Positive Radon Measure Research Articles

Order-boundary characterization of the linear lattice of Riemann 𝜇-integrable functions as a certain completion of the linear lattice of bounded continuous functions

The linear lattice of Riemann μ \mu -integrable functions on a completely regular space with a bounded positive Radon measure μ \mu is viewed as an extension of the linear lattice of bounded continuous functions. To characterize this Riemann extension, the new functional analysis category of c c -latlineals with refinements ( ≡ c r \equiv cr -latlineals) is introduced. On this basis, the procedure of c r cr -completion of certain order-boundary type is defined. The μ \mu -Riemann extension is shown to be the result of applying this procedure to the c r μ cr_{\mu } -latlineal of bounded continuous functions.

Optimal Design of Plane Elastic Membranes Using the Convexified Föppl’s Model

This work puts forth a new optimal design formulation for planar elastic membranes. The goal is to minimize the membrane’s compliance through choosing the material distribution described by a positive Radon measure. The deformation of the membrane itself is governed by the convexified Föppl’s model. The uniqueness of this model lies in the convexity of its variational formulation despite the inherent nonlinearity of the strain–displacement relation. It makes it possible to rewrite the optimization problem as a pair of mutually dual convex variational problems. The primal variables are displacement functions, whilst in the dual one seeks stresses being Radon measures. The pair of problems is analysed: existence and regularity results are provided, together with the system of optimality criteria. To demonstrate the computational potential of the pair, a finite element scheme is developed around it. Upon reformulation to a conic-quadratic & semi-definite programming problem, the method is employed to produce numerical simulations for several load case scenarios.

Hausdorff operators on some classical spaces of analytic functions

Abstract In this note, we start on the study of the sufficient conditions for the boundedness of Hausdorff operators $$ \begin{align*}(\mathcal{H}_{K,\mu}f)(z):=\int_{\mathbb{D}}K(w)f(\sigma_w(z))d\mu(w)\end{align*} $$ on three important function spaces (i.e., derivative Hardy spaces, weighted Dirichlet spaces, and Bloch type spaces), which is a continuation of the previous works of Mirotin et al. Here, $\mu $ is a positive Radon measure, K is a $\mu $ -measurable function on the open unit disk $\mathbb {D}$ , and $\sigma _w(z)$ is the classical Möbius transform of $\mathbb {D}$ .

Spectral asymptotics for generalized Schrödinger operators

Let \(d \in \{3,4,5,\ldots\}\). Consider \(L = -\frac{1}{w} \, \operatorname{div}(A \, \nabla u) + \mu\) over its maximal domain in \(L^2_w(\mathbb{R}^d)\). Under certain conditions on the weight \(w\), the coefficient matrix \(A\) and the positive Radon measure \(\mu\) we obtain upper and lower bounds on \(N(\lambda,L)\)–the number of eigenvalues of \(L\) that are at most \(\lambda \ge 1\). Furthermore we show that the eigenfunctions of \(L\) corresponding to those eigenvalues are exponentially decaying. In the course of proofs, we develop generalized Poincaré and weighted Young convolution inequalities as the main tools for the analysis.

Hausdorff–Berezin operators on weighted spaces

The research presented in this paper is a continuation of the recent studies of new classes of integral operators on the unit disc , which are called Hausdorff–Berezin operators. We consider general operators of Hausdorff–Berezin type constructed with an arbitrary positive Radon measure on the unit disc within the framework of weighted spaces with the so‐called Möbius weights. This is a fairly wide class of weights in the unit disc which includes classical radial weights with singularities or zeroes at the boundary. Sufficient conditions for the boundedness of the Hausdorff–Berezin operators in these spaces are obtained. In the case of nonnegative kernels, a boundedness criterion is obtained in . Approximation properties of operators of Hausdorff–Berezin type are also studied. We also discuss some applications.

