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Let Ω be a C 2 bounded domain in R N ( N ≥ 3 ), δ ( x ) = dist ( x , ∂ Ω ) and C H ( Ω ) be the best constant in the Hardy inequality with respect to Ω. We investigate positive solutions to a boundary value problem for Lane-Emden equations with Hardy potential of the form − Δ u − μ δ 2 u = u p in Ω , u = ρ ν on ∂ Ω , ( P ρ ) where 0 μ C H ( Ω ) , ρ is a positive parameter, ν is a positive Radon measure on ∂Ω with norm 1 and 1 p N μ , with N μ being a critical exponent depending on N and μ. It is known from [22] that there exists a threshold value ρ ⁎ such that problem ( P ρ ) admits a positive solution if 0 ρ ≤ ρ ⁎ , and no positive solution if ρ > ρ ⁎ . In this paper, we go further in the study of the solution set of ( P ρ ) . We show that the problem admits at least two positive solutions if 0 ρ ρ ⁎ and a unique positive solution if ρ = ρ ⁎ . We also prove the existence of at least two positive solutions for Lane-Emden systems { − Δ u − μ δ 2 u = v p in Ω , − Δ v − μ δ 2 v = u q in Ω , u = ρ ν , v = σ τ on ∂ Ω , under the smallness condition on the positive parameters ρ and σ.

Let $\mu$ be a Radon measure on $\mathbb{R}^d$. We define and study conical energies $\mathcal{E}_{\mu,p}(x,V,\alpha)$, which quantify the portion of $\mu$ lying in the cone with vertex $x\in\mathbb{R}^d$, direction $V\in G(d,d-n)$, and aperture $\alpha\in (0,1)$. We use these energies to characterize rectifiability and the big pieces of Lipschitz graphs property. Furthermore, if we assume that $\mu$ has polynomial growth, we give a sufficient condition for $L^2(\mu)$-boundedness of singular integral operators with smooth odd kernels of convolution type., Comment: 40 pages; removed the measurability assumption from Theorem 1.4, many minor improvements; to appear in Rev. Mat. Iberoam

Let $H$ be a bounded and Lipschitz continuous function. We consider discontinuous viscosity solutions of the Hamilton-Jacobi equation $U_{t}+H(U_x)=0$ and signed Radon measure valued entropy solutions of the conservation law $u_{t}+[H(u)]_x=0$. After having proved a precise statement of the formal relation $U_x=u$, we establish estimates for the (strictly positive!) times at which singularities of the solutions disappear. Here singularities are jump discontinuities in case of the Hamilton-Jacobi equation and signed singular measures in case of the conservation law., 33 pages

In this paper, we obtain a version of the strong maximum principle for the spectral Dirichlet Laplacian. Specifically, let $$d \in \{1,2,3,\ldots \}$$ , $$s \in (\frac{1}{2},1)$$ , and $$\Omega \subset \mathbb {R}^d$$ be open, bounded, connected with Lipschitz boundary. Suppose $$u \in L^1(\Omega )$$ satisfies $$u \ge 0$$ a.e. in $$\Omega $$ and $$(-\Delta )^s u$$ is a Radon measure on $$\Omega $$ . Then u has a quasi-continuous representative $${\tilde{u}}$$ . Let $$a \in L^1(\Omega )$$ be such that $$a \ge 0$$ a.e. in $$\Omega $$ . Then if $$\begin{aligned} (-\Delta )^s u + au \ge 0 \quad \text {a.e.} \text { in } \Omega \end{aligned}$$ and $${\tilde{u}} = 0$$ on a subset of positive $$H^s$$ -capacity of $$\Omega $$ , then $$u = 0$$ a.e. in $$\Omega $$ .

The purpose of this work is to study a class of periodic parabolic equations having a critical growth nonlinearity with respect to the gradient and bounded Radon measure. By the main of the sub- and super-solution method, we employ some new technics to prove the existence of a nonnegative weak periodic solution to the studied problems.

This paper is concerned with a borderline case of double phase problems with a finite Radon measure on the right-hand side. We obtain sharp fractional regularity estimates for such non-uniformly elliptic problems.

