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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.

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.

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 .

In this survey we collect some recent results obtained by the authors and collaborators concerning the fine structure of functions of bounded deformation (BD). These maps are $\mathrm{L}^1$-functions with the property that the symmetric part of their distributional derivative is representable as a bounded (matrix-valued) Radon measure. It has been known for a long time that for a (matrix-valued) Radon measure the property of being a symmetrized gradient can be characterized by an under-determined second-order PDE system, the Saint-Venant compatibility conditions. This observation gives rise to a new approach to the fine properties of BD-maps via the theory of PDEs for measures, which complements and partially replaces classical arguments. Starting from elementary observations, here we elucidate the ellipticity arguments underlying this recent progress and give an overview of the state of the art. We also present some open problems., 33 pages

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.