A Short Proof of the Variational Principle for a Action on a Compact Space
Michal Misiurewicz (1975)
Publications mathématiques et informatique de Rennes
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Michal Misiurewicz (1975)
Publications mathématiques et informatique de Rennes
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Bernd Schmidt (2011)
ESAIM: Control, Optimisation and Calculus of Variations
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We provide a detailed analysis of the minimizers of the functional , , subject to the constraint . This problem,, describes the long-time behavior of the parabolic Anderson in probability theory or ground state solutions of a nonlinear Schrödinger equation. While existence can be proved with standard methods, we show that the usual uniqueness results obtained with PDE-methods can be considerably simplified by additional variational arguments. In addition, we investigate qualitative...
Mark A. Peletier, Robert Planqué, Matthias Röger (2007)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
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We derive a new criterion for a real-valued function to be in the Sobolev space . This criterion consists of comparing the value of a functional with the values of the same functional applied to convolutions of with a Dirac sequence. The difference of these values converges to zero as the convolutions approach , and we prove that the rate of convergence to zero is connected to regularity: if and only if the convergence is sufficiently fast. We finally apply our criterium to...
David Brink (2015)
Acta Arithmetica
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We show how the idea behind a formula for π discovered by the Indian mathematician and astronomer Nilakantha (1445-1545) can be developed into a general series acceleration technique which, when applied to the Gregory-Leibniz series, gives the formula with convergence as , in much the same way as the Euler transformation gives with convergence as . Similar transformations lead to other accelerated series for π, including three “BBP-like” formulas, all of which are collected in...
Jerzy Kalina (1983)
Annales Polonici Mathematici
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Marek Galewski (2007)
Bollettino dell'Unione Matematica Italiana
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We provide a duality theory and existence results for a operator equation where is not necessarily a monotone operator. We use the abstract version of the so called dual variational method. The solution is obtained as a limit of a minimizng sequence whose existence and convergence is proved.
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...
Larry Kitchens, Charles Swartz (1974)
Colloquium Mathematicae
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I. Assani (2005)
Colloquium Mathematicae
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We answer a question of H. Furstenberg on the pointwise convergence of the averages , where U and R are positive operators. We also study the pointwise convergence of the averages when T and S are measure preserving transformations.
Sreela Gangopadhyay (1990)
Colloquium Mathematicae
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Aleksandra Orpel (2002)
Annales Polonici Mathematici
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We investigate the existence of solutions of the Dirichlet problem for the differential inclusion for a.e. y ∈ Ω, which is a generalized Euler-Lagrange equation for the functional . We develop a duality theory and formulate the variational principle for this problem. As a consequence of duality, we derive the variational principle for minimizing sequences of J. We consider the case when G is subquadratic at infinity.
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).
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...
Donald E. Myers (1976)
Annales Polonici Mathematici
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Gianni Dal Maso (2013)
Journal of the European Mathematical Society
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We introduce the space of generalized functions of bounded deformation and study the structure properties of these functions: the rectiability and the slicing properties of their jump sets, and the existence of their approximate symmetric gradients. We conclude by proving a compactness results for , which leads to a compactness result for the space of generalized special functions of bounded deformation. The latter is connected to the existence of solutions to a weak formulation...
S. V. Konyagin, V. N. Temlyakov (2003)
Studia Mathematica
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We consider convergence of thresholding type approximations with regard to general complete minimal systems eₙ in a quasi-Banach space X. Thresholding approximations are defined as follows. Let eₙ* ⊂ X* be the conjugate (dual) system to eₙ; then define for ε > 0 and x ∈ X the thresholding approximations as , where . We study a generalized version of that we call the weak thresholding approximation. We modify the in the following way. For ε > 0, t ∈ (0,1) we set and consider...
J. A. Nitsche (1975)
Publications mathématiques et informatique de Rennes
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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.