In this paper, we endow the space of continuous translation invariant valuation on convex sets generated by mixed volumes coupled with a suitable Radon measure on tuples of convex bodies with two appropriate norms. This enables us to construct a continuous extension of the convolution operator on smooth valuations to non-smooth valuations, which are in the completion of the spaces of valuations with respect to these norms. The novelty of our approach lies in the fact that our proof does not rely on the general theory of wave fronts, but on geometric inequalities deduced from optimal transport methods. We apply this result to prove a variant of Minkowski's existence theorem, and generalize a theorem of Favre-Wulcan and Lin in complex dynamics over toric varieties by studying the linear actions on the Banach spaces of valuations and by studying their corresponding eigenspaces., Comment: v3: 39 pages; minor revisions, some corrections on Theorem A, references updated. v2: 38 pages; references updated, typos corrected, Remark 1.2 and Section 6.2.1 added

International audience; We study the semilinear elliptic equation −∆u + g(u)σ = µ with Dirichlet boundary conditions 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 .

In this paper we are going to prove that the Hardy-Litllewood maximal operators on variable Lebesgue spaces Lp(·)(µ) with respect to a probability Borel measure µ, are weak type and strong type for...

In this paper, we prove the existence of Radon measure-valued solutions for nonlinear degenerate parabolic equations with nonnegative bounded Radon measure data. Moreover, we show the uniqueness of the measure-valued solutions when the Radon measure as a forcing term is diffuse with respect to the parabolic capacity and the Radon measure as a initial value is diffuse with respect to the Newtonian capacity. We also deduce that the concentrated part of the solution with respect to the Newtonian capacity depends on time.

We study a class of nonlinear elliptic equations involving measure data −divA(x,Du)=μinΩ,where μ is a Radon measure. Under the main assumption of A(x,ξ) that there exists a constant Λ>0 such that |A(x,ξ)−A(x0,ξ)|≤Λ(a(x)+a(x0))|x−x0|α(|ξ|2+s2)p−12,α∈(0,1],where 0≤a(x)∈Lm(Ω) for some integrable index m>1, we obtain the Calderon–Zygmund estimates in the Sobolev–Morrey spaces for refined fractional‐order derivatives of distributional solutions depending on α.

If $$\mu $$ is an arbitrary bounded Radon measure $$\mu $$ on $${\mathbb {R}}^n,$$ we denote by $${{\widehat{\mu }}}$$ the Fourier–Stieltjes transform of $$\mu $$ and by $$\sigma $$ the pure point part of $$\mu .$$ A closed $$\varLambda \subset {{\mathbb {R}}}^n$$ is a gregarious set if the following property is satisfied: $$\begin{aligned} (\forall \mu )\, {{\widehat{\mu }}}=0\quad \mathrm{on}\, \varLambda \Rightarrow {{\widehat{\sigma }}}=0\, \mathrm{on}\,\varLambda . \end{aligned}$$ Gregarious sets are studied in this essay.

A class of second-order elliptic equations with variable nonlinearity exponents and the right-hand side in the form of the general Radon measure with finite total variation is considered. The existence of a renormalized solution of the Dirichlet problem is proved as a consequence of stability with respect to the convergence of the right-hand side of the equation. Bibliography: 37 titles.

We provide a quick proof of the following known result: the Sobolev space associated with the Euclidean space, endowed with the Euclidean distance and an arbitrary Radon measure, is Hilbert. Our new approach relies upon the properties of the Alberti-Marchese decomposability bundle. As a consequence of our arguments, we also prove that if the Sobolev norm is closable on compactly-supported smooth functions, then the reference measure is absolutely continuous with respect to the Lebesgue measure., Comment: 8 pages

We study the existence and the regularity of solutions for a class of nonlocal equations involving the fractional Laplacian operator with singular nonlinearity and Radon measure data.

We study the kernel function of the operator u $\rightarrow$ L $\mu$ u = --$\Delta$u + $\mu$ |x| 2 u in a bounded smooth domain $\Omega$ $\subset$ R N + such that 0 $\in$ $\partial$$\Omega$, where $\mu$ $\ge$ -- N 2 4 is a constant. We show the existence of a Poisson kernel vanishing at 0 and a singular kernel with a singularity at 0. We prove the existence and uniqueness of weak solutions of L $\mu$ u = 0 in $\Omega$ with boundary data $\nu$ + k$\delta$ 0 , where $\nu$ is a Radon measure on $\partial$$\Omega$ \ {0}, k $\in$ R and show that this boundary data corresponds in a unique way to the boundary trace of positive solution of L $\mu$ u = 0 in $\Omega$.

Let $$\Omega \subset {\mathbb {R}}^3$$ be a domain and let $$f\in BV_{{\text {loc}}}(\Omega ,{\mathbb {R}}^3)$$ be a homeomorphism such that its distributional adjugate is a finite Radon measure. We show that its inverse has bounded variation $$f^{-1}\in BV_{{\text {loc}}}$$ . The condition that the distributional adjugate is finite measure is not only sufficient but also necessary for the weak regularity of the inverse.

