Radon Measure Research Articles

A class of piecewise constant Radon measure solutions to Riemann problems of compressible Euler equations with discontinuous fluxes: pressureless flow versus Chaplygin gas

A class of piecewise constant Radon measure solutions to Riemann problems of compressible Euler equations with discontinuous fluxes: pressureless flow versus Chaplygin gas

A corona theorem for an algebra of Radon measures with an application to exact controllability for linear controlled delayed difference equations

A corona theorem for an algebra of Radon measures with an application to exact controllability for linear controlled delayed difference equations

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.

Tensorial and Hadamard Product Integral Inequalities for Convex Functions of Continuous Fields of Operators in Hilbert Spaces

Let H be a Hilbert space and Ω a locally compact Hausdorff space endowed with a Radon measure μ with ∫_{Ω}1dμ(t)=1. In this paper we show among others that, if f is continuous differentiable convex on the open interval I, (A_{τ})_{τ∈Ω} is a continuous field of positive operators in B(H) such that Sp(A_{τ}) ⊂I for each τ∈Ω and B and operator such that Sp(B)⊂I, then we have ∫_{Ω}(f′(A_{τ})A_{τ})dμ(τ)⊗1-∫_{Ω}f′(A_{τ})dμ(τ)⊗B ≥∫_{Ω}f(A_{τ})dμ(τ)⊗1-1⊗f(B) ≥(∫_{Ω}A_{τ}dμ(τ)⊗1-(1⊗B))(1⊗f′(B)) and the Hadamard product inequality ∫_{Ω}(f′(A_{τ})A_{τ})dμ(τ)∘1-∫_{Ω}f′(A_{τ})dμ(τ)∘B ≥∫_{Ω}f(A_{τ})dμ(τ)∘1-1∘f(B) ≥∫_{Ω}A_{τ}dμ(τ)∘f′(B)-1∘(f′(B)B).

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.

Convex integrals of molecules in Lipschitz-free spaces

We introduce convex integrals of molecules in Lipschitz-free spaces F(M) as a continuous counterpart of convex series considered elsewhere, based on the de Leeuw representation. Using optimal transport theory, we show that these elements are determined by cyclical monotonicity of their supports, and that under certain finiteness conditions they agree with elements of F(M) that are induced by Radon measures on M, or that can be decomposed into positive and negative parts. We also show that convex integrals differ in general from convex series of molecules. Finally, we present some standalone results regarding extensions of Lipschitz functions which, combined with the above, yield applications to the extremal structure of F(M). In particular, we show that all elements of F(M) are convex series of molecules when M is uniformly discrete and identify all extreme points of the unit ball of F(M) in that case.

Radon measure solutions of spherically symmetric isentropic compressible Euler equations

Radon measure solutions of spherically symmetric isentropic compressible Euler equations

Uniqueness of Ricci flows from non‐atomic Radon measures on Riemann surfaces

AbstractIn previous work (Topping and Yin, https://arxiv.org/abs/2107.14686), we established the existence of a Ricci flow starting with a Riemann surface coupled with a non‐atomic Radon measure as a conformal factor. In this paper, we prove uniqueness, settling Conjecture 1.3 of Topping and Yin (https://arxiv.org/abs/2107.14686). Combining these two works yields a canonical smoothing of such rough surfaces that also regularises their geometry at infinity.

Gradient estimates for mixed local and nonlocal parabolic problems with measure data

We obtain a nonlinear Calderón-Zygmund type estimate for the spatial gradient of SOLA (solutions obtained as limits of approximations) to a mixed local and nonlocal parabolic problem involving measure data. Specifically, we consider a non-homogeneous problem with a source term as a Radon measure, where the leading nonlinear local operator has linear growth and the nonlinearity is allowed to be merely measurable. Moreover, the nonlocal operator is of the type fractional Laplacian with only measurable kernel coefficients. Under some smallness assumption on the local operator, we prove that the gradient of SOLA has the same integrability as that of a fractional maximal function of order 1 of the Radon measure. The proof relies on the use of a new type of maximal functions and novel parabolic tail estimates, which to the best of our knowledge, have not been observed in the previous works. Furthermore, to complement the aforementioned gradient estimates, we prove the existence of SOLA to an initial value problem and derive some gradient estimates with respect to the L1-norm.

Towards optimal sensor placement for inverse problems in spaces of measures

The objective of this work is to quantify the reconstruction error in sparse inverse problems with measures and stochastic noise, motivated by optimal sensor placement. To be useful in this context, the error quantities must be explicit in the sensor configuration and robust with respect to the source, yet relatively easy to compute in practice, compared to a direct evaluation of the error by a large number of samples. In particular, we consider the identification of a measure consisting of an unknown linear combination of point sources from a finite number of measurements contaminated by Gaussian noise. The statistical framework for recovery relies on two main ingredients: first, a convex but non-smooth variational Tikhonov point estimator over the space of Radon measures and, second, a suitable mean-squared error based on its Hellinger–Kantorovich distance to the ground truth. To quantify the error, we employ a non-degenerate source condition as well as careful linearization arguments to derive a computable upper bound. This leads to asymptotically sharp error estimates in expectation that are explicit in the sensor configuration. Thus they can be used to estimate the expected reconstruction error for a given sensor configuration and guide the placement of sensors in sparse inverse problems.

