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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 establish local continuity of solutions to the $G$-Laplace equation involving measures, i.e., \begin{align*} -\text{div}\ \bigg(\frac{g(|\nabla u|)}{|\nabla u|}\nabla u\bigg)=\mu,\end{align*} where $\mu$ is a nonnegative Radon measure satisfying $\mu (B_{r}(x_{0}))\leq Cr^{m}$ for any ball $B_{r}(x_{0})\subset\subset \Omega$ with $r\leq 1$ and $m>n-1-\delta\geq 0$. The function $g$ is supposed to be nonnegative and $C^{1}$-continuous on $[0,+\infty)$, satisfying $g(0)=0$ and \begin{align*}\delta\leq \frac{tg'(t)}{g(t)}\leq g_{0}, \forall t>0\end{align*} with positive constants $\delta$ and $g_{0}$, which generalizes the structural conditions of Ladyzhenskaya–Ural'tseva for an elliptic operator.

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.

We study existence and stability of solutions of $$ -\Delta u +\frac{\mu}{|x|^{2 }}u+ g(u)= ν \text{ in }\Omega,\ \ \ u=0\text{ on } ∂Ω,$$ where $\Omega$ is a bounded, smooth domain of $\mathbb{ R}^N$, $N\geq 2$, containing the origin, $\mu\geq-\frac{(N-2)^2}{4}$ is a constant, $g$ is a nondecreasing function satisfying some integral growth assumption and the weak ∆2-condition, and ν is a Radon measure in Ω. We show that the situation differs depending on whether the measure is diffuse or concentrated at the origin. When $g$ is a power function, we introduce a capacity framework to find necessary and sufficient conditions for solvability.

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 competitive exclusion principle in the case of a continuously distributed initial population. We introduce an ordinary differential equations model of species competing for a single resource, whose efficiency is encoded by a continuous variable (the "trait variable") living in a Euclidean space. The differential equation is solved in a measure space to allow the observation of the natural concentration of the species distribution on specific traits. We show the concentration of the distribution on the maximizing set of the fitness function in the sense of the Kantorovitch-Rubinstein metric. In the case when the initial weight of the maximizing set is positive, we give a precise description of the convergence of the orbit, including a formula for the asymptotic distribution. We also investigate precisely the case of a finite number of regular global maxima and show that the initial distribution may have an influence on the support of the eventual distribution. In particular, the natural process of competition is not always selecting a unique species, as it would be predicted by the competitive exclusion principle, but several species may coexist as long as they maximize the fitness function. In many cases it is possible to compute the eventual distribution of the surviving competitors. In some configurations, species that maximize the fitness may still get extinct depending on the shape of the initial distribution and on an other parameter of the model, and we provide a way to characterize when this unexpected extinction happens.

We study existence and uniqueness of solutions of ([Formula: see text]) [Formula: see text] in [Formula: see text], [Formula: see text] on [Formula: see text], where [Formula: see text] is a bounded smooth domain such that [Formula: see text], [Formula: see text] is a constant, [Formula: see text] a continuous nondecreasing function satisfying some integral growth condition and [Formula: see text] and [Formula: see text] two Radon measures, respectively, in [Formula: see text] and on [Formula: see text]. We show that the situation differs considerably according the measure is concentrated at [Formula: see text] or not. When [Formula: see text] is a power we introduce a capacity framework which provides necessary and sufficient conditions for the solvability of problem ([Formula: see text]).