This study investigates the restriction problem for the Riesz potentials of Hardy-Hausdorff spaces HH−γ1 (ℝn) and Q-type spaces Qγ(ℝn). By exploiting a geometric-measure theory generated by the indicatorlike functions of compact sets, it is proved that the Riesz operator Iα continuously maps HH−γ1 (ℝn) into the weak Morrey spaces Lμ,*q,λ induced by a Radon measure μ, which obeys a geometric condition.

The aim of the paper is to study a gradient constrained problem associated with a linear operator. Two types of problems are investigated. The first one is the equivalence between a non-constant gradient constrained problem and a suitable obstacle problem, where the obstacle solves a Hamilton-Jacobi equation in the viscosity sense. The equivalence result is obtained under a condition on the gradient constraint. The second problem is the existence of Lagrange multipliers. We prove that the non-constant gradient constrained problem admits a Lagrange multiplier, which is a Radon measure if the free term of the equation f ∈ L p , p > 1 . If f is a positive constant, we regularize the result, namely we prove that the Lagrange multipliers belong to L 2 .

Made available in DSpace on 2020-12-10T20:00:07Z (GMT). No. of bitstreams: 0 Previous issue date: 2020-06-01 In this work, we present conditions to obtain a global-in-time existence of solutions to a class of nonlinear viscous transport equations describing aggregation phenomena in biology with sufficiently small initial data in Besov-Morrey spaces and gradient potential as a Radon measure. We also study the self-similarity and asymptotic stability of solutions at large times. Univ Estadual Paulista, Dept Matemat, BR-15054000 Sao Jose Do Rio Preto, SP, Brazil Univ Estadual Paulista, Dept Matemat, BR-15054000 Sao Jose Do Rio Preto, SP, Brazil

An evolution inclusion with the right-hand side containing a time-dependent maximal monotone operator and a multivalued mapping with closed nonconvex values is studied in a separable Hilbert space. The dependence of the maximal monotone operator on time is described with the help of the distance between maximal monotone operators in the sense of Vladimirov. This distance as a function of time has bounded variation with an upper bound given by a nonatomic positive Radon measure. By a solution of the inclusion one means a continuous function of bounded variation whose differential measure (Stieltjes measure) is absolutely continuous with respect to the positive Radon measure above, and the values of the density of this differential measure with respect to the Radon measure belong to the right-hand side of the inclusion almost everywhere. Under the traditional assumptions on the perturbation (measurability, Lipschitzianity in the phase variable in the Hausdorff metric, linear growth condition), the existence of solutions is proven and some properties of the solution set are established.

In this paper we address the weak Radon measure-valued solutions associated with the Young measure for a class of nonlinear parabolic equations with initial data as a bounded Radon measure. This problem is described as follows: $$ \textstyle\begin{cases} u_{t}=\alpha u_{xx}+\beta [\varphi (u) ]_{xx}+f(u) &\text{in} \ Q:=\Omega \times (0,T), \\ u=0 &\text{on} \ \partial \Omega \times (0,T), \\ u(x,0)=u_{0}(x) &\text{in} \ \Omega , \end{cases} $$ { u t = α u x x + β [ φ ( u ) ] x x + f ( u ) in Q : = Ω × ( 0 , T ) , u = 0 on ∂ Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) in Ω , where $T>0$ T > 0 , $\Omega \subset \mathbb{R}$ Ω ⊂ R is a bounded interval, $u_{0}$ u 0 is nonnegative bounded Radon measure on Ω, and $\alpha , \beta \geq 0$ α , β ≥ 0 , under suitable assumptions on φ and f. In this work we prove the existence and the decay estimate of suitably defined Radon measure-valued solutions for the problem mentioned above. In particular, we study the asymptotic behavior of these Radon measure-valued solutions.

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 superposition of Dirac masses, whereas the flux is Lipschitz continuous and bounded. The solution class is determined by an additional condition which is needed to prove uniqueness.

The aim of this paper is to prove the weak type vector-valued inequality for the modified Hardy– Littlewood maximal operator for general Radon measure on ℝ n . Earlier, the strong type vector-valued inequality for the same operator and the weak type vector-valued inequality for the dyadic maximal operator were obtained. This paper will supplement these existing results by proving a weak type counterpart.

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.

We consider a class of anisotropic elliptic equations of second order with variable exponents of non-linearity where a special Radon measure is used as the right-hand side. We establish uniqueness of entropy and renormalized solutions of the Dirichlet problem in anisotropic Sobolev spaces with variable exponents of non-linearity for arbitrary domains and certain other their properties. In addition, we prove the equivalence of entropy and renormalized solutions of the problem under consideration.

We investigate the existence of solutions of \begin{equation} divF=\mu,\text{ on } \mathbf{R}^{d}. \tag{*} \end{equation} Here, $\mu\geq0$ is a Radon measure, and we look for a solution $F\in X(\mathbf{R}^{d}\rightarrow\mathbf{R}^{d})$, where $X$ is a rearrangement-invariant space. We first prove the equivalence of the following assertions: (i) (*) has a solution for some nontrivial $\mu$; (ii) the function $x\mapsto |x|^{1-d}1_{B^{c}}(x)$ belongs to $X$. Here, $B$ is the unit ball in $\mathbf{R}^{d}$. We next investigate the solvability of (*) when $\mu$ is fixed. A sufficient condition is that $I_{1}\mu\in X$, where $ I_{1}\mu$ is the 1-Riesz potential of $\mu$. This condition turns out to be also necessary when the Boyd indexes of $X$ belong to $(0,1)$. Our analysis generalizes the one of Phuc and Torres (2008) when $X=L^{p}$.

Rules for calculating the densities of Borel measures which are absolutely continuous with respect to a positive nonatomic Radon measure are considered. The Borel measures are generated by composite functions which depend on continuous functions of bounded variation defined on an interval. Questions related to the absolute continuity of Borel measures generated by composite functions with respect to a positive Radon measure and rules for calculating the densities of Borel measures generated by composite functions with respect to a positive nonatomic Radon measure are studied.

We investigate the problem of optimal transport in the so-called Kantorovich form, i.e. given two Radon measures on two compact sets, we seek an optimal transport plan which is another Radon measure on the product of the sets that has these two measures as marginals and minimizes a certain cost function. We consider quadratic regularization of the problem, which forces the optimal transport plan to be a square integrable function rather than a Radon measure. We derive the dual problem and show strong duality and existence of primal and dual solutions to the regularized problem. Then we derive two algorithms to solve the dual problem of the regularized problem: A Gauss-Seidel method and a semismooth quasi-Newton method and investigate both methods numerically. Our experiments show that the methods perform well even for small regularization parameters. Quadratic regularization is of interest since the resulting optimal transport plans are sparse, i.e. they have a small support (which is not the case for the often used entropic regularization where the optimal transport plan always has full measure)., Ergebnisberichte des Instituts für Angewandte Mathematik;600

In this work, we consider the one dimensional very singular fourth-order equation for solid-on-solid model in attachment-detachment-limit regime with exponential nonlinearity $$h_t = \nabla \cdot (\frac{1}{|\nabla h|} \nabla e^{\frac{\delta E}{\delta h}}) =\nabla \cdot (\frac{1}{|\nabla h|}\nabla e^{- \nabla \cdot (\frac{\nabla h}{|\nabla h|})})$$ where total energy $E=\int |\nabla h|$ is the total variation of $h$. Using a logarithmic correction $E=\int |\nabla h|\ln|\nabla h| d x$ and gradient flow structure with a suitable defined functional, we prove the evolution variational inequality solution preserves a positive gradient $h_x$ which has upper and lower bounds but in BV space. We also obtain the global strong solution to the solid-on-solid model which allows an asymmetric singularity $h_{xx}^+$ happens., Comment: 15 pages

In this paper, we prove the existence of weak solutions to the complex m-Hessian equations in the class $${\mathcal {D}}_{m}(\Omega )$$ on an open subset $$\Omega $$ of $${\mathbb {C}}^n$$ . In the end of the paper we give an example shows that in the unit ball $${\mathbb {B}}^{2}(0,1)\subset {\mathbb {C}}^{2}$$ the complex Monge-Ampere equation $$(dd^{c} .)^{2}=\mu $$ is solvable but the complex Hessian equation $$H_{1}(.)=\mu $$ has not any weak solutions where $$\mu $$ is a nonnegative Radon measure on $${\mathbb {B}}^{2}(0,1)$$ .

We study a nonlinear anisotropic elliptic problem under Fourier type boundary condition governed by a general anisotropic operator with variable exponents and diffuse Radon measure data which does not charge the sets of zero p(·)-capacity. We prove an existence and uniqueness result of entropy or renormalized solution.

We prove a global Calderon–Zygmund type estimate for the gradient of a solution to a nonlinear elliptic problem with nonstandard growth when the right-hand side is a bounded Radon measure. Minimal regularity requirements on both the nonlinearity and the boundary of the domain are investigated for such a gradient estimate.

