We demonstrate a measure theoretical approach to the local regularity of weak supersolutions elliptic and parabolic equations in divergence form. In first part, we show that become lower semicontinuous after redefinition on set zero. The proof relies general principle, i.e. De Giorgi type lemma, which offers unified for wide class equations, including an anisotropic equation, p-Laplace porous m...

Based on some ideas of Greene and Krantz, we study the semicontinuity of automorphism groups of domains in one and several complex variables. We show that semicontinuity fails for domains in C, n > 1, with Lipschitz boundary, but it holds for domains in C1 with Lipschitz boundary. Using the same ideas, we develop some other concepts related to mappings of Lipschitz domains. These include Bergma...

In this paper, we give sufficient conditions for the upper semicontinuity property of the solution mapping of a parametric generalized vector quasiequilibrium problem with mixed relations and moving cones. The main result is proven under the assumption that moving cones have local openness/local closedness properties and set-valued maps are cone-semicontinuous in a sense weaker than the usual s...

In this paper, we use topological methods to study various semicontinuity properties of the local spectrum of singular points of algebraic plane curves and spectrum at infinity of polynomial maps in two variables. Using the Seifert form, the Tristram–Levine signatures of links, and the associated Murasugi-type inequalities, we reprove (in a slightly weaker form) a result obtained by Steenbrink ...

We consider an infinite-horizon optimal control problem with the cost functional described either by an integral over an unbounded interval (a Lebesgue integral) or by a limit of integrals (an improper Lebesgue integral). We prove some theorems on the existence of solutions to such problems. The proofs are based on appropriate lower closure theorems and some extensions of Olech’s theorem on the...

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 18 February 2020Accepted: 28 September 2020Published online: 13 January 2021Keywordsnonlocality, double integrals, relaxation, lower semicontinuity, nonlocal inclusions, Young measuresAMS Subject HeadingsPrimary, 49J45; Secondary, 26B25, 74G65Publication DataISSN (print):...

Abstract: The concepts of upper and lower semicontinuity in pointfree topology were introduced and first studied by Li and Wang in 1997. However Li and Wang’s treatment does not faithfully reflect the original classical notion. In this note, we present algebraic descriptions of upper and lower semicontinuous real functions, in terms of frame homomorphisms, that suggest the right alternative to ...

In this paper we give simpler proofs of the classical continuity and lower semicontinuity theorems of Reshetnyak. 1. Main result In 1968, Reshetnyak [20] proved two important results concerning the continuity and lower semicontinuity of functionals with respect to weak-star convergence of measures. These theorems are used in a variety of areas in the calculus of variations, ranging from problem...

In this work, we study the continuity and topological structural stability of attractors for nonautonomous random differential equations obtained by small bounded perturbations autonomous semilinear problems. First, existence permanence unstable sets hyperbolic solutions. Then, use to establish lower semicontinuity show that gradient structure persists under perturbations. Finally, apply abstra...