Causal variational principles in the infinite-dimensional setting: Existence of minimizers

We provide a method for constructing (possibly non-trivial) measures on non-locally compact Polish subspaces of infinite-dimensional separable Banach spaces which, under suitable assumptions, are minimizers of causal variational principles in the non-locally compact setting. Moreover, for non-trivial minimizers the corresponding Euler–Lagrange equations are derived. The method is to exhaust the underlying Banach space by finite-dimensional subspaces and to prove existence of minimizers of the causal variational principle restricted to these finite-dimensional subsets of the Polish space under suitable assumptions on the Lagrangian. This gives rise to a corresponding sequence of minimizers. Restricting the resulting sequence to countably many compact subsets of the Polish space, by considering the resulting diagonal sequence, we are able to construct a regular measure on the Borel algebra over the whole topological space. For continuous Lagrangians of bounded range, it can be shown that, under suitable assumptions, the obtained measure is a (possibly non-trivial) minimizer under variations of compact support. Under additional assumptions, we prove that the constructed measure is a minimizer under variations of finite volume and solves the corresponding Euler–Lagrange equations. Afterwards, we extend our results to continuous Lagrangians vanishing in entropy. Finally, assuming that the obtained measure is locally finite, topological properties of spacetime are worked out and a connection to dimension theory is established.

Local boundedness for minimizers of convex integral functionals in metric measure spaces

In this paper we consider the convex integral functional $ I := \int _\Omega {\Phi (g_u)\,d\mu } $ in the metric measure space $(X,d,\mu )$, where $X$ is a set, $d$ is a metric, µ is a Borel regular measure satisfying the doubling condition, Ω is a bounded open subset of $X$, $u$ belongs to the Orlicz-Sobolev space $N^{1,\Phi }(\Omega )$, Φ is an N-function satisfying the $\Delta _2$-condition, $g_u$ is the minimal Φ-weak upper gradient of $u$. By improving the corresponding method in the Euclidean space to the metric setting, we establish the local boundedness for minimizers of the convex integral functional under the assumption that $(X,d,\mu )$ satisfies the $(1,1)$-Poincaré inequality. The result of this paper can be applied to the Carnot-Carathéodory space spanned by vector fields satisfying Hörmander's condition.

The obstacle problem for quasilinear stochastic PDEs with Neumann boundary condition

We prove the existence and uniqueness of solution to obstacle problem for quasilinear stochastic partial differential equations with Neumann boundary condition. Our method is based on the analytical techniques coming from parabolic potential theory. The solution is expressed as a pair [Formula: see text] where [Formula: see text] is a predictable continuous process which takes values in a proper Sobolev space and [Formula: see text] is a random regular measure satisfying minimal Skohorod condition.

A note on measure and expansiveness on uniform spaces

<p>We prove that the set of points doubly asymptotic to a point has measure zero with respect to any expansive outer regular measure for a bi-measurable map on a separable uniform space. Consequently, we give a class of measures which cannot be expansive for Denjoy home-omorphisms on S¹. We then investigate the existence of expansive measures for maps with various dynamical notions. We further show that measure expansive (strong measure expansive) homeomorphisms with shadowing have periodic (strong periodic) shadowing. We relate local weak specification and periodic shadowing for strong measure expansive systems.</p>

On a measure of noncompactness in the space of regulated functions and its applications

In this paper we formulate a criterion for relative compactness in the space of functions regulated on a bounded and closed interval. We prove that the mentioned criterion is equivalent to a known criterion obtained earlier by D. Fraňkova, but it turns out to be very convenient in applications. Among others, it creates the basis to construct a regular measure of noncompactness in the space of regulated functions. We show the applicability of the constructed measure of noncompactness in proving the existence of solutions of a quadratic Hammerstein integral equation in the space of regulated functions.

Pointwise Asymptotics for Orthonormal Polynomials at the Endpoints of the Interval via Universality

We show that universality limits and other bounds imply pointwise asymptotics for orthonormal polynomials at the endpoints of the interval of orthonormality. As a consequence, we show that if $\mu $ is a regular measure supported on $\left [ -1,1\right ] $, and in a neighborhood of 1, $ \mu $ is absolutely continuous, while for some $\alpha&gt;-1$, $\mu ^{\prime }\left ( t\right ) =h\left ( t\right ) \left ( 1-t\right )^{\alpha }$, where $ h\left ( t\right ) \rightarrow 1$ as $t\rightarrow 1-$, then the corresponding orthonormal polynomials $\left \{ p_{n}\right \} $ satisfy the asymptotic $$ \lim_{n\rightarrow \infty }\frac{p_{n}\left( 1-\frac{z^{2}}{2n^{2}}\right) }{ p_{n}\left( 1\right) }=\frac{J_{\alpha }^{\ast }\left( z\right) }{J_{\alpha }^{\ast }\left( 0\right) } $$uniformly in compact subsets of the plane. Here $J_{\alpha }^{\ast }\left ( z\right ) =J_{\alpha }\left ( z\right ) /z^{\alpha }$ is the normalized Bessel function of order $\alpha $. These are by far the most general conditions for such endpoint asymptotics.

A remark on Poincaré inequalities on metric measure spaces

We show that, in a complete metric measure space equipped with a doubling Borel regular measure, the Poincaré inequality with upper gradients introduced by Heinonen and Koskela [3] is equivalent to the Poincaré inequality with "approximate Lipschitz constants" used by Semmes in [9].

Existence and Properties of ℎ-Sets

In this note we shall consider the following problem: which conditions should satisfy a function ℎ : (0, 1) → ℝ in order to guarantee the existence of a (regular) measure μ in with compact support and for some positive constants 𝑐2, and 𝑐2 independent of γ ∈ Γ and 𝑟 ∈ (0,1)? The theory of self-similar fractals provides outstanding examples of sets fulfilling (♡) with ℎ(𝑟) = 𝑟𝑑, 0 ≤ 𝑑 ≤ 𝑛, and a suitable measure μ. Analogously, we shall rely on some recent techniques for the construction of pseudo self-similar fractals in order to deal with our more general task.

C0-continuity of the Fröbenius-Perron semigroup

We consider the Fröbenius-Perron semigroup of linear operators associated to a semidynamical system defined in a topological spaceXendowed with a finite or aσ-finite regular measure. We prove that if there exists afaithful invariant measurefor the semidynamical system, then the Fröbenius-Perron semigroup of linear operators isC0-continuous in the spaceLμ 1(X). We also give a geometrical condition which ensuresC0-continuity of the Fröbenius-Perron semigroup of linear operators in the spaceLμ p(X)for1≤p<∞, as well as in the spaceLloc 1.