Let μ be a finite Radon measure in ℝd 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 the so-called β-numbers of Jones, David and Semmes, all singular integral operators with an odd and sufficiently smooth Calderon-Zygmund kernel are bounded in L2(μ). As a corollary, we obtain a lower bound for the Lipschitz harmonic capacity of a compact set in ℝd only in terms of its metric and geometric properties.

In this paper, we prove the existence of an initial trace T u of any positive solution u of the semilinear fractional diffusion equation (H) ∂ t u + (−∆) α u + f (t, x, u) = 0 in R * + $\times$ R N , where N ≥ 1 where the operator (−∆) α with α ∈ (0, 1) is the fractional Laplacian and f : R + $\times$ R N $\times$ R + → R is a Caratheodory function satisfying f (t, x, u)u ≥ 0 for all (t, x, u) ∈ R + $\times$ R N $\times$ R +. We define the regular set of the trace T u as an open subset of R u ⊂ R N carrying a nonnegative Radon measive ν u such that lim t→0 Ru u(t, x)ζ(x)dx = Ru ζdν ∀ζ ∈ C 2 0 (R u), and the singular set S u = R N \ R u as the set points a such that lim sup t→0 Bρ(a) u(t, x)dx = ∞ ∀ρ > 0. We study the reverse problem of constructing a positive solution to (H) with a given initial trace (S, ν) where S ⊂ R N is a closed set and ν is a positive Radon measure on R = R N \ S and develop the case f (t, x, u) = t β u p where β > −1 and p > 1.

We investigate the existence of distributional solutions to nonlinear elliptic problems with general growth when the right-hand side is a bounded Radon measure. An optimal global Calderon-Zygmund type estimate for such problems is obtained by using mapping property of the fractional maximal function of the measure.

[ES] Los espacios libres Lipschitz F(M) son linearizaciones canónicas de espacios métricos M cualesquiera. Más concretamente, F(M) es el único espacio de Banach que contiene una copia isométrica de M que es linearmente densa, y tal que toda aplicación Lipschitz de M en cualquier espacio de Banach X puede extenderse a un operador linear continuo de F(M) en X. Estos espacios suponen una herramienta muy potente para el estudio de la geometría no lineal de espacios de Banach, al permitir la aplicación de las técnicas lineales clásicas, bien conocidas, a problemas no lineales. Pero este esfuerzo sólo merece la pena si se dispone de un conocimiento lo bastante detallado de la estructura de F(M). El estudio sistemático de los espacios libres Lipschitz es bastante reciente y, por ello, dicho conocimiento es todavía más bien limitado. Esta tesis se enmarca en el programa general de estudio de la estructura espacios libres Lipschitz genéricos. Empezamos nuestro estudio desarrollando algunas herramientas básicas para la teoría general de espacios libres Lipschitz. Primero definimos operadores de ponderación en espacios Lipschitz y los usamos para demostrar la conjetura de Weaver de que todos los funcionales normales del bidual F(M)** son débil* continuos. A continuación demostramos el teorema de la intersección, que en esencia dice que la intersección de espacios libres Lipschitz es de nuevo un espacio libre Lipschitz. Este resultado nos permite desarrollar el concepto de soporte de un elemento de F(M), análogo al de soporte de una medida. Además, extendemos el uso de estas herramientas al bidual F(M) y las usamos para establecer una descomposición del bidual en espacios de funcionales que están "concentrados en el infinito" y "separados del infinito", respectivamente. Con estas herramientas en nuestro poder, emprendemos el estudio de dos aspectos concretos de los espacios libres Lipschitz. En primer lugar analizamos la relación entre F(M) y los espacios de medidas sobre M. En particular, obtenemos caracterizaciones de los elementos de F(M) que pueden representarse como la integración con respecto a una medida de Borel (no necesariamente finita) sobre M y viceversa, y probamos que el soporte coincide con el de la medida asociada. También identificamos los espacios métricos M en los cuales todo elemento de F(M) puede ser representado como una medida de Borel. Este análisis se generaliza al bidual F(M)**, utilizando en este caso medidas sobre la compactificación uniforme de M y llegando a resultados similares. Obtenemos también algunas consecuencias para los elementos de F(M) y F(M)** que pueden expresarse como diferencia de dos elementos positivos, como la existencia de un análogo de la descomposición de Jordan para medidas. En segundo lugar, estudiamos la estructura extremal de la bola unidad de F(M) y hacemos algunas contribuciones al programa general consistente en encontrar caracterizaciones puramente geométricas de todos sus elementos extremales. Concretamente, caracterizamos los puntos extremos preservados de la bola, así como aquellos puntos extremos y expuestos que tienen soporte finito. Además damos una descripción completa de la estructura extremal de la parte positiva de la bola unidad. La teoría de los soportes en F(M) desarrollada anteriormente juega un papel crucial en las demostraciones de estos resultados., [CA] Els espais lliures Lipschitz F(M) són linearitzacions canòniques d'espais mètrics M qualssevol. Més concretament, F(M) és l'únic espai de Banach que conté una còpia isomètrica de M que és linealment densa, i tal que tota aplicació Lipschitz de M en qualsevol espai de Banach X pot ser estesa a un operador lineal continu de F(M) en X. Aquests espais són una eina molt potent per a l'estudi de la geometria no lineal d'espais de Banach, ja que permeten l'aplicació de les tècniques lineals clàssiques, ben conegudes, a problemes no lineals. Però aquest esforç nomes val la pena si es disposa d'un coneixement bastant detallat de l'estructura de F(M). L'estudi sistemàtic dels espais lliures Lipschitz és bastant recent i, per això, aquest coneixement és encara prou limitat. Aquesta tesi s'emmarca en el programa general d'estudi de l'estructura dels espais lliures Lipschitz genèrics. Comencem el nostre estudi desenvolupant algunes eines bàsiques per a la teoria general d'espais lliures Lipschitz. Primer definim operadors de ponderació en espais Lipschitz i els fem servir per demostrar la conjectura de Weaver que tots els funcionals normals del bidual F(M)** son feble* continus. A continuació demostrem el teorema de la intersecció, que en essència diu que la intersecció d'espais lliures Lipschitz és de nou un espai lliure Lipschitz. Aquest resultat ens permet desenvolupar el concepte de suport d'un element de F(M), anàleg al de suport d'una mesura. A més, estenem l'ús d'aquestes eines al bidual F(M)** i les fem servir per establir una descomposició del bidual en espais de funcionals que estan "concentrats a l'infinit" i "separats de l'infinit", respectivament. Amb aquestes eines al nostre abast, emprenem l'estudi de dos aspectes concrets dels espais lliures Lipschitz. En primer lloc, analitzem la relació entre F(M) i els espais de mesures sobre M. En particular, obtenim caracteritzacions dels elements de F(M) que poden representar-se com la integració respecte a una mesura de Borel (no necessàriament finita) sobre M i viceversa, i provem que el suport coincideix amb el de la mesura associada. També identifiquem els espais mètrics M on tot element de F(M) pot ser representat com una mesura de Borel. Aquesta anàlisi es generalitza al bidual F(M)**, utilitzant en aquest cas mesures sobre la compactificació uniforme de M i arribant a resultats similars. També obtenim algunes conseqüències per als elements de F(M) i F(M)** que poden expressar-se com a diferència de dos elements positius, com ara l'existència d'un anàleg de la descomposició de Jordan per a mesures. En segon lloc, estudiem l'estructura extremal de la bola unitat de F(M) i fem algunes contribucions al programa general consistent en trobar caracteritzacions purament geomètriques de tots els seus elements extremals. Concretament, caracteritzem els punts extrems preservats de la bola, així com aquells punts extrems i exposats que tenen suport finit. A més fem una descripció completa de l'estructura extremal de la part positiva de la bola unitat. La teoria dels suports en F(M) desenvolupada anteriorment juga un paper crucial en les demostracions d'aquests resultats., [EN] Lipschitz-free spaces F(M) are canonical linearizations of arbitrary complete metric spaces M. More specifically, F(M) is the unique Banach space that contains an isometric copy of M that is linearly dense, and such that any Lipschitz mapping from M into some Banach space X extends to a bounded linear operator from F(M) into X. Those spaces are a very powerful tool for studies of the nonlinear geometry of Banach spaces, as they allow the application of well-known classical linear techniques to nonlinear problems. But this effort is only worthwhile if we have sufficient knowledge about the structure of F(M). The systematic study of Lipschitz-free spaces is rather recent and so the current understanding of their structure is still quite limited. This thesis is framed within the general program of studying the structure of general Lipschitz-free spaces. We start our study by developing some basic tools for the general theory of Lipschitz-free spaces. First we introduce weighting operators and use them to solve Weaver's conjecture that all normal functionals in the bidual F(M)** are weak* continuous. Next we prove the intersection theorem, which essentially says that the intersection of Lipschitz-free spaces is again a Lipschitz-free space. That result allows us to develop the concept of support of an element of F(M), analogous to the support of a measure. Furthermore, we extend the use of these tools to the bidual F(M)** and apply them to establish a decomposition of the bidual into spaces of functionals that are "concentrated at infinity" and "separated from infinity", respectively. With these tools at our disposal, we undertake the study of two particular aspects of Lipschitz-free spaces. First we analyze the relationship between F(M) and spaces of measures on M. In particular, we obtain characterizations of those elements of F(M) that can be represented as integration against a (not necessarily finite) Borel measure on M and vice versa, and we show that their supports agree. We also identify those metric spaces such that every element of F(M) can be represented by a Borel measure. This analysis is generalized to the bidual F(M)**, using measures on the uniform compactification of M in that case and obtaining similar results. We also derive some consequences for those elements of F(M) and F(M)** that can be expressed as the difference between two positive elements, such as the existence of an analog of the Jordan decomposition for measures. Secondly, we study the extremal structure of the unit ball of F(M) and provide some contributions to the general program of finding purely geometric characterizations of all of its extremal elements. Namely, we characterize all of its preserved extreme points, and its extreme and exposed points of finite support. We also give a full description of the extremal structure of the positive unit ball. The theory of supports developed previously plays a crucial role in the proofs of these results, The author would like to thank Marek Cúth, Michal Doucha, Antonio José Guirao, Gilles Lancien and Eva Pernecká for their careful reading and correction of this document or parts of it. Some activities related to this thesis were partially supported by the Spanish Ministry of Economy, Industry and Competitiveness under Grant MTM2017-83262-C2-2-P, and by a travel grant of the Institute of Mathematics (IEMath-GR) of the University of Granada. Part of this research was conducted during visits to the Czech Technical University in Prague in 2018 and 2020, the Laboratoire de Mathématiques de Besançon in 2019, and the University of Granada in 2020. The author wishes to express his gratitude for the hospitality and the excellent working conditions during his visits.

