Nonnegative Measure Research Articles

On a class of forward-backward parabolic equations: Formation of singularities

We study the formation of singularities for the problem{ut=[φ(u)]xx+ε[ψ(u)]txxinΩ×(0,T)φ(u)+ε[ψ(u)]t=0in∂Ω×(0,T)u=u0≥0inΩ×{0}, where ϵ and T are positive constants, Ω a bounded interval, u0 a nonnegative Radon measure on Ω, φ a nonmonotone and nonnegative function with φ(0)=φ(∞)=0, and ψ an increasing bounded function. We show that if u0 is a bounded or continuous function, singularities may appear spontaneously. The class of singularities which can arise in finite time is remarkably large, and includes infinitely many Dirac masses and singular continuous measures.

Existence of a unique positive entropy solution to a singular fractional Laplacian

In this paper, we study the existence of a positive solution to the elliptic problem: Here (N>2s) is an open bounded domain with smooth boundary, and . For , we take advantage of the convexity of Ω. The operator indicates the restricted fractional Laplacian, and μ is a non-negative Radon measure as a source term. The assumptions on f and h will be precised later. Besides, we will discuss the notion of entropy solution and its uniqueness for some specific measures.

The Hopf Lemma for the Schrödinger Operator

Abstract We prove the Hopf boundary point lemma for solutions of the Dirichlet problem involving the Schrödinger operator - Δ + V {-\Delta+V} with a nonnegative potential V which merely belongs to L loc 1 ⁢ ( Ω ) {L_{\mathrm{loc}}^{1}(\Omega)} . More precisely, if u ∈ W 0 1 , 2 ⁢ ( Ω ) ∩ L 2 ⁢ ( Ω ; V ⁢ d ⁢ x ) {u\in W_{0}^{1,2}(\Omega)\cap L^{2}(\Omega;V\mathop{}\!\mathrm{d}{x})} satisfies - Δ ⁢ u + V ⁢ u = f {-\Delta u+Vu=f} on Ω for some nonnegative datum f ∈ L ∞ ⁢ ( Ω ) {f\in L^{\infty}(\Omega)} , f ≢ 0 {f\not\equiv 0} , then we show that at every point a ∈ ∂ ⁡ Ω {a\in\partial\Omega} where the classical normal derivative ∂ ⁡ u ⁢ ( a ) ∂ ⁡ n {\frac{\partial u(a)}{\partial n}} exists and satisfies the Poisson representation formula, one has ∂ ⁡ u ⁢ ( a ) ∂ ⁡ n > 0 {\frac{\partial u(a)}{\partial n}>0} if and only if the boundary value problem { - Δ ⁢ v + V ⁢ v = 0 in Ω , v = ν on ∂ ⁡ Ω , \begin{dcases}\begin{aligned} \displaystyle-\Delta v+Vv&\displaystyle=0&&% \displaystyle\phantom{}\text{in ${\Omega}$,}\\ \displaystyle v&\displaystyle=\nu&&\displaystyle\phantom{}\text{on ${\partial% \Omega}$,}\end{aligned}\end{dcases} involving the Dirac measure ν = δ a {\nu=\delta_{a}} has a solution. More generally, we characterize the nonnegative finite Borel measures ν on ∂ ⁡ Ω {\partial\Omega} for which the boundary value problem above has a solution in terms of the set where the Hopf lemma fails.

Existence of minimal solutions to quasilinear elliptic equations with several sub-natural growth terms

We study the existence of positive solutions to quasilinear elliptic equations of the type −Δpu=σuq+μinRn,in the sub-natural growth case 0<q<p−1, where Δpu=∇⋅(|∇u|p−2∇u) is the p-Laplacian with 1<p<n, and σ and μ are nonnegative Radon measures on Rn. We construct minimal generalized solutions under certain generalized energy conditions on σ and μ. To prove this, we give new estimates for interaction between measures. We also construct solutions to equations with several sub-natural growth terms using the same methods.

