Radon measure
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On singular elliptic equations with measure sources Open
We prove existence of solutions for a class of singular elliptic problems\nwith a general measure as source term whose model is\n $$\\begin{cases} -\\Delta u = \\frac{f(x)}{u^{\\gamma}} +\\mu & \\text{in}\\ \\Omega,\nu=0 &\\text{on}\\ \\pa…
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A new optimal transport distance on the space of finite Radon measures Open
We introduce a new optimal transport distance between nonnegative finite Radon measures with possibly different masses. The construction is based on non-conservative continuity equations and a corresponding modified Benamou-Brenier formula…
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A new optimal transport distance on the space of finite Radon measures Open
We introduce a new optimal transport distance between nonnegative finite Radon measures with possibly different masses. The construction is based on non-conservative continuity equations and a corresponding modified Benamou-Brenier formula…
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Discontinuous sweeping process with prox-regular sets Open
In this paper, we study the well−posedness (in the sense of existence and uniqueness of a solution) of a discontinuous sweeping process involving prox-regular sets in Hilbert spaces. The variation of the moving set is controlled by a posit…
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Multiscale Analysis of 1-rectifiable Measures II: Characterizations Open
A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the…
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Elliptic equations with general singular lower order term and measure data Open
In this paper we study a nonlinear elliptic boundary value problem with a general singular lower order term, whose model is {-Delta u = H(u)mu, in Omega, u = 0 on partial derivative Omega, u > 0 on Omega, where Omega is an open bounded sub…
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A Lazer-McKenna type problem with measures Open
In this paper we are concerned with a general singular Dirichlet boundary value problem whose model is the following $$ \begin{cases} -Δu = \fracμ{u^γ} & \text{in}\ Ω, u=0 &\text{on}\ \partialΩ, u>0 &\text{on}\ Ω\,. \end{cases} $$ Here $μ$…
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Relaxation and Integral Representation forFunctionals of Linear Growth on MetricMeasure spaces Open
This article studies an integral representation of functionals of linear growth on metric measure spaces with a doubling measure and a Poincaré inequality. Such a functional is defined via relaxation, and it defines a Radon measure on the …
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Existence results for some nonlinear elliptic equations with measure data in Orlicz-Sobolev spaces Open
We prove the existence results in the setting of Orlicz spaces for the following nonlinear elliptic equation: $$A(u)+g(x,u,Du)=\mu, $$ where A is a Leray-Lions operator defined on $D(A)\subset W_{0}^{1}L_{M}(\Omega)$ , while g is a nonline…
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Weak solutions of semilinear elliptic equations with Leray-Hardy potentials and measure data Open
We study existence and stability of solutions of (E 1) --$\\Delta$u + $\\mu$\n|x| 2 u + g(u) = $\\nu$ in $\\Omega$, u = 0 on $\\partial$$\\Omega$, where $\\Omega$\nis a bounded, smooth domain of R N , N $\\ge$ 2, containing the origin, $\\…
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Le problème de Minkowski équivariant dans l’espace de Minkowski Open
The classical Minkowski problem in Minkowski space asks, for a positive function $\phi$ on $\mathbb{H}^d$, for a convex set $K$ in Minkowski space with $C^2$ space-like boundary $S$, such that $\phi(\eta)^{-1}$ is the Gauss--Kronecker curv…
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Regularity estimates for quasilinear elliptic equations with variable growth involving measure data Open
We investigate a quasilinear elliptic equation with variable growth in a bounded nonsmooth domain involving a signed Radon measure. We obtain an optimal global Calderón–Zygmund type estimate for such a measure data problem, by proving that…
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Fractional heat equations with subcritical absorption having a measure as initial data Open
Nonlinear Analysis, Theory, Methods and Applications, to appear.
