Open Subset Of Rn Research Articles

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

Regularity of minimizers of quasi perimeters with a volume constraint

We study the regularity of the boundary of sets minimizing a quasi perimeter T(E) = P(E,Ω) + G(E) with a volume constraint. Here Ω is any open subset of Rn with n > 2, and G is a lower semicontinuous function on sets of finite perimeter satisfying G(E) 6 G(F ) + C|E 4 F |β for all sets E,F of finite perimeter with equal volume. We show that under the condition β > 1 − 1/n, any volume constrained minimizer E of the quasi perimeter T has both interior points and exterior points, and E is indeed a quasi minimizer of perimeter without the volume constraint. Using a well known regularity result for quasi minimizers of perimeter, we get the classical C1,α regularity for the reduced boundary of E.

Comparison theorems for second-order elliptic inequalities

We study non-negative solutions of the elliptic inequalities∑i,j=1naij(x)∂2u∂xi∂xj+∑i=1nbi(x)∂u∂xi⩾F(x,u)inΩ,where Ω≠0 is an open subset of Rn,n⩾2, and F:Ω×[0,∞)→[0,∞) is some function. In particular, the influence of the geometric properties of Ω in the behavior of solutions is investigated.

Regularity of Solutions to the Dirichlet Problem for Degenerate Elliptic Equation

In this paper, the author studies the regularity of solutions to the Dirichlet problem for equation Lu = f, where L is a second order degenerate elliptic operator in divergence form in Ω, a bounded open subset of Rn(n ≥ 3).

Measures of non–compactness of classical embeddings of Sobolev spaces

AbstractLet Ω be an open subset of ℝn and let p ∈ [1, n). We prove that the measure of non–compactness of the Sobolev embedding Wk,p0(Ω) → Lp*(Ω) is equal to its norm. This means that the entropy numbers of this embedding are constant and equal to the norm. The same is true, when λn(Ω) is small enough, for the embedding of W1,n0(Ω) into the Orlicz space with Young function exp(tn/(n−1) − 1. The position is different for the embedding of W1,p0(J) in C0,1−1/p($ \bar J $), J equals; (0, 1), when p ∈ (1,∞): in this case the measure of non–compactness is less than the norm. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On Closed Ideals in Smooth Classes

We study closedness properties of ideals generated by real – analytic functions in some subrings 𝒞 of C∞(Ω), where Ω is an open subset of ℝn. In contrast with the case 𝒞 = C∞(Ω), which has been clarified by famous works of Hörmander, Lojasiewicz and Malgrange, it turns out that such ideals are generally not closed when 𝒞 is an ultradifferentiable class. If 𝒞 is sufficiently regular and non – quasianalytic, and under the assumption that the real zero locus of the ideal reduces to a single point, we obtain a sharp sufficient condition of closedness, expressed in terms of the geometry of common complex zeros for the germs of the generators at this point. This condition is shown to be also necessary in dimension 2, when the ideal is principal. Some related questions about rings of ultradifferentiable germs and about ultradistributions are discussed.

On Closed Ideals in Smooth Classes

We study closedness properties of ideals generated by real – analytic functions in some subrings 𝒞 of C∞(Ω), where Ω is an open subset of ℝn. In contrast with the case 𝒞 = C∞(Ω), which has been clarified by famous works of Hörmander, Lojasiewicz and Malgrange, it turns out that such ideals are generally not closed when 𝒞 is an ultradifferentiable class. If 𝒞 is sufficiently regular and non – quasianalytic, and under the assumption that the real zero locus of the ideal reduces to a single point, we obtain a sharp sufficient condition of closedness, expressed in terms of the geometry of common complex zeros for the germs of the generators at this point. This condition is shown to be also necessary in dimension 2, when the ideal is principal. Some related questions about rings of ultradifferentiable germs and about ultradistributions are discussed.

Mappings of finite distortion: Monotonicity and continuity

We study mappings f = ( f1, ..., fn) : Ω → Rn in the Sobolev space W loc (Ω,R n), where Ω is a connected, open subset of Rn with n ