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This paper is devoted to the existence and Lipschitz continuity of global conservative weak solutions in time for the modified two-component Camassa–Holm system on the real line. We obtain the global weak solutions via a coordinate transformation into the Lagrangian coordinates. The key ingredients in our analysis are the energy density given by the positive Radon measure and the proposed new distance functions as well.

This article shows that if H L p , κ and H L μ q , λ are ( p , κ ) -Hardy–Morrey space based on the standard Lebesgue measure and ( q , λ ) -Hardy–Morrey space induced by the non-negative Radon measure μ respectively then one has the following trace law for the Hardy–Morrey–Sobolev space I α ( H L p , κ ) of Riesz potentials of order α of H L p , κ -functions: sup 0 ‖ f ‖ H L p , κ ∞ ⁡ ‖ I α f ‖ H L μ q , λ ‖ f ‖ H L p , κ − 1 ∞ ⟺ sup ( x , r ) ∈ R n × ( 0 , ∞ ) ⁡ r − β μ ( B ( x , r ) ) ∞ under { 0 λ ≤ κ ≤ n ; 0 α n ; 0 p κ / α ; n − α p β ≤ n ; 0 q = p ( β + λ − n ) / ( κ − α p ) .

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.

We investigate a quasilinear elliptic equation with variable growth in a bounded nonsmooth domain involving a signed Radon measure. We obtain an optimal global Calderon–Zygmund type estimate for such a measure data problem, by proving that the gradient of a very weak solution to the problem is as globally integrable as the first order maximal function of the associated measure, up to a correct power, under minimal regularity requirements on the nonlinearity, the variable exponent and the boundary of the domain.

We study the existence of solutions of the semilinear parabolic equation ∂ t u − Δ u = u p + Λ in D ′ ( R N × I ) . Here N ≥ 2 , p > 1 , ( N − 2 ) p N and I ⊂ R is an open interval. It is assumed that Λ ∈ D ′ ( R N × I ) is supported on { ( ξ ( t ) , t ) ∈ R N + 1 ; t ∈ I } for some ξ ∈ C 1 / 2 ( I ‾ ; R N ) and is given by Λ ( φ ) = ∫ I φ ( ξ ( t ) , t ) d μ ( t ) , where μ is a Radon measure on I satisfying μ ( ( a , b ) ) ≤ ϕ ( b − a ) ( a , b ∈ I ) for some positive function ϕ ( η ) . In the case where I is bounded, we give conditions on the behavior of ϕ ( η ) near η = 0 for the existence and nonexistence of solutions. The existence for I = R is also shown under additional conditions on p and the behavior of ϕ ( η ) near η = ∞ . Moreover, the singularity of solutions at x = ξ ( t ) is examined.

Let $$\{X_t\}_{t \ge 0}$$ be the symmetric $$\alpha $$ -stable process with generator $$H = (-\Delta )^{\alpha /2}$$ for $$0 < \alpha \le 2$$ . For a positive Radon measure $$\mu $$ , we define the Schrodinger operator $$H^\mu = H - \mu $$ and consider the fundamental solution of the equation $$\partial u/\partial t = - H^{\mu } u$$ . If $$\mu $$ is critical, the behavior of the fundamental solution is different from that of the transition density function of $$\{X_t\}_{t \ge 0}$$ . In this paper, we give a certain ergodic-type limit theorem for fundamental solutions of critical Schrodinger operators.

We prove existence of weak solutions to certain refraction problems in the setting of materials with negative refractive index, both in the near and far field. Solutions are obtained when the target measure is discrete, as well as when it is given by a general Radon measure. The far field problem is treated using techniques from optimal mass transport. As such a fully nonlinear PDE of Monge–Ampere type arises in this setting, and we calculate the coefficients explicitly. Finally, we produce an example in which the A3w condition holds but the A3s condition does not.