Minimum Riesz energy problems with external fields

The paper deals with minimum energy problems in the presence of external fields with respect to the Riesz kernels |x−y|α−n, 0<α<n, on Rn, n⩾2. For quite a general (not necessarily lower semicontinuous) external field f, we obtain necessary and/or sufficient conditions for the existence of λA,f minimizing the Gauss functional∫|x−y|α−nd(μ⊗μ)(x,y)+2∫fdμ over all positive Radon measures μ with μ(Rn)=1, concentrated on quite a general (not necessarily closed) A⊂Rn. We also provide various alternative characterizations of the minimizer λA,f, analyze the continuity of both λA,f and the modified Robin constant for monotone families of sets, and give a description of the support of λA,f. The significant improvement of the theory in question thereby achieved is due to a new approach based on the close interaction between the strong and the vague topologies, as well as on the theory of inner balayage, developed recently by the author.

A Sweeping Process Control Problem Subject To Mixed Constraints

In this study, we investigate optimal control problems that involve sweeping processes with a drift term and mixed inequality constraints. Our goal is to establish necessary optimality conditions for these problems. We address the challenges that arise due to the combination of sweeping processes and inequality mixed constraints in two contexts: regular and non-regular. This requires working with different types of multipliers, such as finite positive Radon measures for the sweeping term and integrable functions for regular mixed constraints. For non-regular mixed constraints, the multipliers correspond to purely finitely additive set functions.

Does strong repulsion lead to smooth solutions in a repulsion-attraction chemotaxis system even when starting with highly irregular initial data?

It has been well established that, in attraction-repulsion Keller–Segel systems of the form$ \begin{equation*} \left\{ \begin{aligned} u_t &amp; = \Delta u - \chi \nabla \cdot (u\nabla v) + \xi \nabla \cdot (u\nabla w), \\ \tau v_t &amp; = \Delta v + \alpha u - \beta v, \\ \tau w_t &amp; = \Delta w + \gamma u - \delta w \end{aligned} \right. \end{equation*} $in a smooth bounded domain $ \Omega \subseteq \mathbb{R}^n $, $ n\in\mathbb{N} $, with Neumann boundary conditions and parameters $ \chi, \xi \geq 0 $, $ \alpha, \beta, \gamma, \delta &gt; 0 $ and $ \tau \in \{0, 1\} $, finite-time blow-up can be ruled out in many scenarios given sufficiently smooth initial data if the repulsive chemotaxis is sufficiently stronger than its attractive counterpart. In this paper, we will go - in a sense - a step further than this by studying the same system with initial data that could already be understood as being in a blown-up state (e.g. a positive Radon measure for the first solution component) and then ask the question whether sufficiently strong repulsion has enough of a regularizing effect to lead to the existence of a smooth solution, which is still connected to said initial data in a sensible fashion. Regarding this, we in fact establish that the construction of such a solution is possible in the two-dimensional parabolic-parabolic system and the two- and three-dimensional parabolic-elliptic system under appropriate assumptions on the interaction of repulsion and attraction as well as the initial data.

Distributed-order time-fractional wave equations

Distributed-order time-fractional wave equations appear in the modeling of wave propagation in viscoelastic media. The material characteristics of the medium are modeled through constitutive functions or distributions in the distributed-order constitutive law. In this work we propose to take positive Radon measures for the constitutive "functions". First, we derive a thermodynamical restriction on the constitutive measures which is easy to check, and therefore suitable for applications. Then we prove that the setting with measures in combination with the derived thermodynamical restriction guarantee existence and uniqueness of solutions for the distributed-order fractional wave equation. We further discuss the support and regularity of the fundamental solution, and conclude with a discussion on wave velocities.

Weak Solutions to Monge–Ampère Type Equations on Compact Hermitian Manifold with Boundary

We prove the bounded subsolution theorem for the complex Monge–Ampère type equation, with the right-hand side being a positive Radon measure, on a compact Hermitian manifold with boundary.

Bochner Representable Operators on Spaces of Continuous Functions

Let C_b(X) be the Banach lattice of all bounded continuous real-valued functions on a completely regular Hausdorff space X and beta denote the natural strict topology on C_b(X). For a Banach space (E,Vert cdot Vert _E), a linear operator T:C_b(X)rightarrow E is said to be tight if Vert T(u_alpha )Vert _Erightarrow 0 whenever (u_alpha ) is a uniformly bounded net in C_b(X) such that u_alpha rightarrow 0 uniformly on all compact sets in X. It is shown that a linear operator T:C_b(X)rightarrow E is nuclear tight if and only if T is a nuclear operator between the locally convex space (C_b(X),beta ) and a Banach space E and if and only if T is Bochner representable, that is, there exist a positive Radon measure mu on X and a E-valued mu -Bochner integrable function g on X so that T(u)=int _X u(x)g(x)dmu for all uin C_b(X).

