On the Monge-Ampere differential equation
B. N. Rachajsky (1969)
Matematički Vesnik
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B. N. Rachajsky (1969)
Matematički Vesnik
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Slimane Benelkourchi, Vincent Guedj, Ahmed Zeriahi (2008)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
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Let be a compact Kähler manifold and be a smooth closed form of bidegree which is nonnegative and big. We study the classes of -plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight 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 . This is done by...
Michal Misiurewicz (1975)
Publications mathématiques et informatique de Rennes
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Pham Hoang Hiep (2006)
Annales Polonici Mathematici
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We establish the comparison principle in the class . The result obtained is applied to the Dirichlet problem in .
Pham Hoang Hiep (2007)
Annales Polonici Mathematici
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We study boundary values of functions in Cegrell’s class .
Per Åhag, Rafał Czyż, Pham Hoàng Hiêp (2007)
Annales Polonici Mathematici
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The energy class is studied for 0 < p < 1. A characterization of certain bounded plurisubharmonic functions in terms of and its pluricomplex p-energy is proved.
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 then the convergence is in .
Z. Kominek (1974)
Annales Polonici Mathematici
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C. T. Ng (1973)
Annales Polonici Mathematici
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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 be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on with right hand side, . The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range of the complex Monge-Ampère operator acting on -plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Kołodziej’s result that measures with -density belong to and proving that...
H. Światak (1967)
Annales Polonici Mathematici
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M. Malenica (1982)
Matematički Vesnik
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Yasutaka Sibuya (1984)
Recherche Coopérative sur Programme n°25
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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 consinsting of all derivatives of hölder-continuous functions in where is a domain in not necessarily bounded. We characterize, moreover the domain of the operator and the interpolation spaces between this and the space . We prove also that the spaces can be considered as extrapolation spaces relative to suitable non-variational...
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) , where is the element of the partition containing and τₙ(x) is the return time of the trajectory of x to the cylinder αⁿ(x).
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 -growth with exponents and show for the scalar case that locally bounded local minimizers are of class . Note that to our knowledge the only -results without imposing a relation between and 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.
Min Zhang, Jianguo Si (2014)
Annales Polonici Mathematici
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This work deals with Feigenbaum’s functional equation ⎧ , ⎨ ⎩ g(0) = 1, -1 ≤ g(x) ≤ 1, x∈[-1,1] where p ≥ 2 is an integer, 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.
Mats Andersson, Elizabeth Wulcan (2014)
Annales de l’institut Fourier
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Let be a coherent ideal sheaf on a complex manifold with zero set , and let be a plurisubharmonic function such that locally at , where is a tuple of holomorphic functions that defines . We give a meaning to the Monge-Ampère products for , and prove that the Lelong numbers of the currents at coincide with the so-called Segre numbers of at , introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we show that satisfy a certain...
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 whose boundary is formed by two circles , with the same center and radii , , where . On one circle the homogeneous Dirichlet boundary condition and on the other one the nonhomogeneous Neumann boundary condition are prescribed. Both possibilities for are considered. The standard finite elements satisfying the minimum angle condition are in this case inconvenient; thus...
Andrea Malchiodi, Luca Martinazzi (2014)
Journal of the European Mathematical Society
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On the unit disk we study the Moser-Trudinger functional and its restrictions , where for . We prove that if a sequence of positive critical points of (for some ) blows up as , then , and weakly in and strongly in . Using this fact we also prove that when is large enough, then has no positive critical point, complementing previous existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe.
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: ℝ → ℝ|φ is continuous, , . 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...