Nonnegative Radon Measure Research Articles

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.

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.

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.

Structured population models on Polish spaces: A unified approach including graphs, Riemannian manifolds and measure spaces to describe dynamics of heterogeneous populations

This paper presents a mathematical framework for modeling the dynamics of heterogeneous populations. Models describing local and non-local growth and transport processes appear in a variety of applications, such as crowd dynamics, tissue regeneration, cancer development and coagulation-fragmentation processes. The diverse applications pose a common challenge to mathematicians due to the multiscale nature of the structures that underlie the system’s self-organization and control. Similar abstract mathematical problems arise when formulating problems in the language of measure evolution on a multi-faceted state space. Motivated by these observations, we propose a general mathematical framework for nonlinear structured population models on abstract metric spaces, which are only assumed to be separable and complete. We exploit the structure of the space of non-negative Radon measures with the dual bounded Lipschitz distance (flat metric), which is a generalization of the Wasserstein distance, capable of addressing non-conservative problems. The formulation of models on general metric spaces allows considering infinite-dimensional state spaces or graphs and coupling discrete and continuous state transitions. This opens up exciting possibilities for modeling single-cell data, crowd dynamics or coagulation-fragmentation processes.

Gamma-convergence of quadratic functionals perturbed by bounded linear functionals

Given a bounded open set Ω⊂Rn, we study sequences of quadratic functionals on the Sobolev space H01(Ω), perturbed by sequences of bounded linear functionals. We prove that their Γ-limits, in the weak topology of H01(Ω), can always be written as the sum of a quadratic functional, a linear functional, and a non-positive constant. The classical theory of G- and H-convergence completely characterises the quadratic and linear parts of the Γ-limit and shows that their coefficients do not depend on Ω. The constant, which instead depends on Ω and will be denoted by −ν(Ω), plays an important role in the study of the limit behaviour of the energies of the solutions. The main result of this paper is that, passing to a subsequence, we can prove that ν coincides with a non-negative Radon measure on a sufficiently large collection of bounded open sets Ω. Moreover, we exhibit an example that shows that the previous result cannot be obtained for every bounded open set. The specific form of this example shows that the compactness theorem for the localisation method in Γ-convergence cannot be easily improved.

Existence of nonminimal solutions to an inhomogeneous elliptic equation with supercritical nonlinearity

Abstract In our previous paper [K. Ishige, S. Okabe, and T. Sato, A supercritical scalar field equation with a forcing term, J. Math. Pures Appl. 128 (2019), pp. 183–212], we proved the existence of a threshold κ ∗ &gt; 0 {\kappa }^{\ast }\gt 0 such that the elliptic problem for an inhomogeneous elliptic equation − Δ u + u = u p + κ μ -\Delta u+u={u}^{p}+\kappa \mu in R N {{\bf{R}}}^{N} possesses a positive minimal solution decaying at the space infinity if and only if 0 &lt; κ ≤ κ ∗ 0\lt \kappa \le {\kappa }^{\ast } . Here, N ≥ 2 N\ge 2 , μ \mu is a nontrivial nonnegative Radon measure in R N {{\bf{R}}}^{N} with a compact support, and p &gt; 1 p\gt 1 is in the Joseph-Lundgren subcritical case. In this article, we prove the existence of nonminimal positive solutions to the elliptic problem. Our arguments are also applicable to inhomogeneous semilinear elliptic equations with exponential nonlinearity.

A stability result for elliptic equations with singular nonlinearity and its applications to homogenization problems

We consider model semilinear elliptic equations of the type{−div(A(x)∇u)=σu−λ,u>0inΩ,u∈H01(Ω), where Ω is a bounded domain in RN, N≥1, A∈L∞(Ω)N×N is a coercive matrix, 0<λ≤1 and σ is a nonnegative function in Lloc1(Ω), or more generally, nonnegative Radon measure on Ω. We discuss H01(Ω)-stability of u under a minimal assumption on σ and apply the result to homogenization problems.