Quasilinear elliptic equations with exponential nonlinearity and measure data

Let be either a bounded domain or the entire space. This paper concerns existence and global pointwise estimates in terms of Wolff potentials of solutions to quasilinear and Hessian equations of Lane‐Emden type: under minimal assumptions on σ and ω, where σ is an A∞ weight and ω is a nonnegative Borel measure, and with γ&gt;0, β ≥ 1. Our methods are based on systematic application of Wolff potentials and good‐λ inequality for fractional maximal potential.

Regularity of the minimiser of one-dimensional interaction energies

We consider both the minimisation of a class of nonlocal interaction energies over non-negative measures with unit mass and a class of singular integral equations of the first kind of Fredholm type. Our setting covers applications to dislocation pile-ups, contact problems, fracture mechanics and random matrix theory. Our main result shows that both the minimisation problems and the related singular integral equations have the same unique solution, which provides new regularity results on the minimiser of the energy and new positivity results on the solutions to singular integral equations.

A tumor growth model of Hele-Shaw type as a gradient flow

In this paper, we characterize a degenerate PDE as the gradient flow in the space of nonnegative measures endowed with an optimaltransport-growthmetric. The PDE of concern, of Hele-Shaw type, was introduced by Perthameet. al. as a mechanical model for tumor growth and the metric was introduced recently in several articles as the analogue of the Wasserstein metric for nonnegative measures. We show existence of solutions usingminimizing movementsand show uniqueness of solutions on convex domains by proving theEvolutional Variational Inequality. Our analysis does not require any regularity assumption on the initial condition. We also derive a numerical scheme based on the discretization of the gradient flow and the idea of entropic regularization. We assess the convergence of the scheme on explicit solutions. In doing this analysis, we prove several new properties of the optimal transport-growth metric, which generally have a known counterpart for the Wasserstein metric.

Subdifferential Calculus Rules for Possibly Nonconvex Integral Functions

We are concerned with the subdifferentials of integral functionals and functions given in the form $E_f(x)=\int_{T} f(t,x)d\mu$, for a possibly nonconvex normal integrand f defined on a separable Banach with separable dual and a nonnegative sigma-finite measure mu. We establish some limit-based estimates for the Fréchet and the limiting subdifferentials of E_f, covering the cases of Lipschitz and non-Lipschitz integrands.

An inexact primal-dual algorithm for semi-infinite programming

This paper considers an inexact primal-dual algorithm for semi-infinite programming (SIP) for which it provides general error bounds. To implement the dual variable update, we create a new prox function for nonnegative measures which turns out to be a generalization of the Kullback-Leibler divergence for probability distributions. We show that under suitable conditions on the error, this algorithm achieves an $\mathcal{O}(1/\sqrt{K})$ rate of convergence in terms of the optimality gap and constraint violation. We then use our general error bounds to analyze the convergence and sample complexity of a specific primal-dual SIP algorithm based on Monte Carlo integration. Finally, we provide numerical experiments to demonstrate the performance of our algorithm.

Iterated Commutators for Multilinear Singular Integral Operators on Morrey Space with Non-Doubling Measures

Let μ be a non-negative Radon measure on Rd which only satisfies the following growth condition that there exists a positive constant C such that μ(B(x,r)) ≤ Crn for all x∈ Rd, r > 0 and some fixed n ∈ (0,d]. This paper is interested in the properties of the iterated commutators of multilinear singular integral operators on Morrey spaces .Precisely speaking, we show that the iterated commutators generated by multilinear singular integrals operators are bounded from to where (Regular Bounded Mean Oscillation space) and 1 qj ≤ pj ∞ with 1/p = 1/p1 + ... + 1/pm and 1/q = 1/q1+ ... + 1/qm.

Discrete-time population dynamics on the state space of measures.

