Subset Of Rn Research Articles

A RELATIVE ISOPERIMETRIC INEQUALITY

We prove that, if Ω is an open subset of ℝN with finite measure, there exists a hyperplane H through 0 such that the measure of Ω ∩ H is less than the measure of B ∩ H, where B is the open ball with center 0 having the same measure as Ω. An application is given to the optimal Poincaré inequality on BV(Ω).

The Hemivariational Inequalities for an Upper Semicontinuous Set-valued Mapping

We establish some existence results for hemivariational inequalities of Stampacchia type involving an upper semicontinuous set-valued mapping on a bounded, closed and convex subset in ℝn. We also derive a sufficient condition for the existence and boundedness of solution, without assuming boundedness of the constraint set.

Optimization problems, first order approximated optimization problems and their connections

In this paper, we attach to the optimization problem ... where X is a subset of Rn, f : X → R, g = (g1, ..., gm) : X → Rm and h = (h1, ..., hq) : X → Rq are functions, the (0, 1) − η− approximated optimization problem (AP). We will study the connections between the optimal solutions for Problem (AP), the saddle points for Problem (AP), optimal solutions for Problem (P) and saddle points for Problem (P).

Optimization problems and (0, 2)-η-approximated optimization problems

In this paper, we attach to the optimization problem ... where X is a subset of Rn, f : X → R, g : X → Rm and h : X → Rq are three functions, m, n, q ∈ N, a (0, 2)-η-approximated optimization problem (AP). We will study the connections between the feasible solutions of the η-approximated problem and the feasible solutions of the original problem. Then we will study the connections between the optimal solutions of Problem (AP) and the optimal solutions of Problem (P) via the saddle points of the two problems.

Flat forms, bi-Lipschitz parametrizations, and smoothability of manifolds

We give a sufficient condition for a metric (homology) manifold to be locally bi-Lipschitz equivalent to an open subset in R n . The condition is a Sobolev condition for a measurable coframe of flat 1-forms. In combination with an earlier work of D. Sullivan, our methods also yield an analytic characterization for smoothability of a Lipschitz manifold in terms of a Sobolev regularity for frames in a cotangent structure. In the proofs, we exploit the duality between flat chains and flat forms, and recently established differential analysis on metric measure spaces. When specialized to R n , our result gives a kind of asymptotic and Lipschitz version of the measurable Riemann mapping theorem as suggested by Sullivan.

ON THE SOLUTION SETS OF FOUR-POINT BOUNDARY VALUE PROBLEMS FOR NONCONVEX DIFFERENTIAL INCLUSIONS

We consider the multivalued problem [Formula: see text] under four boundary conditions u(0) = x0, u(η) = u(θ) = u(T) where 0 < η < θ < T and for F is a multifunctions from [0, T] × ℝn× ℝnto the nonempty compact subsets of ℝnnot necessary convex. We give a lemma which is useful in the study of four boundary problems for the differential equations and the differential inclusions. Further we have results that improve earlier theorems.

Injective mappings and solvable vector fields

We establish a sufficient condition for injectivity in a class of mappings defined on open connected subsets of Rn , for arbitrary n. The result relates solvability of the appropriate vector fields with injectivity of the mapping and extends a result proved by the first author for n < 3 . Furthermore, we extend the result to connected paracompact smooth oriented manifolds and show that the convexity condition imposes strong topological restrictions on the manifold.

An approach via symmetrization methods to nonlinear elliptic problems with a lower order term

In this paper we consider a class of nonlinear elliptic problems of the type $$ \left\{ \begin{gathered} - div(a(x,\nabla u)) - div(\Phi (x,u)) = fin\Omega \hfill \\ u = 0on\partial \Omega , \hfill \\ \end{gathered} \right. $$ where Ω is a bounded open subset of R N , N ≥ 2, f is a L 1 (Ω) function or a Radon measure with bounded total variation. We fix some structural conditions on a and Φ to prove uniqueness results when f ∈ L 1 (Ω).

FULL MEREOGEOMETRIES

We analyze and compare geometrical theories based on mereology (mereogeometries). Most theories in this area lack in formalization, and this prevents any systematic logical analysis. To overcome this problem, we concentrate on specific interpretations for the primitives and use them to isolate comparable models for each theory. Relying on the chosen interpretations, we introduce the notion of environment structure, that is, a minimal structure that contains a (sub)structure for each theory. In particular, in the case of mereogeometries, the domain of an environment structure is composed of particular subsets of Rn. The comparison of mereogeometrical theories within these environment structures shows dependencies among primitives and provides (relative) definitional equivalences. With one exception, we show that all the theories considered are equivalent in these environment structures.

Exponential Rates of Convergence in the Ergodic Theorem: A Constructive Approach

We prove that, once an algorithm of perfect simulation for a stationary and ergodic random field F taking values in \(S^{\mathbb{Z}^{d}}\), S a bounded subset of Rn, is provided, the speed of convergence in the mean ergodic theorem occurs exponentially fast for F. Applications from (non-equilibrium) statistical mechanics and interacting particle systems are presented.

Duality results involving functions associated to nonempty subsets of locally convex spaces

In many papers on consumer theory and production analysis duality results between profit, revenue, cost, input, output and shortage functions are established. This functions are associated to certain subsets of ℝn. The aim of this paper is to study in a systematic way such duality results in locally convex spaces and to derive them under minimal hypotheses.

