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Except a closed set $${\mathfrak {S}}^c$$ with zero parabolic Hausdorff measure, the weak limit $$(\rho ,\mathbf{u})$$ of approximate solutions is a renormalized weak solution with finite energy of three dimensional compressible Navier–Stokes equations for $$\gamma \in (6/5, 3/2]$$ as constructed by Lions and Feireisl et al. in the Leray sense. The key novelty of the paper is the improved integrability of pressure by localization, which is based on the faster decay of the gradient of velocity and the higher integrability of the Riesz potentials of both density and momentum.

This paper is devoted to the partial regularity of suitable weak solutions to the system of the incompressible shear-thinning flow in a bounded domain Ω ⊂ ℝn, n ≥ 2. It is proved that there exists a suitable weak solution of the shear-thinning fluid in the n-D smooth bounded domain (for n ≥ 2). For 3D model, it is proved that the singular points are concentrated on a closed set whose 1 dimensional Hausdorff measure is zero.

The power moments of a positive measure on the real line or the circle are characterized by the non-negativity of an infinite matrix, Hankel, respectively Toeplitz, attached to the data. Except some fortunate configurations, in higher dimensions there are no non-negativity criteria for the power moments of a measure to be supported by a prescribed closed set. We combine two well studied fortunate situations, specifically a class of curves in two dimensions classified by Scheiderer and Plaumann, and compact, basic semi-algebraic sets, with the aim at enlarging the realm of geometric shapes on which the power moment problem is accessible and solvable by non-negativity certificates.

We study the asymptotic behavior of the Navier–Stokes system with Dirichlet boundary conditions posed in a domain Ω e = Ω ∖ T e . Here Ω ⊂ R 3 is a bounded open set and T e is the union of many disjoint closed sets of size e 3 and density of order 1 ∕ e 3 , with e a small positive parameter. Similarly to the periodic case we get a limit system corresponding to a Brinkman flow. The difference is that now the Brinkman matrix is not homogeneous, it depends on the density of the closed sets composing T e . The result is obtained through an adaptation of the two-scale convergence method.

We study the capacity in the sense of Beurling–Deny associated with the Dirichlet space D ( μ ) where μ is a finite positive Borel measure on the unit circle. First, we obtain a sharp asymptotic estimate of the norm of the reproducing kernel of D ( μ ) . It allows us to give an estimate of the capacity of points and arcs of the unit circle. We also provide new conditions on closed sets to be polar. Our method is based on sharp estimates of norms of some outer functions, which allow us to transfer these problems to an estimate of the reproducing kernel of an appropriate weighted Sobolev space.

The Sturm – Liouville differential operators on closed sets of the real line are considered. Properties of their spectral characteristics are obtained and the inverse problem of recovering the operators from their spectra is studied. An algorithm for the solution of the inverse problem is developed and the uniqueness of the solution is established. The statement and the study of inverse spectral problems essentially depend on the structure of the closed set. We consider an important subclass of closed sets when the set is a unification of a finite number of closed intervals and isolated points. In order to solve the inverse spectral problem for this class of closed sets, we develop ideas of the method of spectral mappings. We also establish and use connections between the Weyl-type functions related to different subsets of the main closed set. Using these ideas and properties we obtain a global constructive procedure for the solution of the nonlinear inverse problem considered, and we establish the uniqueness of the solution of the inverse problem.

This paper studies set invariance and contractivity in hybrid systems modeled by hybrid inclusions using barrier functions. After introducing the notion of a multiple barrier functions, we investigate the tightest possible sufficient conditions to guarantee different forward invariance and contractivity notions of a closed set for hybrid systems with nonuniqueness of solutions and solutions terminating prematurely. More precisely, we consider forward (pre-)invariance of sets, which guarantees solutions to stay in a set, and (pre-)contractivity, which further requires solutions that reach the boundary of the set to evolve (continuously or discretely) towards its interior. Our conditions for forward invariance and contractivity involve infinitesimal conditions in terms of multiple barrier functions. Examples illustrate the results. Keywords: Forward invariance, contractivity, barrier functions, hybrid dynamical systems., Technical report accompanying the paper entitled: Sufficient conditions for forward invariance and contractivity in hybrid inclusions using barrier functions, submitted to Automatica, 2019

In this paper, we will establish the global existence of a suitable weak solution to the Erickson--Leslie system modeling hydrodynamics of nematic liquid crystal flows with kinematic transports for molecules of various shapes in ${\mathbb{R}^3}$, which is smooth away from a closed set of (parabolic) Hausdorff dimension at most $\frac{15}{7}$., 42 pages