A Note on the Differentiability of the Hellinger–Kantorovich Distances

This paper will deal with differentiability properties of the class of Hellinger–Kantorovich distances textsf{HK}_{Lambda , Sigma } (Lambda , Sigma > 0) which was recently introduced on the space {mathcal {M}}({mathbb {R}}^d) of finite nonnegative Radon measures. The derivatives of tmapsto textsf{HK}_{Lambda , Sigma }(mu _t, nu _t)^2, for absolutely continuous curves (mu _t)_t, (nu _t)_t in ({mathcal {M}}({mathbb {R}}^d),textsf{HK}_{Lambda , Sigma }), will be computed {mathscr {L}}^1-a.e.. The characterization of absolutely continuous curves in ({mathcal {M}}({mathbb {R}}^d), textsf{HK}_{Lambda , Sigma }) will be refined.

Weak solutions to the complex m-Hessian type equations for arbitrary nonnegative Radon measures on open subsets of $${\mathbb {C}}^n$$

Weak solutions to the complex m-Hessian type equations for arbitrary nonnegative Radon measures on open subsets of $${\mathbb {C}}^n$$

Optimal singularities of initial functions for solvability of a semilinear parabolic system

Let $(u, v)$ be a nonnegative solution to the semilinear parabolic system $$ \mbox{(P)} \qquad \left\{ \begin{array}{ll} \partial_{t} u = D_{1} \Delta u + v^{p}, \quad x \in \mathbf{R}^{N}, \ t > 0,\\ \partial_{t} v = D_{2} \Delta v + u^{q}, \quad x \in \mathbf{R}^{N}, \ t > 0,\\ (u(\cdot,0), v(\cdot,0)) = (\mu, \nu), \quad x \in \mathbf{R}^{N}, \end{array} \right. $$ where $D_{1}$, $D_{2} > 0$, $0 < p \leq q$ with $pq > 1$ and $(\mu, \nu)$ is a pair of nonnegative Radon measures or nonnegative measurable functions in $\mathbf{R}^{N}$. In this paper we study sufficient conditions on the initial data for the solvability of problem (P) and clarify optimal singularities of the initial functions for the solvability.

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.

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.

The Dirichlet problem for sublinear elliptic equations with source

We present a necessary and sufficient condition on nonnegative Radon measures μ and ν for the existence of a positive continuous solution of the Dirichlet problem for the sublinear elliptic equation −Δu=μuq+ν with prescribed nonnegative continuous boundary data in a general domain. Moreover, two-sided pointwise estimates of Brezis–Kamin type for positive bounded solutions and the uniqueness of a positive continuous Lq-solution are investigated.

End Point Estimate of Littlewood-Paley Operator Associated to the Generalized Schrödinger Operator

Let L = − Δ + μ be the generalized Schrödinger operator on ℝ d , d ≥ 3 , where μ ≠ 0 is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. In this work, we give a new BMO space associated to the generalized Schrödinger operator L , BM O θ , L , which is bigger than the BMO spaces related to the classical Schrödinger operators A = − Δ + V , with V a potential satisfying a reverse Hölder inequality introduced by Dziubański et al. in 2005. Besides, the boundedness of the Littlewood-Paley operators associated to L in BM O θ , L also be proved.

A minimizing Movement approach to a class of scalar reaction–diffusion equations

The purpose of this paper is to introduce a Minimizing Movement approach to scalar reaction–diffusion equations of the form [see formula in PDF] with parameters Λ, Σ &gt; 0 and no-flux boundary condition [see formula in PDF] which is built on their gradient-flow-like structure in the space [see formula in PDF] of finite nonnegative Radon measures on [see formula in PDF], endowed with the recently introduced Hellinger-Kantorovich distance HKΛ,Σ. It is proved that, under natural general assumptions on [see formula in PDF] and [see formula in PDF], the Minimizing Movement scheme [see formula in PDF] for [see formula in PDF] yields weak solutions to the above equation as the discrete time step size τ ↓ 0. Moreover, a superdifferentiability property of the Hellinger-Kantorovich distance HKΛ,Σ, which will play an important role in this context, is established in the general setting of a separable Hilbert space; that result will constitute a starting point for the study of the differentiability of HKΛ,Σ along absolutely continuous curves which will be carried out in a subsequent paper.