Generalized Schrödinger operators on the Heisenberg group and Hardy spaces

Let L=−ΔHn+μ be a generalized Schrödinger operator on the Heisenberg group Hn, where ΔHn is the sub-Laplacian, and μ is a nonnegative Radon measure satisfying certain conditions. In this paper, we first establish some estimates of the fundamental solution and the heat kernel of L. Applying these estimates, we then study the Hardy spaces HL1(Hn) introduced in terms of the maximal function associated with the heat semigroup e−tL; in particular, we obtain an atomic decomposition of HL1(Hn), and prove the Riesz transform characterization of HL1(Hn). The dual space of HL1(Hn) is also studied.

Mixed-norm of orthogonal projections and analytic interpolation on dimensions of measures

Suppose that \mu and \nu are compactly supported Radon measures on \mathbb{R}^{d} , V\in G(d,n) is an n -dimensional subspace, and let \pi_{V}\colon \mathbb{R}^{d}\rightarrow V denote the orthogonal projection. In this paper, we study the mixed-norm \int \|\pi^{y}\mu\|_{L^p(G(d,n))}^{q} d\nu(y) , where \pi^{y}\mu(V):=\int_{y+V^\perp}\mu \, d\mathcal{H}^{d-n}=\pi_{V} \mu(\pi_{V}y), assuming \mu has continuous density. When n=d-1 and p=q , our result significantly improves a previous result of Orponen on radial projections. We also discuss about consequences including jump discontinuities in the range of p , and m -planes determined by a set of given Hausdorff dimension. In the proof, we run analytic interpolation not only on p and q , but also on dimensions of measures. This is partially inspired by previous work of Greenleaf and Iosevich on Falconer-type problems. We also introduce a new quantity called s -amplitude, that is crucial for our interpolation and gives an alternative definition of Hausdorff dimension.

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 $\mathbb {D}$ .

带有混合奇异项和测度项的分数阶<i>p</i>-Laplace方程解的存在性问题

带有混合奇异项和测度项的分数阶<i>p</i>-Laplace方程解的存在性问题

Linear inverse problems with nonnegativity constraints: Singularity of optimisers

We look at continuum solutions in optimisation problems associated to linear inverse problems $ y = Ax $ with non-negativity constraint $ x \geq 0 $. We focus on the case where the noise model leads to maximum likelihood estimation through general divergences, which cover a wide range of common noise statistics such as Gaussian and Poisson. Considering $ x $ as a Radon measure over the domain on which the reconstruction is taking place, we show a general singularity result. In the high noise regime corresponding to $ y \notin\{A x \mid x \geq 0\} $ and under a key assumption on the divergence as well as on the operator $ A $, any optimiser has a singular part with respect to the Lebesgue measure. We hence provide an explanation as to why any possible algorithm successfully solving the optimisation problem will lead to undesirably spiky-looking images when the image resolution gets finer, a phenomenon well documented in the literature. We illustrate these results with several numerical examples inspired by medical imaging.

Quasilinear elliptic equations involving measure valued absorption terms and measure data

AbstractLet 1 < p < N and Ω ⊂ ℝN be an open bounded domain. We study the existence of solutions to equation $$(E) - {\Delta _p}u + g(u)\sigma = \mu $$ ( E ) − Δ p u + g ( u ) σ = μ in Ω, where g ∈ C(ℝ) is a nondecreasing function, μ is a bounded Radon measure on Ω and σ is a nonnegative Radon measure on ℝN. We show that if σ belongs to some Morrey space of signed measures, then we may investigate the existence of solutions to equation (E) in the framework of renormalized solutions. Furthermore, imposing a subcritical integral condition on g, we prove that equation (E) admits a renormalized solution for any bounded Radon measure μ. When $$g(t) = |t{|^{q - 1}}t$$ g ( t ) = | t | q − 1 t with q > p − 1, we give various sufficient conditions for the existence of renormalized solutions to (E). These sufficient conditions are expressed in terms of Bessel capacities.

Min-max harmonic maps and a new characterization of conformal eigenvalues

Given a surface M and a fixed conformal class c one defines \Lambda_{k}(M,c) to be the supremum of the k -th nontrivial Laplacian eigenvalue over all metrics g\in c of unit volume. It has been observed by Nadirashvili that the metrics achieving \Lambda_{k}(M,c) are closely related to harmonic maps to spheres. In the present paper, we identify \Lambda_{1}(M,c) and \Lambda_{2}(M,c) with min-max quantities associated to the energy functional for sphere-valued maps. As an application, we obtain several new eigenvalue bounds, including a sharp isoperimetric inequality for the first two Steklov eigenvalues. This characterization also yields an alternative proof of the existence of maximal metrics realizing \Lambda_{1}(M,c) , \Lambda_{2}(M,c) , and moreover allows us to obtain a regularity theorem for maximal Radon measures satisfying a natural compactness condition.

Minimal [formula omitted]-solutions to singular sublinear elliptic problems

We solve the existence problem for the minimal positive solutions u∈Lp(Ω,dx) to the Dirichlet problems for sublinear elliptic equations of the form Lu=σuq+μinΩ,lim infx→yu(x)=0y∈∂∞Ω,where 0<q<1 and Lu≔−div(A(x)∇u) is a linear uniformly elliptic operator with bounded measurable coefficients. The coefficient σ and data μ are nonnegative Radon measures on an arbitrary domain Ω⊂Rn with a positive Green function associated with L. Our techniques are based on the use of sharp Green potential pointwise estimates, weighted norm inequalities, and norm estimates in terms of generalized energy.

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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.