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Geometric conditions for the $L^2$-boundedness of singular integral operators with odd kernels with respect to measures with polynomial growth in $\mathbb{R}^d$ Open
Let $μ$ be a finite Radon measure in $\mathbb{R}^d$ with polynomial growth of degree $n$, although not necessarily $n$-AD-regular. We prove that under some geometric conditions on $μ$ that are closely related to rectifiability and involve …
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Necessary Condition for Rectifiability Involving Wasserstein Distance W2 Open
A Radon measure $\mu $ is $n$-rectifiable if it is absolutely continuous with respect to $n$-dimensional Hausdorff measure and $\mu $-almost all of ${\operatorname{supp}}\mu $ can be covered by Lipschitz images of $\mathbb{R}^n$. In this p…
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Differentiability in perturbation parameter of measure solutions to perturbed transport equation Open
We consider a linear perturbation in the velocity field of the transport equation. We investigate solutions in the space of bounded Radon measures and show that they are differentiable with respect to the perturbation parameter in a proper…
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Using Deep Learning to Model the Groundwater Tracer Radon in Coastal Waters Open
Submarine groundwater discharge (SGD) is an important driver of coastal biogeochemical budgets worldwide. Radon ( 222 Rn) has been widely used as a natural geochemical tracer to quantify SGD, but field measurements are time consuming and c…
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Pointwise estimates via parabolic potentials for a class of doubly nonlinear parabolic equations with measure data Open
On a cylindrical domain $$E_T$$ , we consider doubly nonlinear parabolic equations, whose prototype is $$\partial _t u - \mathrm{div}(|u|^{m-1}|Du|^{p-2}Du) = \mu ,$$ where $$\mu $$ is a non-negative Radon measure having finite total mass …
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On BV functions and essentially bounded divergence-measure fields in metric spaces Open
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) eq…
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An Intrinsic Characterization of Moment Functionals in the Compact Case Open
We consider the class of all linear functionals $L$ on a unital commutative real algebra $A$ that can be represented as an integral w.r.t. to a Radon measure with compact support in the character space of $A$. Exploiting a recent generaliz…
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Quasilinear and Hessian Lane-Emden type systems with measure data Open
We study nonlinear systems of the form $-{\Delta }_{p}u=v^{q_{1}}+\mu , -{\Delta }_{p}v=u^{q_{2}}+\eta $ and $F_{k}[-u]=v^{s_{1}}+\mu , F_{k}[-v]=u^{s_{2}}+\eta $ in a bounded domain Ω or in $\mathbb {R}^{N}$ where μ and η are nonnegative …
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A direct approach to Plateau's problem Open
We provide a compactness principle which is applicable to different formulations of Plateau’s problem in codimension one and which is exclusively based on the theory of Radon measures and elementary comparison arguments. Exploiting some ad…
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Characterization of rectifiable measures in terms of $\alpha$-numbers Open
We characterize Radon measures $\\mu$ in $\\mathbb{R}^{n}$ that are\n$d$-rectifiable in the sense that their supports are covered up to\n$\\mu$-measure zero by countably many $d$-dimensional Lipschitz graphs and $\\mu\n\\ll \\mathcal{H}^{d…
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Fast reaction limit and forward-backward diffusion: A Radon–Nikodym approach Open
We consider two singular limits: a fast reaction limit with a non-monotone nonlinearity and a regularization of the forward-backward diffusion equation. We derive pointwise identities satisfied by the Young measure generated by these probl…
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Good Radon measure for anisotropic problems with variable exponent Open
We study nonlinear anisotropic problems with bounded Radon diffuse measure and variable exponent. We prove the existence and uniqueness of entropy solution.
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Radon Transforms in Hyperbolic Spaces and Their Discrete Counterparts Open
In the hyperbolic disc (or more generally in real hyperbolic spaces) we consider the horospherical Radon transform R and the geodesic Radon transform X . Composition with their respective dual operators yields two convolution operators on …
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Signed Radon measure-valued solutions of flux saturated scalar conservation laws Open
We prove existence and uniqueness for a class of signed Radon measure-valued entropy solutions of the Cauchy problem for a first order scalar hyperbolic conservation law in one space dimension. The initial data of the problem is a finite s…
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Lipschitz continuity results for a class of obstacle problems Open
We prove Lipschitz continuity results for solutions to a class of obstacle problems under standard growth conditions of p -type, p \geq 2 . The main novelty is the use of a linearization technique going back to [28] in order to interpret o…
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Pointwise Multipliers on BMO Spaces with Non-doubling Measures Open
Let $\\mu$ be a non-negative Radon measure satisfying the polynomial growth condition. In this paper, the authors characterize the set of pointwise multipliers on a BMO type space $\\operatorname{RBMO}(\\mu)$ introduced by Tolsa.
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Nonlinear multivalued problems with variable exponent and diffuse measure data in anisotropic space Open
We study a nonlinear elliptic problem governed by a general anisotropic operator with variable exponents and Radon diffuse measure data. We start by proving the existence of a renormalized solution and we establish that the notion of renor…