If the individual state space of a structured population is given by a metric space S, measures μ on the σ-algebra of Borel subsets T of S offer a modeling tool with a natural interpretation: μ(T) is the number of individuals with structural characteristics in the set T. A discrete-time population model is given by a population turnover map F on the cone of finite nonnegative Borel measures that maps the structural population distribution of a given year to the one of the next year. Under suitable assumptions, F has a first order approximation at the zero measure (the extinction fixed point), which is a positive linear operator on the ordered vector space of real measures and can be interpreted as a basic population turnover operator. For a semelparous population, it can be identified with the next generation operator. A spectral radius can be defined by the usual Gelfand formula.We investigate in how far it serves as a threshold parameter between population extinction and population persistence. The variation norm on the space of measures is too strong to give the basic turnover operator enough compactness that its spectral radius is an eigenvalue associated with a positive eigenmeasure. A suitable alternative is the flat norm (also known as (dual) bounded Lipschitz norm), which, as a trade-off, makes the basic turnover operator only continuous on the cone of nonnegative measures but not on the whole space of real measures.

Schur analysis of matricial Hausdorff moment sequences

We develop the algebraic instance of an algorithmic approach to the matricial Hausdorff moment problem on a compact interval [α,β] of the real axis. Our considerations are along the lines of the classical Schur algorithm and the treatment of the Hamburger moment problem on the real axis by Nevanlinna. More precisely, a transformation of matrix sequences is constructed, which transforms Hausdorff moment sequences into Hausdorff moment sequences reduced by 1 in length. It is shown that this transformation corresponds essentially to the left shift of the associated sequences of canonical moments. As an application, we show that a matricial version of the arcsine distribution can be used to characterize a certain centrality property of non-negative Hermitian measures on [α,β].

A note on generalized averaged Gaussian formulas for a class of weight functions

In the recent paper Notaris (Numer. Math., 142:129–147, 2019) it has been introduced a new and useful class of nonnegative measures for which the well-known Gauss–Kronrod quadrature formulae coincide with the generalized averaged Gaussian quadrature formulas. In such a case, the given generalized averaged Gaussian quadrature formulas are of the higher degree of precision, and can be numerically constructed by an effective and simple method; see Spalevic (Math. Comp., 76:1483–1492, 2007). Moreover, as almost immediate consequence of our results from Spalevic (Math. Comp.,76:1483–1492, 2007) and that theory, we prove the main statements in Notaris (Numer. Math.,142:129–147, 2019) in a different manner, by means of the Jacobi tridiagonal matrix approach.

Nonlinear elliptic equations with measure valued absorption potentials

We study the semilinear elliptic equation −∆u + g(u)σ = µ with Dirichlet boundary condition in a smooth bounded domain where σ is a nonnegative Radon measure, µ a Radon measure and g is an absorbing nonlinearity. We show that the problem is well posed if we assume that σ belongs to some Morrey class. Under this condition we give a general existence result for any bounded measure provided g satisfies a subcritical integral assumption. We study also the supercritical case when g(r) = |r| ^{q−1} r, with q > 1 and µ satisfies an absolute continuity condition expressed in terms of some capacities involving σ. 2010 Mathematics Subject Classification. 35 J 61; 31 B 15; 28 C 05 .

Efficient mass and stiffness matrix assembly via weighted Gaussian quadrature rules for B-splines

Calabrò et al. (2017) changed the paradigm of the mass and stiffness computation from the traditional element-wise assembly to a row-wise concept, showing that the latter one offers integration that may be orders of magnitude faster. Considering a B-spline basis function as a non-negative measure, each mass matrix row is integrated by its own quadrature rule with respect to that measure. Each rule is easy to compute as it leads to a linear system of equations, however, the quadrature rules are of the Newton–Cotes type, that is, they require a number of quadrature points that is equal to the dimension of the spline space. In this work, we propose weighted quadrature rules of Gaussian type which require the minimum number of quadrature points while guaranteeing exactness of integration with respect to the weight function. The weighted Gaussian rules arise as solutions of non-linear systems of equations. We derive rules for the mass and stiffness matrices for uniform C1 quadratic and C2 cubic isogeometric discretizations. In each parameter direction, our rules require locally only p+1 quadrature points, p being the polynomial degree. While the nodes cannot be reused for various weight functions as in Calabrò et al. (2017), the computational cost of the mass and stiffness matrix assembly is comparable.