The translative intersection formula of integral geometry yields an expression for the mean Euler characteristic of a stationary random closed set intersected with a fixed observation window. We formulate this result in the setting of sets with positive reach and using flag measures which yield curvature measures as marginals. As an application, we consider excursion sets of stationary random fields with $C^{1,1}$ realizations, in particular, stationary Gaussian fields, and obtain results which extend those known from the literature., A new section (Polytopal window) has been added

We develop a variational method for constructing positive entropy invariant measures of Lagrangian systems without assuming transversal intersections of stable and unstable manifolds, and without restrictions to the size of non-integrable perturbations. We apply it to a family of two and a half degrees of freedom a-priori unstable Lagrangians, and show that if we assume that there is no topological obstruction to diffusion (precisely formulated in terms of topological non-degeneracy of minima of the Peierl's barrier function), then there exists a vast family of "horsheshoes", such as "shadowing" ergodic positive entropy measures having precisely any closed set of invariant tori in its support. Furthermore, we give bounds on the topological entropy and the "drift acceleration" in any part of a region of instability in terms of a certain extremal value of the Fr\'{e}chet derivative of the action functional, generalizing the angle of splitting of separatrices. The method of construction is new, and relies on study of formally gradient dynamics of the action (coupled parabolic semilinear partial differential equations on unbounded domains). We apply recently developed techniques of precise control of the local evolution of energy (in this case the Lagrangian action), energy dissipation and flux. In Part II of the paper we will apply the theory to obtain sharp bounds for topological entropy and drift acceleration for the same class of equations in the case of small perturbations., Comment: Version 2: corrected typos

Weakly forward (pre-)invariant sets guarantee the existence of at least one maximal solution, when starting from any point in the set, that stays in that set. As a continuation to prior works and using multiple barrier functions, this paper studies weak forward invariance in hybrid systems modeled by constrained inclusions. We propose sufficient conditions to guarantee weak forward invariance of a closed set generated by the intersection of the zero-sublevel sets of the different components of a (vector) function called multiple barrier function. Our sufficient conditions are in terms of the multiple barrier function generating the set. Moreover, along the flow part of the hybrid system, our conditions are of two types. The first type of flow conditions need to hold only at the boundary of the set and weak forward invariance is shown by imposing transversality conditions on the intersection between the zero-sublevel sets of the components of the barrier function. The second type of conditions require both transversality and flow conditions on an external complement of the boundary of the considered set. Examples throughout the paper illustrate the results.

If a closed subset of a Riemann surface is a set of uniform meromorphic approximation, then its boundary is shown to be a set of tangential meromorphic approximation.

International audience; In a separable Banach space E, we study the invariance of a closed set K under the action of the evolution equation associated with a maximal dissipative linear operator A perturbed by a quasi-dissipative continuous term B. Using the distance to the closed set, we give a general necessary and sufficient condition for the invariance of K. Then, we apply our result to several examples of partial differential equations in Banach and Hilbert spaces.

We study α-sets in Euclidean space ℝ n . The notion of α-set is introduced as a generalization of a convex closed set in ℝ n . This notion appeared in the study of reachable sets and integral funnels of nonlinear control systems in Euclidean spaces. Reachable sets of nonlinear dynamic systems are usually nonconvex, and the degree of their nonconvexity is different in different systems. This circumstance prompted the introduction of a classification of sets in ℝ n according to the degree of their nonconvexity. Such a classification stems from control theory and is presented here in terms of α-sets in ℝ n .

Consider a function u ∈ BV2loc(R2) such that ∇u takes values in a fixed set of 3 vectors almost everywhere. This condition implies that u is piecewise affine away from a closed set of vanishing 1-dimensional Hausdorff measure. Furthermore, there is some rigidity in the sense that away from the exceptional set, small perturbations of u will result only in controllable changes of the structure.

We investigate nonlinear irregular equations in Hilbert space with a priori constraints. The differentiability of the problem's operator is not assumed. The constraints are described by a bounded closed set D that is part of an extended source representation class in terms of a given linear operator. The unique solvability of the problem is not assumed. It is established that solutions to the problem form a cluster of diameter ρ diam ( D ) with ρ ∈ ( 0 , 1 ) . The value ρ depends on the problem parameters and can be arbitrarily small. We also justify the approximation properties of the quasi–solution method in relation to the solution set of the original problem.

The aim of this work is to study new sets of soft closed and soft open sets by using soft pre-open sets with collection of a soft ideal which is said to be an a soft-ᶅ-pre-g-closed and soft-ᶅ- pre-g-open sets with some properties, examples and relationships between these concepts with soft pre-g * - closed and soft pre-g *-open. New concepts of soft open and soft closed functions are defined by using these sets with some examples and scheme. Also, many continuous function by using these with examples and schemes are explained.

Latest generality appearance of nano weakly πg-closed sets, nano weakly πg-open sets which are weaker forms of nano πg-closed sets, nano πg-open sets respectively. The relationships among them related generalized closed sets is also investigated in this paper.