We consider degenerate and singular parabolic equations with $p$-Laplacian structure in bounded nonsmooth domains when the right-hand side is a signed Radon measure with finite total mass. We develop a new tool that allows global regularity estimates for the spatial gradient of solutions to such parabolic measure data problems, by introducing the (intrinsic) fractional maximal function of a given measure., Comment: 32 pages

In the setting of $\R^d$ with an $n-$dimensional measure $\mu,$ we give several characterizations of Lipschitz spaces in terms of mean oscillations involving $\mu.$ We also show that Lipschitz spaces are preserved by those Calderon-Zygmund operators $T$ associated to the measure $\mu$ for which T(1) is the Lipschitz class $0.$, Comment: 10 pages

In this paper the new procedure for a construction of an approximated solution to initial data problem for one-dimensional pressureless gas dynamics system is introduced. The procedure is based on solving the Riemann problems and tracking singular wave interactions. For that system the new problem with initial data containing Dirac delta function is solved whenever two waves interact. Use of the shadow waves as singular solutions to such problems enables us to easily solve the interaction problems. That permits us to make a simple extension of the well known Wave Front Tracking algorithm. A non-standard part of the new algorithm is dealing with delta functions as a part of a solution. In the final part of the paper we show that the approximated solution has a subsequence converging to a signed Radon measure.

We consider a standard Navier–Stokes system on the n-D torus $${\mathbb {T}}^n={\mathbb {R}}^n/{\mathbb {Z}}^n$$ , valid when the flow is not very compressible and the temperature does not vary too much. We construct a sequence of approximate solutions that tend to satisfy the equations in a weak sense for arbitrary physical initial conditions. By weak compactness, we obtain Radon measure solutions in density and momentum when the velocity of the flow is finite, as numerically observed in all tests, and the absence of void regions in the flow. We notice that the method also applies without viscosity and complements results on other systems of evolution equations such as the systems of isothermal and isentropic gas flows considered in Colombeau (Z Angew Math Phys 66(5):2575–2599, 2015).

