Radon Measure Research Articles

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.

Regularity estimates for singular parabolic measure data problems with sharp growth

We prove global gradient estimates for parabolic p-Laplace type equations with measure data, whose model isut−div(|Du|p−2Du)=μinΩ×(0,T)⊂Rn×R,where μ is a signed Radon measure with finite total mass. We consider the singular case2nn+1<p≤2−1n+1and give possibly minimal conditions on the nonlinearity and the boundary of Ω, which guarantee the regularity results for such measure data problems.

The effect of new building regulations on indoor radon in radonprone municipalities

Radon is an important contributor to public radiation dose and it is important to monitor levels in homes and introduce measures to reduce radon concentration levels, both overall and where levels are especially high. In Norway, new building regulations were introduced in 2010, which required balanced ventilation and preventive measures to reduce indoor radon levels, including a radon barrier toward the ground and pressure reducing features beneath the building that prevent soil gas from entering (radon sump). Investigations of randomly selected homes all across Norway have shown that houses built under these new regulations have significantly lower radon levels. However, a few municipalities in Norway are especially radon-prone and have houses with particularly high levels. It is crucial to verify the effect of the new regulations in these municipalities, which we have done in this study. Here, we show that both preventive radon measures and balanced ventilation and the building regulations of 2010 have significant effects on reducing the radon levels in the houses of the public. Noticeably for management, houses with a well-ventilated crawl space, which have been exempt from the required preventive measures, still in some cases have levels above action and maximum recommended levels

Relaxation of functionals with linear growth: Interactions of emerging measures and free discontinuities

Abstract For an integral functional defined on functions ( u , v ) ∈ W 1 , 1 × L 1 {(u,v)\in W^{1,1}\times L^{1}} featuring a prototypical strong interaction term between u and v, we calculate its relaxation in the space of functions with bounded variations and Radon measures. Interplay between measures and discontinuities brings various additional difficulties, and concentration effects in recovery sequences play a major role for the relaxed functional even if the limit measures are absolutely continuous with respect to the Lebesgue one.

A Convex Duality Approach for Assigning Probability Distributions to the State of Nonlinear Stochastic Systems

In order to optimally assign a desired probability distribution to the state of a nonlinear stochastic system, a convex duality approach is proposed to arrive at the associated optimality conditions. For a general class of stochastic systems governed by controlled It&#x00F4; differential equations and subject to constraints on the probability distribution of the state at a fixed terminal time, a measure theoretic formulation is presented and it is shown that the original problem is embedded in a convex linear program on the space of Radon measures and that the embedding is tight, i.e., the optimal solution of both the original and the convex relaxation problems are equal. By exploiting the duality relationship between the space of continuous functions and that of measures, the associated optimality conditions are identified in the form of Hamilton-Jacobi problems where the optimization objective, in addition to the value function evaluation at the initial conditions, includes an extra term which is the integral of the product of the value function at the terminal time and the desired probability distribution. Numerical examples are provided to illustrate the results.

Gridless 3D Recovery of Image Sources From Room Impulse Responses

Given a sound field generated by a sparse distribution of impulse image sources, can the continuous 3D positions and amplitudes of these sources be recovered from discrete, bandlimited measurements of the field at a finite set of locations, e.g., a multichannel room impulse response? Borrowing from recent advances in super-resolution imaging, it is shown that this nonlinear, non-convex inverse problem can be efficiently relaxed into a convex linear inverse problem over the space of Radon measures in R3. The linear operator introduced here stems from the fundamental solution of the free-field inhomogenous wave equation combined with the receivers' responses. An adaptation of the Sliding Frank-Wolfe algorithm is proposed to numerically solve the problem off-the-grid, i.e., in continuous 3D space. Simulated experiments show that the approach achieves near-exact recovery of hundreds of image sources using an arbitrarily placed compact 32-channel spherical microphone array in random rectangular rooms. The impact of noise, sampling rate and array diameter on these results is also examined.

On two-dimensional steady hypersonic-limit Euler flows passing ramps and radon measure solutions of compressible Euler equations

We proposed rigorous definitions of Radon measure solutions for boundary value problems of steady compressible Euler equations which modeling hypersonic-limit inviscid flows passing two-dimensional ramps, and their interactions with still gas and pressureless jets. We proved the Newton-Busemann pressure law of drags of body in hypersonic flow, and constructed various physically interesting measure solutions with density containing Dirac measures supported on curves, also exhibited examples of blow up of certain measure solutions. This established a mathematical foundation for applications in engineering and further studies of measure solutions of compressible Euler equations.

Existence and stability of solutions to nonlinear parabolic problems with perturbed gradient and measure data

In this paper we prove the existence of an entropy solution to nonlinear parabolic equations with diffuse Radon measure data which does not charge the sets of zero p(⋅)-capacity and nonhomogeneous Neumann boundary condition. By a time discretization technique we analyze existence, the uniqueness and the stability questions. The functional setting involves Lebesgue and Sobolev spaces with variable exponents.