We extend to Hausdorff continua the result of Camargo and Uzcategui that says that if X is a metric continuum, Jones' set function T is continuous on singletons and T is idempotent on closed sets, then G = { T ( { x } ) | x ∈ X } is a decomposition of X . We also present important implications of this result, a couple of them answer questions of the celebrated Houston Problem Book . We study Hausdorff continua with the property of Kelley and with the property of Kelley weakly. We establish a Hausdorff version of Jones' Aposyndetic Decomposition Theorem. To this end, we introduce the uniform property of Effros.

The upper and lower complex interpolation of -spaces and -spaces are considered. When , and u are finite, we prove that . Otherwise, the space is a closed subspace of . We also establish interpolation results of certain closed subspaces of -spaces and -spaces, which are the closure of linear subspaces satisfying the lattice condition.

The concept of replacement of the initial stationary optimization problem with some nonstationary mechanical system tending with time to the position of equilibrium, which coincides with the solution of the initial problem, makes it possible to construct effective numerical algorithms. First, differential equations of the movement should be derived. Then we pass to the difference scheme and define the iteration algorithm. There is a wide class of optimization methods constructed in such a way. One of the most known representatives of this class is the heavy ball method. As a rule, such type of algorithms includes parameters that highly affect the convergence rate. In this paper, the charged ball method, belonging to this class, is proposed and investigated. It is a new effective optimization method that allows solving some computational geometry problems. A problem of orthogonal projection of a point onto a convex closed set with a smooth boundary and the problem of finding the minimum distance between two such sets are considered in detail. The convergence theorems are proved, and the estimates for the convergence rate are found. Numerical examples illustrating the work of the proposed algorithms are given.

For continuously differentiable vector fields, we characterize the omega limit set of a trajectory converging to a compact curve Gamma subset of R-n. In particular, the limit set is either a fixed point or a continuum of fixed points if Gamma is a simple open curve; otherwise, the limit set can in addition be either a closed orbit or a number of fixed points with compatibly oriented orbits connecting them. An implication of the result is a tightened-up version of the Poincare-Bendixson theorem. (C) 2017 Elsevier Ltd. All rights reserved.

In this paper, in terms of the Frechet normal cone, we establish exact separation results for finitely many disjoint closed sets in an Asplund space, which supplement the extremal principle and some fuzzy separation theorems. As an application, we provide a new optimality condition for a constraint optimization problem in terms of Frechet subdifferential and Frechet normal cone.

In this paper, we introduce and investigate the notion of almost contra βθ -continuous functions by utilizing βθ -closed sets. We obtain fundamental properties of almost contra βθ -continuous functions and discuss the relationships between almost contra βθ -continuity and other related functions.

For a closed set S and a bounded closed convex set U in a real normed vector space, we give exact subdifferential formulas of an optimal value function $$\mathrm {I}\!\Gamma _{S|U}$$ whose definition is based on the Minkowski function of U. $$\mathrm {I}\!\Gamma _{S|U}$$ covers distance function and indicator function as special cases. The main contribution is dropping two important assumptions of some main results in the literature.

The gradient projection method and Newton’s method are generalized to the case of nonconvex constraint sets representing the set-theoretic intersection of a spherical surface with a convex closed set. Necessary extremum conditions are examined, and the convergence of the methods is analyzed.

As a continuation of [1] and using multiple barrier functions, this paper studies forward invariance in hybrid systems modeled by hybrid inclusions. After introducing the notion of a multiple barrier function, we propose sufficient conditions to guarantee different forward invariance properties of a closed set for hybrid systems with nonuniqueness of solutions, solutions terminating prematurely, and Zeno solutions. More precisely, we consider forward (pre-)invariance of sets, which guarantees solutions to stay in a set, and (pre-)contractivity, which further requires solutions that stay in the boundary of the set to evolve (continuously or discretely) towards its interior. Our conditions for forward invariance involve infinitesimal conditions in terms of multiple barrier functions while our conditions for pre-contractivity (and contractivity) involve Minkowski functionals. Examples illustrate the results.

In this paper the small-amplitude motion of multiple superposed viscous fluids is studied as a linearized initial-value problem. The analysis results in a closed set of equations for the Laplace transformed amplitudes of the interfaces that can be inverted numerically. The derived equations also contain the general normal mode equations, which can be used to determine the asymptotic growth-rates of the systems directly. After derivation, the equations are used to study two different problems involving three fluid layer. The first problem is the effect of initial phase difference on the development of a Rayleigh-Taylor instability and the second is the damping effect of a thin, highly viscous, surface layer., 22 pages and 9 figures

An existence and uniqueness result, up to fattening, for crystalline mean curvature flows with forcing and arbitrary (convex) mobilities, is proven. This is achieved by introducing a new notion of solution to the corresponding level set formulation. Such solutions satisfy a comparison principle and stability properties with respect to the approximation by suitably regularized problems. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. The approach accounts for the possible presence of a time-dependent bounded forcing term, with spatial Lipschitz continuity. As a result of our analysis, we deduce the convergence of a minimizing movement scheme proposed by Almgren, Taylor, and Wang (1993) to a unique (up to fattening) “flat flow” in the case of general, including crystalline, anisotropies, solving a long-standing open question.

