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Gaussian estimates for Schrödinger perturbations

Krzysztof Bogdan, Karol Szczypkowski (2014)

Studia Mathematica

We propose a new general method of estimating Schrödinger perturbations of transition densities using an auxiliary transition density as a majorant of the perturbation series. We present applications to Gaussian bounds by proving an optimal inequality involving four Gaussian kernels, which we call the 4G Theorem. The applications come with honest control of constants in estimates of Schrödinger perturbations of Gaussian-type heat kernels and also allow for specific non-Kato perturbations.

Generalized c -almost periodic type functions in ℝ n

M. Kostić (2021)

Archivum Mathematicum

In this paper, we analyze multi-dimensional quasi-asymptotically c -almost periodic functions and their Stepanov generalizations as well as multi-dimensional Weyl c -almost periodic type functions. We also analyze several important subclasses of the class of multi-dimensional quasi-asymptotically c -almost periodic functions and reconsider the notion of semi- c -periodicity in the multi-dimensional setting, working in the general framework of Lebesgue spaces with variable exponent. We provide certain...

Generators of Brownian motions on abstract Wiener spaces

Kei Harada (2010)

Banach Center Publications

We prove that Brownian motion on an abstract Wiener space B generates a locally equicontinuous semigroup on C b ( B ) equipped with the T t -topology introduced by L. Le Cam. Hence we obtain a “Laplace operator” as its infinitesimal generator. Using this Laplacian, we discuss Poisson’s equation and heat equation, and study its properties, especially the difference from the Gross Laplacian.

Global Existence and Boundedness of Solutions to a Model of Chemotaxis

J. Dyson, R. Villella-Bressan, G. F. Webb (2008)

Mathematical Modelling of Natural Phenomena

A model of chemotaxis is analyzed that prevents blow-up of solutions. The model consists of a system of nonlinear partial differential equations for the spatial population density of a species and the spatial concentration of a chemoattractant in n-dimensional space. We prove the existence of solutions, which exist globally, and are L∞-bounded on finite time intervals. The hypotheses require nonlocal conditions on the species-induced production of the chemoattractant.

Global existence for a Riccati equation arising in a boundary control problem for distributed parameters

Franco Flandoli (1982)

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

Si prova resistenza globale della soluzione di una equazione di Riccati collegata alla sintesi di un problema di controllo ottimale. Il problema considerato rappresenta la versione astratta di alcuni problemi governati da equazioni paraboliche con il controllo sulla frontiera.

Global Schauder estimates for a class of degenerate Kolmogorov equations

Enrico Priola (2009)

Studia Mathematica

We consider a class of possibly degenerate second order elliptic operators on ℝⁿ. This class includes hypoelliptic Ornstein-Uhlenbeck type operators having an additional first order term with unbounded coefficients. We establish global Schauder estimates in Hölder spaces both for elliptic equations and for parabolic Cauchy problems involving . The Hölder spaces in question are defined with respect to a possibly non-Euclidean metric related to the operator . Schauder estimates are deduced by sharp...

Gradient estimates in parabolic problems with unbounded coefficients

M. Bertoldi, S. Fornaro (2004)

Studia Mathematica

We study, with purely analytic tools, existence, uniqueness and gradient estimates of the solutions to the Neumann problems associated with a second order elliptic operator with unbounded coefficients in spaces of continuous functions in an unbounded open set Ω in ℝ N .

Growth of semigroups in discrete and continuous time

Alexander Gomilko, Hans Zwart, Niels Besseling (2011)

Studia Mathematica

We show that the growth rates of solutions of the abstract differential equations ẋ(t) = Ax(t), ẋ ( t ) = A - 1 x ( t ) , and the difference equation x d ( n + 1 ) = ( A + I ) ( A - I ) - 1 x d ( n ) are closely related. Assuming that A generates an exponentially stable semigroup, we show that on a general Banach space the lowest growth rate of the semigroup ( e A - 1 t ) t ≥ 0 is O(∜t), and for ( ( A + I ) ( A - I ) - 1 ) ⁿ it is O(∜n). The similarity in growth holds for all Banach spaces. In particular, for Hilbert spaces the best estimates are O(log(t)) and O(log(n)), respectively. Furthermore, we give conditions...

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