Existence of solution for a system involving a singular-nonlocal operator, a singularity and a Radon measure

The existence of a nontrivial solution in some ‘weaker’ sense of the system of equations: is proved. Here, , , , l, w are bounded, nonnegative real-valued functions in Ω, μ is a nonnegative Radon measure and k>1 belongs to a certain range.

Barycenters in the Hellinger–Kantorovich Space

Recently, Liero, Mielke and Savare introduced the Hellinger–Kantorovich distance on the space of nonnegative Radon measures of a metric space X. We prove that Hellinger–Kantorovich barycenters always exist for a class of metric spaces containing of compact spaces and Polish $$\mathrm{CAT}(1)$$ spaces; and if we assume further some conditions on the data, such barycenters are unique. We also introduce homogeneous multimarginal problems and illustrate some relations between their solutions and Hellinger–Kantorovich barycenters. Our results are analogous to the work of Agueh and Carlier for Wasserstein barycenters.

Stability/nonstability properties of renormalized/entropy solutions for degenerate parabolic equations with $$L^{1}$$/measure data

We study the possibility to give a formulation to the degenerate parabolic problems modeled by $$\begin{aligned} (\mathcal {P}_{b}^{1})\ {\left\{ \begin{array}{ll} u_{t}-\text {div}\left[ |\nabla u|^{p-2}\nabla u)\text {/}(1+|u|)^{\theta (p-1)}\right] =\mu \text { in } (0,T)\times \Omega ,\\ u(0,x)=u_{0}(x)\text { in }\Omega ,\quad u(t,x)=0\text { on }(0,T)\times \partial \Omega , \end{array}\right. } \end{aligned}$$ where $$\theta >0$$ , $$u_{0}\in L^{1}(\Omega )$$ and $$\mu $$ is a general (nonnegative) Radon measure. We also investigate the strong stability of solutions for noncoercive absorption problems whose model $$\begin{aligned}(\mathcal {P}_{b}^{2})\ {\left\{ \begin{array}{ll} u_{t}-\text {div}\left[ |\nabla u|^{p-2}\nabla u)\text {/}(1+|u|)^{\theta (p-1)}\right] +|u|^{q-1}u=f\text { in }(0,T)\times \Omega ,\\ u(0,x)=0\text { in }\Omega ,\quad u(t,x)=0\hbox { on }(0,T)\times \partial \Omega \end{array}\right. }\end{aligned}$$ where $$q>r(p-1)[1+\theta (p-1)]\text {/}(r-p)$$ and $$f\in L^{1}_{\text {loc}}(Q\backslash K)$$ with K is a compact subset of Q of zero r-capacity (or, is a measure concentrated on a set of r-capacity zero). We prove the convergence of approximate solutions $$u_{n}$$ (related to a regular approximation $$\mu _{n}$$ of $$\mu $$ ) towards a renormalized solutions u of $$(\mathcal {P}_{b}^{1})$$ , and we extend the previous known-results on the nonstability of entropy solutions for problems $$(\mathcal {P}_{b}^{2})$$ .

Asymptotic stability of peakons for the Novikov equation

The Novikov equation is an integrable Camassa-Holm type equation with a cubic nonlinearity. One of the most important features of this equation is the existence of peaked traveling waves, also called peakons. This paper aims to prove the asymptotic stability of peakon solutions under H1(R)-perturbations satisfying that their associated momentum density defines a non-negative Radon measure. Motivated by Molinet's work [24–26], we shall first prove a Liouville property for H1(R) global solutions belonging to a certain class of almost localized functions. More precisely, we show that such solutions have to be a peakon. The main novelty in our analysis in comparison to the Camassa-Holm equation comes from the fact that in our present case the momentum is not a conserved quantity and may be unbounded along the trajectory. In this regard, to prove the Liouville property, we used a new Lyapunov functional not related to the (not conserved) momentum of the equation.

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.