We develop new efficient algorithms for a class of inverse problems of gravimetry to recover an anomalous volume mass distribution (measure) in the sense that we design fast local level-set methods to simultaneously reconstruct both unknown domain and varying density of the anomalous measure from modulus of gravity force rather than from gravity force itself. The equivalent-source principle of gravitational potential forces us to consider only measures of the form \begin{document}$ \mu = f\,\chi_{D} $\end{document} , where \begin{document}$ f $\end{document} is a density function and \begin{document}$ D $\end{document} is a domain inside a closed set in \begin{document}$ \bf{R}^n $\end{document} . Accordingly, various constraints are imposed upon both the density function and the domain so that well-posedness theories can be developed for the corresponding inverse problems, such as the domain inverse problem, the density inverse problem, and the domain-density inverse problem. Starting from uniqueness theorems for the domain-density inverse problem, we derive a new gradient from the misfit functional to enforce the directional-independence constraint of the density function and we further introduce a new labeling function into the level-set method to enforce the geometrical constraint of the corresponding domain; consequently, we are able to recover simultaneously both unknown domain and varying density from given modulus of gravity force. Our fast level-set method is built upon localizing the level-set evolution around a narrow band near the zero level-set and upon accelerating numerical modeling by novel low-rank matrix multiplication. Numerical results demonstrate that uniqueness theorems are crucial for solving the inverse problem of gravimetry and will be impactful on gravity prospecting. To the best of our knowledge, our inversion algorithm is the first of such for the domain-density inverse problem since it is based upon the conditional well-posedness theory of the inverse problem.

An evolution inclusion with the right-hand side containing the difference of subdifferentials of proper convex lower semicontinuous functions and a multivalued perturbation whose values are nonconvex closed sets is considered in a separable Hilbert space. In addition to the original inclusion, we consider an inclusion with convexified perturbation and a perturbation whose values are extremal points of the convexified perturbation that also belong to the values of the original perturbation. Questions of the existence of solutions under various perturbations are studied and relations between solutions are established. The primary focus is on the weakening of assumptions on the perturbation as compared to the known assumptions under which existence and relaxation theorems are valid. All our assumptions, in contrast to the known assumptions, concern the convexified rather than original perturbation.

We consider a general minimal time problem with a convex constant dynamics and a lower semicontinuous extended real-valued target function defined on a Banach space. If the target function is the indicator function of a closed set, this problem is a minimal time problem for a target set, studied previously in particular by Colombo, Goncharov and Mordukhovich. We investigate several properties of the Frechet and proximal subdifferentials for the infimum time function. Also explicit expressions of the above mentioned subdifferentials as well as various directional derivatives are obtained. We provide some examples to show the essentiality of assumptions of our theorems.

On a bounded domain Omega in the Euclidean space R-n, we study the homogeneous Dirichlet problem for the eikonal equation associated with a system of smooth vector fields, which satisfies Hormander's bracket generating condition. We prove that the solution is smooth in the complement of a closed set of Lebesgue measure zero. (C) 2018 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

International audience; This paper provides a new characterization of the stochastic invariance of a closed subset of R^d with respect to a diffusion. We extend the well-known inward pointing Stratonovich drift condition to the case where the diffusion matrix can fail to be differentiable: we only assume that the covariance matrix is. In particular, our result can be directly applied to construct affine diffusions and polynomial preserving diffusions on any arbitrary closed set.

We prove a sufficient condition for a time-dependent closed set to be viable with respect to a delay evolution inclusion subjected to an implicit nonlocal initial condition. An application to a comparison problem is also included.

We prove the existence and continuity of solutions to obstacle problems with p ( x ) -type nonstandard growth, and establish the conditions of removability of closed sets for Holder continuous solutions of the elliptic equations in the framework of variable exponent Sobolev spaces.

The estimation of the mean density of random closed sets in $$\mathbb {R}^d$$Rd with integer Hausdorff dimension $$n

The nonlinear impulsive control system with trajectories of bounded variation is considered. The notions of strong and weak V-invariance of a closed set relative to the impulsive control system are introduced. These notions are adapted to the system nonautonomy in the so-called "fast" time, in which the impulsive dynamics occurs. Invariance criteria in the form of the proximal Hamilton-Jacobi equalities were proved.