International audience; 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 positive Radon measure and the perturbation is assumed to satisfy a Lipschitz property. The existence of a solution with bounded variation is achieved thanks to the Moreau’s catching-up algorithm adapted to this kind of problem. Various properties and estimates of jumps of the solution are also provided. We give sufficient conditions to ensure the uniform prox-regularity when the moving set is described by inequality constraints. As an application, we consider a nonlinear differential complementarity system which is a combination of an ordinary differential equation with a nonlinear complementarily condition. Such problems appear in many areas such as nonsmooth mechanics, nonregular electrical circuits and control systems.

A quasi-perfect map is a continuous, closed function such that the preimage of every point is countably compact. An ambitious old problem due to van Douwen [1] is whether every first countable regular space of cardinality ≤ c is a quasi-perfect image of a locally compact space. Here we construct locally compact, normal, quasi-perfect preimages for all stationary, co-stationary subsets of ω 1 . These subsets are ω 1 -compact but not σ-countably compact. Quasi-perfect preimages preserve these properties, and the preimages constructed here are all of cardinality b . This provides a new upper bound for the following problem: What is the least cardinality of a ZFC example of a locally compact, ω 1 -compact space that is not σ-countably compact?

The paper deals with stationary equation governed by the operator − ∇ ⋅ A ( x , ∇ u ) = μ in the case where A ( x , ξ ) is a maximal monotone graph and μ is a Radon measure. Our main interest concerns the typical situation where A ( x , . ) is defined only in a bounded region of I R n ; so that A ( x , . ) does not satisfy the standard polynomial growth control condition.

Актуальнiсть теми роботи зумовлена необхiднiсю побудови розвиненого апарату нескiнченновимiрного аналiзу та знаходження його застосувань в iнших областях математики, таких як нескiнченновимiрна диференцiальна геометрiя та теорiя випадкових процесiв. Мета роботи полягає в дослiдженнi оператора дивергенцiї полiвекторних полiв на банахових многовидах. Об’єктом дослiдження є полiвекторнi поля на скiнченновимiрних та нескiнченновимiрних многовидах. Предметом дослiдження є дивергенцiя полiвекторних полiв на банахових многовидах з радоновою мiрою та її основнi властивостi. У ходi дослiдження активно використовуються поняття та методи полiлiнiйної алгебри, аналiзу на многовидах, функцiонального аналiзу, диференцiальної геометрiї та топологiї. The relevance of the topic of this work is due to the necessity of building developed machinery of infinite-dimensional analysis and finding applications of the latter to other areas of mathematics, such as infinite-dimensional differential geometry and the theory of stochastic processes. The aim of this work is to study the divergence operator on multivector fields on Banach manifolds. The object of the study is multivector fields on finite-dimensional and infinite-dimensional manifolds. The subject of the study is the divergence of multivector fields on Banach manifolds with a Radon measure and its main properties. In the course of the study we actively employ notions and methods of multilinear algebra, analysis an manifolds, functional analysis, differential geometry and topology.

In the context of Euclidean spaces equipped with an arbitrary Radon measure, we prove the equivalence among several different notions of Sobolev space present in the literature and we characterise the minimal weak upper gradient of all Lipschitz functions., Comment: 39 pages

We study existence of nonnegative solutions to a nonlinear parabolic boundary value problem with a general singular lower order term and a nonnegative measure as nonhomogeneous datum, of the form u t − Δ p u = h ( u ) f + μ in Ω × ( 0 , T ) , u = 0 on ∂ Ω × ( 0 , T ) , u = u 0 in Ω × { 0 } , where Ω is an open bounded subset of R N ( N ≥ 2 ), u 0 is a nonnegative integrable function, Δ p is the p -Laplace operator, μ is a nonnegative bounded Radon measure on Ω × ( 0 , T ) and f is a nonnegative function of L 1 ( Ω × ( 0 , T ) ) . The term h is a positive continuous function possibly blowing up at the origin. Furthermore, we show uniqueness of finite energy solutions in presence of a nonincreasing h .