Balayage of measures on a locally compact space

We develop a theory of inner balayage of a positive Radon measure μ of finite energy on a locally compact space X to arbitrary A ⊂ X, thereby generalizing Cartan’s theory of Newtonian inner balayage on ℝn, n ⩾ 3, to a suitable function kernel on X. As an application of the theory developed, we show that if a locally compact space X is σ-compact and perfectly normal, then a recent result by B. Fuglede (Anal. Math., 2016) on outer balayage of μ to quasiclosed A remains valid for arbitrary Borel A. We give in particular various alternative definitions of inner (outer) balayage, provide a formula for evaluation of its total mass, and prove convergence theorems for inner (outer) swept measures and their potentials. The results obtained hold true (and are partly new) for most classical kernels on ℝn, n ⩾ 2, which is important in applications.

Representation formulae for nonhomogeneous differential operators and applications to PDEs

We obtain integral representation formulae for functions u∈Lloc1(RN) which satisfy P(−Δ)u=μ in the sense of distributions in RN, where μ is a positive Radon measure on RN and P(z)=zm(zσ+aσ−1zσ−1+⋯+a1z+a0) is a nonconstant real polynomial whose roots belong to the interval (−∞,0], m,σ≥0, N>2m. Further, we apply these results to a number of nonhomogeneous higher order differential inequalities.

A Markov process for an infinite age-structured population

For an infinite system of particles arriving in and departing from a habitat $X$ -- a locally compact Polish space with a positive Radon measure $\chi$ -- a Markov process is constructed in an explicit way. Along with its location $x\in X$, each particle is characterized by age $\alpha\geq 0$ -- time since arriving. As the state space one takes the set of marked configurations $\widehat{\Gamma}$, equipped with a metric that makes it a complete and separable metric space. The stochastic evolution of the system is described by a Kolmogorov operator $L$, expressed through the measure $\chi$ and a departure rate $m(x,\alpha)\geq 0$, and acting on bounded continuous functions $F:\widehat{\Gamma}\to \mathds{R}$. For this operator, we pose the martingale problem and show that it has a unique solution, explicitly constructed in the paper. We also prove that the corresponding process has a unique stationary state and is temporarily egrodic if the rate of departure is separated away from zero.

Multiplicity and uniqueness for Lane-Emden equations and systems with Hardy potential and measure data

Let Ω be a C2 bounded domain in RN (N≥3), δ(x)=dist(x,∂Ω) and CH(Ω) be the best constant in the Hardy inequality with respect to Ω. We investigate positive solutions to a boundary value problem for Lane-Emden equations with Hardy potential of the form−Δu−μδ2u=up in Ω,u=ρν on ∂Ω,(Pρ) where 0<μ<CH(Ω), ρ is a positive parameter, ν is a positive Radon measure on ∂Ω with norm 1 and 1<p<Nμ, with Nμ being a critical exponent depending on N and μ. It is known from [22] that there exists a threshold value ρ⁎ such that problem (Pρ) admits a positive solution if 0<ρ≤ρ⁎, and no positive solution if ρ>ρ⁎. In this paper, we go further in the study of the solution set of (Pρ). We show that the problem admits at least two positive solutions if 0<ρ<ρ⁎ and a unique positive solution if ρ=ρ⁎.We also prove the existence of at least two positive solutions for Lane-Emden systems{−Δu−μδ2u=vp in Ω,−Δv−μδ2v=uq in Ω,u=ρν,v=στ on ∂Ω, under the smallness condition on the positive parameters ρ and σ.

Doob’s ω-Transform on Local Dirichlet Spaces

In this article, we study the existence and uniqueness of nonnegative solutions for the following parabolic problem where T > 0, X is a locally compact separable metric space, Ht is a operator (not necessarily selfadjoint) associated with a regular (not necessarily symmetric)local Dirichlet form the initial value and m is a positive Radon measure on Borel subset U of X such that m(U) > 0. Our technical approach for uniqueness is based on Täcklind type uniqueness theorem.