The structure on the real field generated by the standard part map on an o-minimal expansion of a real closed field

Let R be a sufficiently saturated o-minimal expansion of a real closed field, let \( \mathcal{O} \) be the convex hull of ℚ in R, and let st: \( \mathcal{O}^n \) → ℝn be the standard part map. For X ⊆ Rn define st X:= st (X ∩ \( \mathcal{O}^n \)). We let ℝind be the structure with underlying set ℝ and expanded by all sets st X, where X ⊆ Rn is definable in R and n = 1, 2,.... We show that the subsets of ℝn that are definable in ℝind are exactly the finite unions of sets st X st Y, where X, Y ⊆ Rn are definable in R. A consequence of the proof is a partial answer to a question by Hrushovski, Peterzil and Pillay about the existence of measures with certain invariance properties on the lattice of bounded definable sets in Rn.

Blow-up of solutions of nonlinear parabolic inequalities

We study nonnegative solutions u ( x , t ) u(x,t) of the nonlinear parabolic inequalities \[ a u λ ≤ u t − Δ u ≤ u λ au^\lambda \le u_t - \Delta u \le u^\lambda \] in various subsets of R n × R \textbf {R}^n\times \textbf {R} , where λ > n + 2 n \lambda >\frac {n+2}{n} and a ∈ ( 0 , 1 ) a\in (0,1) are constants. We show that changing the value of a a in the open interval ( 0 , 1 ) (0,1) can dramatically affect the blow-up of these solutions.

Hyperspace of max-plus convex compact sets

The hyperspace mpcc(ℝn) of max-plus convex compact subsets of ℝn, n ≥ 2, is considered. The main result is as follows: this hyperspace is a contractible manifold modeled on the Hilbert cube Q. It is also shown that the projection mapping mpcc(ℝn) → mpcc(ℝm), n ≥ m, is open. Moreover, it is proved that the hyperspace mpcc(\( I^{\omega _1 } \) ) of the Tikhonov [Tychonoff] cube \( I^{\omega _1 } \) is homeomorphic to \( I^{\omega _1 } \) .

Tug-of-war and the infinity Laplacian

We prove that every bounded Lipschitz function F F on a subset Y Y of a length space X X admits a tautest extension to X X , i.e., a unique Lipschitz extension u : X → R u:X \rightarrow \mathbb {R} for which Lip U ⁡ u = Lip ∂ U ⁡ u \operatorname {Lip}_U u =\operatorname {Lip}_{\partial U} u for all open U ⊂ X ∖ Y U \subset X\smallsetminus Y . This was previously known only for bounded domains in R n \mathbb {R}^n , in which case u u is infinity harmonic; that is, a viscosity solution to Δ ∞ u = 0 \Delta _\infty u = 0 , where \[ Δ ∞ u = | ∇ u | − 2 ∑ i , j u x i u x i x j u x j . \Delta _\infty u = |\nabla u|^{-2} \sum _{i,j} u_{x_i} u_{x_ix_j} u_{x_j}. \] We also prove the first general uniqueness results for Δ ∞ u = g \Delta _{\infty } u = g on bounded subsets of R n \mathbb {R}^n (when g g is uniformly continuous and bounded away from 0 0 ) and analogous results for bounded length spaces. The proofs rely on a new game-theoretic description of u u . Let u ε ( x ) u^\varepsilon (x) be the value of the following two-player zero-sum game, called tug-of-war: fix x 0 = x ∈ X ∖ Y x_0=x\in X \smallsetminus Y . At the k t h k^{\mathrm {th}} turn, the players toss a coin and the winner chooses an x k x_k with d ( x k , x k − 1 ) > ε d(x_k, x_{k-1})> \varepsilon . The game ends when x k ∈ Y x_k \in Y , and player I’s payoff is F ( x k ) − ε 2 2 ∑ i = 0 k − 1 g ( x i ) F(x_k) - \frac {\varepsilon ^2}{2}\sum _{i=0}^{k-1} g(x_i) . We show that ‖ u ε − u ‖ ∞ → 0 \|u^\varepsilon - u\|_{\infty } \to 0 . Even for bounded domains in R n \mathbb {R}^n , the game theoretic description of infinity harmonic functions yields new intuition and estimates; for instance, we prove power law bounds for infinity harmonic functions in the unit disk with boundary values supported in a δ \delta -neighborhood of a Cantor set on the unit circle.

Universal Taylor series on open subsets of Rn

Universal Taylor series on open subsets of R^<i>n</i>

Pointwise characterization of Sobolev classes

We prove that a function f is in the Sobolev class W loc (ℝ n ) or W m,p (Q) for some cube Q ⊂ ℝ n if and only if the formal (m − 1)-Taylor remainder R m−1 f(x,y) of f satisfies the pointwise inequality |R m−1 f(x,y)| ≤ |x − y| m [a(x) + a(y)] for some a e L p (Q) outside a set N ⊂ Q of null Lebesgue measure. This is analogous to H. Whitney’s Taylor remainder condition characterizing the traces of smooth functions on closed subsets of ℝ n .

Study of the Singular Yamabe Problem in Some Bounded Domain of ℝn

Abstract In this paper, we extend the result of R. Mazzeo and F. Pacard in the following direction: Given Ω any bounded open regular subset of ℝn, n ≥ 3 and given x1, x2 ∈ Ω, we give a sufficient condition on x1, x2, for the problem to have a positive weak solution in Ω with 0 boundary data, which is singular at each xi.

A fixed point theorem for a class of Sobolev maps

We prove an analog of the Brouwer fixed point theorem for a map whose differential and adjoint are integrable with exponents n−1 and n/(n−1) respectively. Here Ω is a convex bounded open subset of R n .

Universal Taylor series on open subsets of Rn

In the case of the complex plane, several notions of universal Taylor series have been introduced, ([10, 2, 19, 20]). The purpose of the present article is to establish the existence of universal Taylor series of