1,721,067 research outputs found
On the Dirichlet problem with several volume constraints on the level sets
No full textWe consider minimization problems involving the Dirichlet integral under an arbitrary number of volume constraints on the level sets and a generalized boundary condition. More precisely, given a bounded open domain Ω ⊂ Rn with smooth boundary, we study the problem of minimizing fΩ |∇u|2 among all those functions u ∈ H1 that simultaneously satisfy n-dimensional measure constraints on the level sets of the kind |{u = li}| = α1, i = 1,..., k, and a generalized boundary condition u ∈. Here, Κ is a closed convex subset of H1 such that Κ + H01 = Κ; the invariance of Κ under H01 provides that the condition u ∈ Κ actually depends only on the trace of u along δΩ. By a penalization approach, we prove the existence of minimizers and their Hölder continuity, generalizing previous results that are not applicable when a boundary condition is prescribed. Finally, in the case of just two volume constraints, we investigate the Γ-convergence of the above (rescaled) functionals when the total measure of the two prescribed level sets tends to saturate the whole domain Ω. It turns out that the resulting Γ-limit functional can be split into two distinct parts: the perimeter of the interface due to the Dirichlet energy that concentrates along the jump, and a boundary integral term due to the constraint u ∈ Κ. In the particular case where Κ = H1 (i.e. when no boundary condition is prescribed), the boundary term vanishes and we recover a previous result due to Ambrosio et al
Relaxation and convexity of functionals with pointwise nonlocality
No full textIt is shown that the relaxation of the integral functional involving argument deviations I(u):=∫f(x{ui(gij))}k,li,j=1)d μΩ(x), in weak topology of a Lebesgue space (Lp(⊖, μμ))k (where (Ω, ∑(Ω), μΩ) and (⊖, ∑(⊖),μ⊖) are standard measure spaces, the latter with nonatomic measure), coincides with its convexification whenever the matrix of measurable functions gij: Ω → ⊖ satisfies the special condition, called unifiability, which can be regarded as collective nonergodicity or commensurability property, and is automatically satisfied only if k = l = 1. If, however, either k > 1 or l > 1, then it is shown that as opposed to the classical case without argument deviations, for nonunifiable function matrix {gij} one can always construct an integrand f so that the functional I itself is already weakly lower semicontinuous but not convex
Representation of atomic operators and extension problems
No full textThe notion of an atomic operator between spaces of measurable functions was introduced in 2002 in a paper by Drakhlin, Ponosov and Stepanov in order to provide a reasonable generalization of local operators useful for applications. It has been shown that, roughly speaking, atomic operators amount to compositions of local operators with shifts. A natural problem is then when a continuous-in-measure atomic operator can be represented as a composition of a Nemytskiǐ (composition) operator generated by a Carathéodory function, and a shift operator. In this paper we will show that the answer to this question is inherently related to the possibility of extending an atomic operator with continuity from a space of functions measurable with respect to some σ-algebra to a larger space of functions measurable with respect to a larger σ-algebra, as well as to the possibility of extending any σ-homomorphism from a smaller-measure algebra to a σ-homomorphism on a larger-measure algebra. We characterize precisely the condition on the respective σ-algebras which provides such possibilities and induces the positive answer to the above representation problem
Rectifiability of metric flat chains and fractional masses
No full textWe prove that every real flat chain T of finite mass in a complete separable metric space E is rectifiable when Mα(T) < + ∞ for some α ∈ [0, 1), where Mα(T) is the α-mass of T. Bibliography: 12 titles. © 2010 Springer Science+Business Media, Inc
The saga of a fish: from a survival guide to closing lemmas
No full textIn the paper by D. Burago, S. Ivanov and A. Novikov, “A survival guide for feeble fish”, it has been shown that a fish with limited velocity can reach any point in the (possibly unbounded)ocean provided that the fluid velocity field is incompressible, bounded and has vanishing mean drift. This result extends some known global controllability theorems though being substantially nonconstructive. We give a fish a different recipe of how to survive in a turbulent ocean, and show its relationship to structural stability of dynamical systems by providing a constructive way to change slightly the velocity field to produce conservative (in the sense of not having wandering sets of positive measure)dynamics. In particular, this leads to the extension of C. Pugh's closing lemma to incompressible vector fields over unbounded domains. The results are based on an extension of the Poincaré recurrence theorem to some σ-finite measures and on specially constructed Newtonian potentials
Atomic operators, random dynamical systems and invariant measures
No full textIt is proved that the existence of invariant measures for families of the so-called atomic operators (nonlinear generalized weighted shifts) defined over spaces of measurable functions follows from the existence of appropriate invariant bounded sets. Typically, such operators come from infinite-dimensional stochastic differential equations generating not necessarily regular solution flows, for instance, from stochastic differential equations with time delay in the diffusion term (regular solution flows called also Carathéodory flows are those almost surely continuous with respect to the initial data). Thus, it is proved that to ensure the existence of an invariant measure for a stochastic solution flow it suffices to find a bounded invariant subset, and no regularity requirement for the flow is necessary. This result is based on the possibility to extend atomic operators by continuity to a suitable set of Young measures, which is proved in the paper. A motivating example giving a new result on the existence of an invariant measure for a possibly nonregular solution flow of some model stochastic differential equation is also provided
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