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.

Gradient of the single layer potential and quantitative rectifiability for general Radon measures

We identify a set of sufficient local conditions under which a significant portion of a Radon measure μ on Rn+1 with compact support can be covered by an uniformly n-rectifiable set, at the level of a ball B⊂Rn+1 such that μ(B)≈r(B)n. This result involves a flatness condition, formulated in terms of the so-called β1-number of B, and the L2(μ|B)-boundedness, as well as a control on the mean oscillation on the ball, of the operatorTμf(x)=∫∇xE(x,y)f(y)dμ(y). Here E(⋅,⋅) is the fundamental solution for a uniformly elliptic operator in divergence form associated with an (n+1)×(n+1) matrix with Hölder continuous coefficients. This generalizes a work by Girela-Sarrión and Tolsa for the n-Riesz transform. The motivation for our result stems from a two-phase problem for the elliptic harmonic measure.

Microlocal approach to the Hausdorff dimension of measures

In this paper we study the dependence of geometric properties of Radon measures, such as Hausdorff dimension and rectifiability of singular sets, on the wavefront set. We prove our results by adapting the method of Brummelhuis to the non-analytic case. As an application we obtain a general form of the Uncertainty Principle for measures on the complex sphere. This instance of the UP generalizes the celebrated theorem of Aleksandrov and Forelli concerning regularity of pluriharmonic measures.

Existence and asymptotic behavior of Radon measure-valued solutions for a class of nonlinear parabolic equations

In this paper we address the weak Radon measure-valued solutions associated with the Young measure for a class of nonlinear parabolic equations with initial data as a bounded Radon measure. This problem is described as follows: {ut=αuxx+β[φ(u)]xx+f(u)inQ:=Ω×(0,T),u=0on∂Ω×(0,T),u(x,0)=u0(x)inΩ,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} u_{t}=\\alpha u_{xx}+\\beta [\\varphi (u) ]_{xx}+f(u) &\\text{in} \\ Q:=\\Omega \\times (0,T), \\\\ u=0 &\\text{on} \\ \\partial \\Omega \\times (0,T), \\\\ u(x,0)=u_{0}(x) &\\text{in} \\ \\Omega , \\end{cases} $$\\end{document} where T>0, Omega subset mathbb{R} is a bounded interval, u_{0} is nonnegative bounded Radon measure on Ω, and alpha , beta geq 0, under suitable assumptions on φ and f. In this work we prove the existence and the decay estimate of suitably defined Radon measure-valued solutions for the problem mentioned above. In particular, we study the asymptotic behavior of these Radon measure-valued solutions.

Partial isometries, duality, and determinantal point processes

A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures [Formula: see text] on a space S with measure [Formula: see text], whose correlation functions are all given by determinants specified by an integral kernel K called the correlation kernel. We consider a pair of Hilbert spaces, [Formula: see text], which are assumed to be realized as [Formula: see text]-spaces, [Formula: see text], [Formula: see text], and introduce a bounded linear operator [Formula: see text] and its adjoint [Formula: see text]. We show that if [Formula: see text] is a partial isometry of locally Hilbert–Schmidt class, then we have a unique DPP [Formula: see text] associated with [Formula: see text]. In addition, if [Formula: see text] is also of locally Hilbert–Schmidt class, then we have a unique pair of DPPs, [Formula: see text], [Formula: see text]. We also give a practical framework which makes [Formula: see text] and [Formula: see text] satisfy the above conditions. Our framework to construct pairs of DPPs implies useful duality relations between DPPs making pairs. For a correlation kernel of a given DPP our formula can provide plural different expressions, which reveal different aspects of the DPP. In order to demonstrate these advantages of our framework as well as to show that the class of DPPs obtained by this method is large enough to study universal structures in a variety of DPPs, we report plenty of examples of DPPs in one-, two- and higher-dimensional spaces S, where several types of weak convergence from finite DPPs to infinite DPPs are given. One-parameter ([Formula: see text]) series of infinite DPPs on [Formula: see text] and [Formula: see text] are discussed, which we call the Euclidean and the Heisenberg families of DPPs, respectively, following the terminologies of Zelditch.

