Displaying similar documents to “A variational characterization of condenser capacities in C n within a class of plurisubharmonic functions”

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

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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 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 χ ( X , ω ) . This is done by...

Concerning the energy class p for 0 < p < 1

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

Annales Polonici Mathematici

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

The gradient lemma

Urban Cegrell (2007)

Annales Polonici Mathematici

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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 .

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

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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...

Analytic semigroups generated on a functional extrapolation space by variational elliptic equations

Vincenzo Vespri (1988)

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

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We prove that any elliptic operator of second order in variational form is the infinitesimal generator of an analytic semigroup in the functional space C - 1 , α ( Ω ) consinsting of all derivatives of hölder-continuous functions in Ω where Ω is a domain in n not necessarily bounded. We characterize, moreover the domain of the operator and the interpolation spaces between this and the space C - 1 , α ( Ω ) . We prove also that the spaces C - 1 , α ( Ω ) can be considered as extrapolation spaces relative to suitable non-variational...

Pressure and recurrence

Véronique Maume-Deschamps, Bernard Schmitt, Mariusz Urbański, Anna Zdunik (2003)

Fundamenta Mathematicae

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We deal with a subshift of finite type and an equilibrium state μ for a Hölder continuous function. Let αⁿ be the partition into cylinders of length n. We compute (in particular we show the existence of the limit) l i m n n - 1 l o g j = 0 τ ( x ) μ ( α ( T j ( x ) ) ) , where α ( T j ( x ) ) is the element of the partition containing T j ( x ) and τₙ(x) is the return time of the trajectory of x to the cylinder αⁿ(x).

A regularity theory for scalar local minimizers of splitting-type variational integrals

Michael Bildhauer, Martin Fuchs, Xiao Zhong (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of ( p , q ) -growth with exponents p q &lt; and show for the scalar case that locally bounded local minimizers are of class C 1 , μ . Note that to our knowledge the only C 1 , μ -results without imposing a relation between p and q concern the case of two independent variables as it is outlined in Marcellini’s paper [15], Theorem A, and later on in the work of Fusco and Sbordone [10], Theorem 4.2.

Solutions for the p-order Feigenbaum’s functional equation h ( g ( x ) ) = g p ( h ( x ) )

Min Zhang, Jianguo Si (2014)

Annales Polonici Mathematici

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This work deals with Feigenbaum’s functional equation ⎧ h ( g ( x ) ) = g p ( h ( x ) ) , ⎨ ⎩ g(0) = 1, -1 ≤ g(x) ≤ 1, x∈[-1,1] where p ≥ 2 is an integer, g p is the p-fold iteration of g, and h is a strictly monotone odd continuous function on [-1,1] with h(0) = 0 and |h(x)| < |x| (x ∈ [-1,1], x ≠ 0). Using a constructive method, we discuss the existence of continuous unimodal even solutions of the above equation.

Green functions, Segre numbers, and King’s formula

Mats Andersson, Elizabeth Wulcan (2014)

Annales de l’institut Fourier

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Let 𝒥 be a coherent ideal sheaf on a complex manifold X with zero set Z , and let G be a plurisubharmonic function such that G = log | f | + 𝒪 ( 1 ) locally at Z , where f is a tuple of holomorphic functions that defines 𝒥 . We give a meaning to the Monge-Ampère products ( d d c G ) k for k = 0 , 1 , 2 , ... , and prove that the Lelong numbers of the currents M k 𝒥 : = 1 Z ( d d c G ) k at x coincide with the so-called Segre numbers of J at x , introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we show that M k 𝒥 satisfy a certain...

Finite element variational crimes in the case of semiregular elements

Alexander Ženíšek (1996)

Applications of Mathematics

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The finite element method for a strongly elliptic mixed boundary value problem is analyzed in the domain Ω whose boundary Ω is formed by two circles Γ 1 , Γ 2 with the same center S 0 and radii R 1 , R 2 = R 1 + ϱ , where ϱ R 1 . On one circle the homogeneous Dirichlet boundary condition and on the other one the nonhomogeneous Neumann boundary condition are prescribed. Both possibilities for u = 0 are considered. The standard finite elements satisfying the minimum angle condition are in this case inconvenient; thus...

Critical points of the Moser-Trudinger functional on a disk

Andrea Malchiodi, Luca Martinazzi (2014)

Journal of the European Mathematical Society

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On the unit disk B 1 2 we study the Moser-Trudinger functional E ( u ) = B 1 e u 2 - 1 d x , u H 0 1 ( B 1 ) and its restrictions E | M Λ , where M Λ : = { u H 0 1 ( B 1 ) : u H 0 1 2 = Λ } for Λ > 0 . We prove that if a sequence u k of positive critical points of E | M Λ k (for some Λ k > 0 ) blows up as k , then Λ k 4 π , and u k 0 weakly in H 0 1 ( B 1 ) and strongly in C loc 1 ( B ¯ 1 { 0 } ) . Using this fact we also prove that when Λ is large enough, then E | M Λ has no positive critical point, complementing previous existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe.

On Probability Distribution Solutions of a Functional Equation

Janusz Morawiec, Ludwig Reich (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let 0 < β < α < 1 and let p ∈ (0,1). We consider the functional equation φ(x) = pφ (x-β)/(1-β) + (1-p)φ(minx/α, (x(α-β)+β(1-α))/α(1-β)) and its solutions in two classes of functions, namely ℐ = φ: ℝ → ℝ|φ is increasing, φ | ( - , 0 ] = 0 , φ | [ 1 , ) = 1 , = φ: ℝ → ℝ|φ is continuous, φ | ( - , 0 ] = 0 , φ | [ 1 , ) = 1 . We prove that the above equation has at most one solution in and that for some parameters α,β and p such a solution exists, and for some it does not. We also determine all solutions of the equation in ℐ and we show the...