We prove the existence of a positive {\it SOLA (Solutions Obtained as Limits of Approximations)} to the following PDE involving fractional power of Laplacian \begin{equation} \begin{split} (-\Delta)^su&= \frac{1}{u^\gamma}+\lambda u^{2_s^*-1}+\mu ~\text{in}~\Omega, u&>0~\text{in}~\Omega, u&= 0~\text{in}~\mathbb{R}^N\setminus\Omega. \end{split} \end{equation} Here, $\Omega$ is a bounded domain of $\mathbb{R}^N$, $s\in (0,1)$, $2s

In this paper, we establish the coincidence of two classes of $L^p$-Kato class measures in the framework of symmetric Markov processes admitting upper and lower estimates of heat kernel under mild conditions. One class of $L^p$-Kato class measures is defined by the $p$-th power of positive order resolvent kernel, another is defined in terms of the $p$-th power of Green kernel depending on some exponents related to the heat kernel estimates. We also prove that $q$-th integrable functions on balls with radius $1$ having uniformity of its norm with respect to centers are of $L^p$-Kato class if $q$ is greater than a constant related to $p$ and the constants appeared in the upper and lower estimates of the heat kernel. These are complete extensions of some results by Aizenman-Simon and the recent results by the second named author in the framework of Brownian motions on Euclidean space. We further give necessary and sufficient conditions for a Radon measure with Ahlfors regularity to belong to $L^p$-Kato class. Our results can be applicable to many examples, for instance, symmetric (relativistic) stable processes, jump processes on $d$-sets, Brownian motions on Riemannian manifolds, diffusions on fractals and so on., Comment: 25 pages, minor corrections in the proof of Theorem 4.1

Let $M$ be a complete non-compact Riemannian manifold and let $\sigma $ be a Radon measure on $M$. We study the problem of existence or non-existence of positive solutions to a semilinear elliptic inequaliy \begin{equation*} -\Delta u\geq \sigma u^{q}\quad \text{in}\,\,M, \end{equation*} where $q>1$. We obtain necessary and sufficent criteria for existence of positive solutions in terms of Green function of $\Delta $. In particular, explicit necessary and sufficient conditions are given when $M$ has nonnegative Ricci curvature everywhere in $M$, or more generally when Green's function satisfies the 3G-inequality., Comment: 25 pages

We study the formation of singularities for the problem { u t = [ φ ( u ) ] x x + e [ ψ ( u ) ] t x x in Ω × ( 0 , T ) φ ( u ) + e [ ψ ( u ) ] t = 0 in ∂ Ω × ( 0 , T ) u = u 0 ≥ 0 in Ω × { 0 } , where ϵ and T are positive constants, Ω a bounded interval, u 0 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 u 0 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.

We show that for every non-elementary hyperbolic group the Bowen–Margulis current associated with a strongly hyperbolic metric forms a unique group-invariant Radon measure class of maximal Hausdorff dimension on the boundary square. Applications include a characterization of roughly similar hyperbolic metrics via mean distortion.

In this paper, we establish several integral inequalities of Chebyshev type for bounded continuous fields of Hermitian operators concerning Tracy-Singh products and weighted Pythagorean means. The weighted Pythagorean means considered here are parametrization versions of three symmetric means: the arithmetic mean, the geometric mean, and the harmonic mean. Every continuous field considered here is parametrized by a locally compact Hausdorff space equipped with a finite Radon measure. Tracy-Singh product versions of the Chebyshev-Gr&uuml, ss inequality via oscillations are also obtained. Such integral inequalities reduce to discrete inequalities when the space is a finite space equipped with the counting measure. Moreover, our results include Chebyshev-type inequalities for tensor product of operators and Tracy-Singh/Kronecker products of matrices.

We study differentiability properties of a potential of the type $K\star \mu$, where $\mu$ is a finite Radon measure in $\mathbb{R}^N$ and the kernel $K$ satisfies $|\nabla^j K(x)| \le C\, |x|^{-(N-1+j)}, \quad j=0,1,2.$ We introduce a notion of differentiability in the capacity sense, where capacity is classical capacity in the de la Vall\'ee Poussin sense associated with the kernel $|x|^{-(N-1)}.$ We require that the first order remainder at a point is small when measured by means of a normalized weak capacity "norm" in balls of small radii centered at the point. This implies weak $L^{N/(N-1)}$ differentiability and thus $L^{p}$ differentiability in the Calder\'on--Zygmund sense for $1\le p < N/(N-1)$. We show that $K\star \mu$ is a.e. differentiable in the capacity sense, thus strengthening a recent result by Ambrosio, Ponce and Rodiac. We also present an alternative proof of a quantitative theorem of the authors just mentioned, giving pointwise Lipschitz estimates for $K\star \mu.$ As an application, we study level sets of newtonian potentials of finite Radon measures., Comment: 13 pages