BV solutions of a convex sweeping process with a composed perturbation

&lt;p style='text-indent:20px;'&gt;A measurable sweeping process with a composed perturbation is considered in a separable Hilbert space. The values of the moving set generating the sweeping process are closed, convex sets. The retraction of the sweeping process is bounded by a positive Radon measure. The perturbation is the sum of two multivalued mappings. The values of the first one are closed, bounded, not necessarily convex sets. It is measurable in the time variable, is Lipschitz continuous in the phase variable, and satisfies a conventional growth condition. The values of the second one are closed, convex, not necessarily bounded sets. We assume that this mapping has a closed with respect to the phase variable graph.&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;The remaining assumptions concern the intersection of the second mapping and the multivalued mapping defined by the growth conditions. We suppose that this intersection has a measurable selector and has certain compactness properties.&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;We prove the existence of solutions for our inclusion. The proof is based on the author's theorem on continuous with respect to a parameter selectors passing through fixed points of contraction multivalued maps with closed, nonconvex, decomposable values depending on the parameter, and the classical Ky Fan fixed point theorem. The results which we obtain are new.&lt;/p&gt;

Infinitesimal Hilbertianity of Locally mathrm{CAT}(kappa )-Spaces

We show that, given a metric space (mathrm{Y},textsf {d} ) of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure mu on mathrm{Y} giving finite mass to bounded sets, the resulting metric measure space (mathrm{Y},textsf {d} ,mu ) is infinitesimally Hilbertian, i.e. the Sobolev space W^{1,2}(mathrm{Y},textsf {d} ,mu ) is a Hilbert space. The result is obtained by constructing an isometric embedding of the ‘abstract and analytical’ space of derivations into the ‘concrete and geometrical’ bundle whose fibre at xin mathrm{Y} is the tangent cone at x of mathrm{Y}. The conclusion then follows from the fact that for every xin mathrm{Y} such a cone is a mathrm{CAT}(0) space and, as such, has a Hilbert-like structure.

Duality properties of metric Sobolev spaces and capacity

We study the properties of the dual Sobolev space $H^{-1, q}(\mathbb{X}) = \big(H^{1, p}(\mathbb{X})\big)'$ on a complete extended metric-topological measure space $\mathbb{X} = (X, \tau, \rm{d}, \rm{m})$ for $p\in (1, \infty)$. We will show that a crucial role is played by the strong closure $H_{{\rm{pd}}}^{ - 1, q}\left({\mathbb{X}} \right)$ of $L^q(X, \rm{m})$ in the dual $H^{-1, q}(\mathbb{X})$, which can be identified with the predual of $H^{1, p}(\mathbb{X})$. We will show that positive functionals in $H^{-1, q}(\mathbb{X})$ can be represented as a positive Radon measure and we will charaterize their dual norm in terms of a suitable energy functional on nonparametric dynamic plans. As a byproduct, we will show that for every Radon measure $\mu$ with finite dual Sobolev energy, Cap_<i>p</i>-negligible sets are also $\mu$-negligible and good representatives of Sobolev functions belong to $L^1(X, \mu)$. We eventually show that the Newtonian-Sobolev capacity Cap_<i>p</i> admits a natural dual representation in terms of such a class of Radon measures.

BV Continuous Solutions of an Evolution Inclusion with Maximal Monotone Operator and Nonconvex-Valued Perturbation. Existence Theorem

An evolution inclusion with the right-hand side containing a time-dependent maximal monotone operator and a multivalued mapping with closed nonconvex values is studied in a separable Hilbert space. The dependence of the maximal monotone operator on time is described with the help of the distance between maximal monotone operators in the sense of Vladimirov. This distance as a function of time has bounded variation with an upper bound given by a nonatomic positive Radon measure. By a solution of the inclusion one means a continuous function of bounded variation whose differential measure (Stieltjes measure) is absolutely continuous with respect to the positive Radon measure above, and the values of the density of this differential measure with respect to the Radon measure belong to the right-hand side of the inclusion almost everywhere. Under the traditional assumptions on the perturbation (measurability, Lipschitzianity in the phase variable in the Hausdorff metric, linear growth condition), the existence of solutions is proven and some properties of the solution set are established.