In this paper, we study the good- λ type bounds for renormalized solutions to nonlinear elliptic problem: − d i v ( A ( x , ∇ u ) ) = μ in Ω , u = 0 on ∂ Ω . where Ω ⊂ R n , μ is a finite Radon measure and A is a monotone Carathedory vector valued function defined on W 0 1 , p ( Ω ) . The operator A satisfies growth and monotonicity conditions, and the p -capacity uniform thickness condition is imposed on R n ∖ Ω , for the singular case 3 n − 2 2 n − 1 p ≤ 2 − 1 n . In fact, the same good- λ type estimates were also studied by Quoc-Hung Nguyen and Nguyen Cong Phuc. For instance, in Nguyen and Phuc (0000) and Nguyen (0000) authors’ method was also confined to the case of 3 n − 2 2 n − 1 p ≤ 2 − 1 n but under the assumption of Ω is the Reifenberg flat domain and the coefficients of A have small BMO (bounded mean oscillation) semi-norms. Otherwise, the same problem was considered in Phuc (2014) in the regular case of p > 2 − 1 n . In this paper, we extend their results, taking into account the case 3 n − 2 2 n − 1 p ≤ 2 − 1 n and without the hypothesis of Reifenberg flat domain on Ω and small BMO semi-norms of A . Moreover, in rest of this paper, we also give the proof of the boundedness property of maximal function on Lorentz spaces and also the global gradient estimates of solution.

Let $\mathcal{L}=-\Delta+\mu$ be the generalized Schrödinger operator on $\mathbb{R}^{n},n\geq3$ , where $\Delta$ is the Laplacian and $\mu\notequiv0$ is a nonnegative Radon measure on $\mathbb{R}^{n}$ . In this article, we introduce two families of Carleson measures $\{d\nu_{h,k}\}$ and $\{d\nu_{P,k}\}$ generated by the heat semigroup $\{e^{-t\mathcal{L}}\}$ and the Poisson semigroup $\{e^{-t\sqrt{\mathcal{L}}}\}$ , respectively. By the regularities of semigroups, we establish the Carleson measure characterizations of BMO-type spaces $\mathrm{BMO}_{\mathcal{L}}(\mathbb{R}^{n})$ associated with the generalized Schrödinger operators.

In this paper, we first prove the local $L^{s}$Ls norm estimate of composite operators $\triangle^{k}G^{m}(u)$△kGm(u) by use of the $L^{s}$Ls norm of u. Then we establish the local and global higher norm inequalities of $\triangle^{k}G^{m}(u)$△kGm(u). Simultaneously, we also give a global higher norm estimate with Radon measure. Finally, as applications of these results, we give two examples to estimate the higher norm of $\triangle^{k}G^{m}(u)$△kGm(u).

We study the properties of a parameterized sequence of countably additive vector measures with densities defined on a compact space T with a nonnegative nonatomic Radon measure μ and taking values in a separable Banach space. Each vector measure of this sequence depends continuously on a parameter belonging to a metric space. We assume that a countable locally finite open cover and a partition of unity inscribed into this cover are given in the metric space of parameters. We prove that, for each value of the parameter, there exists a sequence of μ-measurable subsets of the space T which is a partition of T. In addition, this sequence depends uniformly continuously on the parameter and, for each value of the parameter and each element of the initial parameterized sequence of vector measures, the relative value of the measure of the corresponding subset in the partition of T can be uniformly approximated by the corresponding value of the corresponding partition of unity function.

In this article we consider weak solutions of the Euler-α equations in the full plane. We take, as initial unfiltered vorticity, an arbitrary nonnegative, compactly supported, bounded Radon measure. Global well-posedness for the corresponding initial value problem is due M. Oliver and S. Shkoller. We show that, for all time, the support of the unfiltered vorticity is contained in a disk whose radius grows no faster than O ( ( t log ⁡ t ) 1 / 4 ) . This result is an adaptation of the corresponding result for the incompressible 2D Euler equations with initial vorticity compactly supported, nonnegative, and p-th power integrable, p > 2 , due to D. Iftimie, T. Sideris and P. Gamblin and, independently, to Ph. Serfati.

Let $\mathcal{L}=-\Delta+\mu$ be the generalized Schrödinger operator on $\mathbb{R}^{n}$ , $n\geq3$ , where $\Delta$ is the Laplacian and $\mu\nequiv0$ is a nonnegative Radon measure on $\mathbb{R}^{n}$ . In this article, we give a characterization of $\mathrm{BMO}_{\mathcal{L}}$ in terms of Carleson measures, where $\mathrm{BMO}_{\mathcal{L}}$ is the $\mathrm{BMO}$ -type space associated with the generalized Schrödinger operator.

We consider a nonlinear parabolic equation with measurable nonlinearity in a nonsmooth bounded domain when the right-hand side is a finite signed Radon measure. Under optimal regularity assumptions on the nonlinearity and the boundary of the domain, we prove a global Calderon–Zygmund type estimate in weighted Orlicz spaces. As an application we obtain such an estimate in variable exponent spaces, which gives an alternative proof for this new result in the literature.