$C([0,1];\mathbb R)$ and $\mathbb R^2 \, ( \, = \, C(\{0,1\};\mathbb R) \, )$are simple examples of a real commutative Banach algebra. $C([0,1];\mathbb C)$ and $\mathbb C^2 \, ( \, = \, C(\{0,1\};\mathbb C) \, )$are simple examples of a commutative $C^*$-algebra. Here, we consider $C([0,1];\mathbb R)$,which is the set of all real-valued continuous functionson the bounded closed interval $[0,1]$.The interval is a contractible compact Hausdorff space.The direct product space $(C([0,1];\mathbb R))^n \, ( \, = \, C([0,1];\mathbb R^n) \, )$is a real Banach space and a free $C([0,1];\mathbb R)$-module. A $C^1$-mapping from an open set of $C([0,1];\mathbb R^n)$to $C([0,1];\mathbb R)$ is said to be $C([0,1];\mathbb R)$-smooth, if its Frechet derivatives are $C([0,1];\mathbb R)$-linear. Then, we can define a concept of an $n$-dimensional smooth $C([0,1];\mathbb R)$-manifold. In this memo, we see existence of a connected metrizable $1$-dimensional smooth $C([0,1];\mathbb R)$-manifold that cannot be embedded in $C([0,1];\mathbb R^n)$. In Section 1, as a primitive observation, we see that embeddability in the Cartesian space $C([0,1];\mathbb R^n)$as a smooth $C([0,1];\mathbb R)$-submanifoldimplies existence of an analog of a bump function. Then, in Section 2, we construct an example where no such analog exists.&nbsp

Let S be a non-empty, closed subspace of a locally compact group G that is a subsemigroup of G. Suppose that X, Y , and Z are Banach lattices that are vector sublattices of the order dual C c (S,R) ∼ Cc(S,R)∼ of the real-valued, continuous functions with compact support on S, and where Z is Dedekind complete. Suppose that ∗ : X × Y → Z is a positive bilinear map such that supp (x ∗ y) ⊆ supp x ⋅ supp y for all x ∈ X+ and y ∈ Y+ with compact support. We show that, under mild conditions, the canonically associated map from X into the vector lattice of regular operators from Y into Z is then a lattice homomorphism. Applications of this result are given in the context of convolutions, answering questions previously posed in the literature.As a preparation, we show that the order dual of the continuous, compactly supported functions on a closed subspace of a locally compact space can be canonically viewed as an order ideal of the order dual of the continuous, compactly supported functions on the larger space.As another preparation, we show that Lp-spaces and Banach lattices of measures on a locally compact space can be embedded as vector sublattices of the order dual of the continuous, compactly supported functions on that space.

In this work, we study the inverse problem stability of the continuous-in-time model which is designed to be used for the finances of public institutions. We discuss this study with determining the Loan measure from algebraic spending measure in Radon measure space $$\mathcal{M}([t_I,\;\Theta_{\text{max}}])$$ , and in Hilbert space $$\mathbb{L}^{2}([t_I,\;\Theta_{\text{max}}])$$ when they are density measures. For this inverse problem we prove the uniqueness theorem, obtain a procedure for constructing the solution and provide necessary and sufficient conditions for the solvability of the inverse problem in $$\mathbb{L}^{2}([t_I,\;\Theta_{\text{max}}])$$ .

