with the zero Dirichlet boundary condition at $x = 0$ . We find a positive selfsimilar solution $u$ which blows up in a finite time $T$ at $x = 0$ while $u (x, T)$ remains bounded for $x > 0$ .

Concerning the energy class $_{p}$ for 0 < p < 1

Per Åhag, Rafał Czyż, Pham Hoàng Hiêp (2007)

Annales Polonici Mathematici

Similarity:

The energy class $_{p}$ is studied for 0 < p < 1. A characterization of certain bounded plurisubharmonic functions in terms of $ℱ_{p}$ and its pluricomplex p-energy is proved.

Potentials with respect to the pluricomplex Green function

Urban Cegrell (2012)

Annales Polonici Mathematici

Similarity:

For μ a positive measure, we estimate the pluricomplex potential of μ, $P_{μ} (x) = \int_{Ω} g (x, y) d μ (y)$ , where g(x,y) is the pluricomplex Green function (relative to Ω) with pole at y.

Envelope of Dirichlet problems on a domain in $C^{n}$

Eric Bedford (1985)

Annales Polonici Mathematici

Similarity:

Continuous pluriharmonic boundary values

Per Åhag, Rafał Czyż (2007)

Annales Polonici Mathematici

Similarity:

Let $D_{j}$ be a bounded hyperconvex domain in $ℂ^{n_{j}}$ and set $D = D ₁ \times \dots \times D_{s}$ , j=1,...,s, s≥ 3. Also let ₙ be the symmetrized polydisc in ℂⁿ, n ≥ 3. We characterize those real-valued continuous functions defined on the boundary of D or ₙ which can be extended to the inside to a pluriharmonic function. As an application a complete characterization of the compliant functions is obtained.

The gradient lemma

Urban Cegrell (2007)

Annales Polonici Mathematici

Similarity:

We show that if a decreasing sequence of subharmonic functions converges to a function in $W_{l o c}^{1, 2}$ then the convergence is in $W_{l o c}^{1, 2}$ .

Estimates near the boundary for second order derivatives of solutions of the Dirichlet problem for the biharmonic equation

Vladimir A. Kondratiev, Olga A. Oleinik (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Similarity:

Per ogni soluzione della (1) nel dominio limitato $\Omega$ ,, appartenente a $H_{0}^{2}(\Omega)$ e soddisfacente le condizioni (2), si dimostra la maggiorazione (5), valida nell'intorno di ogni punto $x^{0}$ del contorno; si consente a $\partial\Omega$ di essere singolare in $x^{0}$ .

A variational characterization of condenser capacities in $C^{n}$ within a class of plurisubharmonic functions

Jerzy Kalina (1983)

Annales Polonici Mathematici

Similarity:

IP-Dirichlet measures and IP-rigid dynamical systems: an approach via generalized Riesz products

Sophie Grivaux (2013)

Studia Mathematica

Similarity:

If ${(n_{k})}_{k \geq 1}$ is a strictly increasing sequence of integers, a continuous probability measure σ on the unit circle is said to be IP-Dirichlet with respect to ${(n_{k})}_{k \geq 1}$ if $σ ̂ (\sum_{k \in F} n_{k}) \to 1$ as F runs over all non-empty finite subsets F of ℕ and the minimum of F tends to infinity. IP-Dirichlet measures and their connections with IP-rigid dynamical systems have recently been investigated by Aaronson, Hosseini and Lemańczyk. We simplify and generalize some of their results, using an approach involving generalized Riesz...

A Monge-Ampère equation in conformal geometry

Matthew J. Gursky (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

We consider the Monge-Ampère-type equation $det (A + λ g) = const .$ , where $A$ is the Schouten tensor of a conformally related metric and $λ > 0$ is a suitably chosen constant. When the scalar curvature is non-positive we give necessary and sufficient conditions for the existence of solutions. When the scalar curvature is positive and the first Betti number of the manifold is non-zero we also establish existence. Moreover, by adapting a construction of Schoen, we show that solutions are in general not unique. ...

Weak and strong density results for the Dirichlet energy

Mariano Giaquinta, Domenico Mucci (2004)

Journal of the European Mathematical Society

Similarity:

Let $𝒴$ be a smooth oriented Riemannian manifold which is compact, connected, without boundary and with second homology group without torsion. In this paper we characterize the sequential weak closure of smooth graphs in $B^{n} \times 𝒴$ with equibounded Dirichlet energies, $B^{n}$ being the unit ball in $ℝ^{n}$ . More precisely, weak limits of graphs of smooth maps $u_{k} : B^{n} \to 𝒴$ with equibounded Dirichlet integral give rise to elements of the space ${cart}^{2, 1} (B^{n} \times 𝒴)$ (cf. [4], [5], [6]). In this paper we prove that every element $T$ in ${cart}^{2, 1} (B^{n} \times 𝒴)$ is the...

Some Banach spaces of Dirichlet series

Maxime Bailleul, Pascal Lefèvre (2015)

Studia Mathematica

Similarity:

The Hardy spaces of Dirichlet series, denoted by $^{p}$ (p ≥ 1), have been studied by Hedenmalm et al. (1997) when p = 2 and by Bayart (2002) in the general case. In this paper we study some $L^{p}$ -generalizations of spaces of Dirichlet series, particularly two families of Bergman spaces, denoted $^{p}$ and $ℬ^{p}$ . Each could appear as a “natural” way to generalize the classical case of the unit disk. We recover classical properties of spaces of analytic functions: boundedness of point evaluation, embeddings...

A note on the regularity of the degenerate complex Monge-Ampère equation

Szymon Pliś (2010)

Annales Polonici Mathematici

Similarity:

We prove the almost $^{1, 1}$ regularity of the degenerate complex Monge-Ampère equation in a special case.