Stirling operators in spatial combinatorics

We define and study a spatial (infinite-dimensional) counterpart of Stirling numbers. In classical combinatorics, the Pochhammer symbol (m)n can be extended from a natural number m∈N to the falling factorials (z)n=z(z−1)⋯(z−n+1) of an argument z from F=R or C, and Stirling numbers of the first and second kinds are the coefficients of the expansions of (z)n through zk, k≤n and vice versa. When taking into account spatial positions of elements in a locally compact Polish space X, we replace N by the space of configurations—discrete Radon measures γ=∑iδxi on X, where δxi is the Dirac measure with mass at xi. The spatial falling factorials (γ)n:=∑i1∑i2≠i1⋯∑in≠i1,…,in≠in−1δ(xi1,xi2,…,xin) can be naturally extended to mappings M(1)(X)∋ω↦(ω)n∈M(n)(X), where M(n)(X) denotes the space of F-valued, symmetric (for n≥2) Radon measures on Xn. There is a natural duality between M(n)(X) and the space CF(n)(X) of F-valued, symmetric continuous functions on Xn with compact support. The Stirling operators of the first and second kind, s(n,k) and S(n,k), are linear operators, acting between spaces CF(n)(X) and CF(k)(X) such that their dual operators, acting from M(k)(X) into M(n)(X), satisfy (ω)n=∑k=1ns(n,k)⁎ω⊗k and ω⊗n=∑k=1nS(n,k)⁎(ω)k, respectively. In the case where X has only a single point, the Stirling operators can be identified with Stirling numbers. We derive combinatorial properties of the Stirling operators, present their connections with a generalization of the Poisson point process and with the Wick ordering under the canonical commutation relations.

Characterizations of countably $n$-rectifiable radon measures by higher-dimensional Menger curvatures

We provide a characterization of countably n-rectifiable measures in terms of $\sigma$-finiteness of the integral Menger curvature. We also prove that a finiteness condition on pointwise Menger curvature can characterize rectifiability of Radon measures. Motivated by the partial converse of Meurer's work by Kolasiński we prove that under suitable density assumptions there is a comparability between pointwise-Menger curvature and the sum over scales of the centered $\beta$-numbers at a point.

On BV functions and essentially bounded divergence-measure fields in metric spaces

By employing the differential structure recently developed by N. Gigli, we first give a notion of functions of bounded variation (BV) in terms of suitable vector fields on a complete and separable metric measure space (\mathbb{X},d,\mu) equipped with a non-negative Radon measure \mu finite on bounded sets. Then, we extend the concept of divergence-measure vector fields \mathcal{DM}^p(\mathbb{X}) for any p\in[1,\infty] and, by simply requiring in addition that the metric space is locally compact, we determine an appropriate class of domains for which it is possible to obtain a Gauss–Green formula in terms of the normal trace of a \mathcal{DM}^\infty(\mathbb{X}) vector field. This differential machinery is also the natural framework to specialize our analysis for \mathsf{RCD}(K,\infty) spaces, where we exploit the underlying geometry to determine the Leibniz rules for \mathcal{DM}^\infty(\mathbb{X}) and ultimately to extend our discussion on the Gauss–Green formulas.

On a certain class of positive definite functions and measures on locally compact Abelian groups and inner-product spaces

Let L be a lattice in Rn and φ the L-periodic Gaussian function on Rn given by φ(x)=∑y∈Le−‖x−y‖2. The paper was motivated by the following observation: φ(x+y)φ(x−y) is a positive definite function of two variables x,y. We say that a positive definite function φ on an Abelian group G is cross positive definite (c.p.d.) if, for each z∈G, the function φ(x+y+z)φ(x−y) of two variables x,y is positive definite. We say that a finite Radon measure on a locally compact Abelian group is c.p.d. if its Fourier transform is a c.p.d. function on the dual group. We investigate properties of c.p.d. functions and measures and give some integral characterizations. It is proved that the classes of c.p.d. functions and measures are closed with respect to certain natural operations. In particular, products, convolutions and Fourier transforms of c.p.d. functions and measures are c.p.d., whenever defined.

Ergodic Decomposition of Dirichlet Forms via Direct Integrals and Applications

We study direct integrals of quadratic and Dirichlet forms. We show that each quasi-regular Dirichlet space over a probability space admits a unique representation as a direct integral of irreducible Dirichlet spaces, quasi-regular for the same underlying topology. The same holds for each quasi-regular strongly local Dirichlet space over a metrizable Luzin σ-finite Radon measure space, and admitting carré du champ operator. In this case, the representation is only projectively unique.

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.

Quasilinear elliptic equations with sub-natural growth terms in bounded domains

We consider the existence of positive solutions to weighted quasilinear elliptic differential equations of the type $$\begin{aligned} {\left\{ \begin{array}{ll} - \Delta _{p, w} u = \sigma u^{q} &{} \text {in }\Omega ,\\ u = 0 &{} \text {on }\partial \Omega \end{array}\right. } \end{aligned}$$ in the sub-natural growth case $$0< q < p - 1$$ , where $$\Omega $$ is a bounded domain in $${\mathbb {R}}^{n}$$ , $$\Delta _{p, w}$$ is a weighted p-Laplacian, and $$\sigma $$ is a nonnegative (locally finite) Radon measure on $$\Omega $$ . We give criteria for the existence problem. For the proof, we investigate various properties of p-superharmonic functions, especially the solvability of Dirichlet problems with infinite measure data.