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., Comment: Final version, minor corrections and adjustments; accepted, to appear in Rev. Mat. Iberoamericana

In this paper we prove generalizations of Lusin-type theorems for gradients due to Giovanni Alberti, where we replace the Lebesgue measure with any Radon measure $\mu$. We apply this to go beyond the known result on the existence of Lipschitz functions which are non-differentiable at $\mu$-almost every point $x$ in any direction which is not contained in the decomposability bundle $V(\mu,x)$, recently introduced by Alberti and the first named author. More precisely, we prove that it is possible to construct a Lipschitz function which attains any prescribed admissible blowup at every point except for a closed set of points of arbitrarily small measure. Here a function is an admissible blowup at a point $x$ if it is null at the origin and it is the sum of a linear function on $V(\mu,x)$ and a Lipschitz function on $V(\mu,x)^{\perp}$., Comment: accepted: Calc. Var. Partial Differential Equations

A Radon measure $\mu$ is $n$-rectifiable if it is absolutely continuous with respect to $\mathcal{H}^n$ and $\mu$-almost all of $\text{supp}\,\mu$ can be covered by Lipschitz images of $\mathbb{R}^n$. In this paper we give two sufficient conditions for rectifiability, both in terms of square functions of flatness-quantifying coefficients. The first condition involves the so-called $\alpha$ and $\beta_2$ numbers. The second one involves $\alpha_2$ numbers -- coefficients quantifying flatness via Wasserstein distance $W_2$. Both conditions are necessary for rectifiability, too -- the first one was shown to be necessary by Tolsa, while the necessity of the $\alpha_2$ condition is established in our recent paper. Thus, we get two new characterizations of rectifiability., Comment: 54 pages, added the proof of Lemma 4.5, minor improvements

Defining a condenser in a locally compact space as a locally finite, countable collection of Borel sets $A_i$, $i\in I$, with the sign $s_i=\pm1$ prescribed such that $A_i\cap A_j=\varnothing$ whenever $s_is_j=-1$, we consider a minimum energy problem with an external field over infinite dimensional vector measures $(\mu^i)_{i\in I}$, where $\mu^i$ is a suitably normalized positive Radon measure carried by $A_i$ and such that $\mu^i\leqslant\xi^i$ for all $i\in I_0$, $I_0\subset I$ and constraints $\xi^i$, $i\in I_0$, being given. If $I_0=\varnothing$, the problem reduces to the (unconstrained) Gauss variational problem, which is in general unsolvable even for a condenser of two closed, oppositely signed plates in $\mathbb R^3$ and the Coulomb kernel. Nevertheless, we provide sufficient conditions for the existence of solutions to the stated problem in its full generality, establish the vague compactness of the solutions, analyze their uniqueness, describe their weighted potentials, and single out their characteristic properties. The strong and the vague convergence of minimizing nets to the minimizers is also studied. The phenomena of non-existence and non-uniqueness of solutions to the problem are illustrated by examples. The results obtained are new even for the classical kernels on $\mathbb R^n$, $n\geqslant2$, and closed $A_i$, $i\in I$, which is important for applications., Comment: 31 pages, 1 figure

We extend a formula for the computation of the shape derivative of an integral cost functional with respect to a class of convex domains, using the so called support functions and gauge functions to express it. This is a priori a formula in shape optimization theory. However, the result also happens to be an extension of a well known formula from the Brunn-Minkowski theory of convex bodies.; Nous généralisons une formule donnant la dérivée de forme d'une fonctionnelle coût par rapport à une classe de domaines convexes, en utilisant ce que l'on appelle fonctions support et fonctions jauge pour l'exprimer. C'est a priori une formule intervenant en optimisation de formes. Cependant, il se trouve qu'elle généralise aussi une formule bien connue dans la théorie de Brunn-Minkowski des corps convexes.

We give a unified method to obtain the conservativeness of a class of Markov processes associated with lower bounded semi-Dirichlet forms on L 2(X;m), including symmetric diffusion processes, some non-symmetric diffusion processes and jump type Markov processes on X, where X is a locally compact separable metric space and m is a positive Radon measure on X with full topological support. Using the method, we give an example in each section, providing the conservativeness of the processes, that are given by the “increasingness of the volume of some sets(balls)” and “that of the coefficients on the sets” of the Markov processes.