Finite difference schemes for a structured population model in the space of measures.

We present two finite-difference methods for approximating solutions to a structured population model in the space of non-negative Radon Measures. The first method is a first-order upwind-based scheme and the second is high-resolution method of second-order. We prove that the two schemes converge to the solution in the Bounded-Lipschitz norm. Several numerical examples demonstrating the order of convergence and behavior of the schemes around singularities are provided. In particular, these numerical results show that for smooth solutions the upwind and high-resolution methods provide a first-order and a second-order approximation, respectively. Furthermore, for singular solutions the second-order high-resolution method is superior to the first-order method.

Sensitivity equations for measure-valued solutions to transport equations.

We consider the following transport equation in the space of bounded, nonnegative Radon measures $\mathcal{M}^+(\mathbb{R}^d)$:$$ ∂_t\mu_t + ∂_x(v(x) \mu_t) = 0.$$We study the sensitivity of the solution $\mu_t$ with respect to a perturbation in the vector field, $v(x)$. In particular, we replace the vector field $v$ with a perturbation of the form $v^h = v_0(x) + h v_1(x)$ and let $\mu^h_t$ be the solution of $$ ∂_t\mu^h_t + ∂_x(v^h(x)\mu^h_t) = 0.$$We derive a partial differential equation that is satisfied by the derivative of $\mu^h_t$ with respect to $h$, $∂artial_h(\mu_t^h)$. We show that this equation has a unique very weak solution on the space $Z$, being the closure of $\mathcal{M}(\mathbb{R}^d)$ endowed with the dual norm $(C^{1,\alpha}(\mathbb{R}^d))^*$. We also extend the result to the nonlinear case where the vector field depends on $\mu_t$, i.e., $v=v[\mu_t](x)$.

The full moment problem on subsets of probabilities and point configurations

The aim of this paper is to study the full K−moment problem for measures supported on some particular non-linear subsets K of an infinite dimensional vector space. We focus on the case of random measures, that is K is a subset of all non-negative Radon measures on Rd. We consider as K the space of sub-probabilities, probabilities and point configurations on Rd. For each of these spaces we provide at least one representation as a generalized basic closed semi-algebraic set to apply the main result in Infusino et al. (2014) [20]. We demonstrate that this main result can be significantly improved by further considerations based on the particular chosen representation of K. In the case when K is a space of point configurations, the correlation functions (also known as factorial moment functions) are easier to handle than the ordinary moment functions. Hence, we additionally express the main results in terms of correlation functions.

Trace formulas and continuous dependence of spectra for the periodic conservative Camassa–Holm flow

This article is concerned with the isospectral problem−f″+14f=zωf+z2υf for the periodic conservative Camassa–Holm flow, where ω is a periodic real distribution in Hloc−1(R) and υ is a periodic non-negative Borel measure on R. We develop basic Floquet theory for this problem, derive trace formulas for the associated spectra and establish continuous dependence of these spectra on the coefficients with respect to a weak⁎ topology.

On isometric embeddings of Wasserstein spaces – the discrete case

The aim of this short paper is to offer a complete characterization of all (not necessarily surjective) isometric embeddings of the Wasserstein space Wp(X), where X is a countable discrete metric space and 0<p<∞ is any parameter value. Roughly speaking, we will prove that any isometric embedding can be described by a special kind of X×(0,1]-indexed family of nonnegative finite measures. Our result implies that a typical non-surjective isometric embedding of Wp(X) splits mass and does not preserve the shape of measures. In order to stress that the lack of surjectivity is what makes things challenging, we will prove alternatively that Wp(X) is isometrically rigid for all 0<p<∞.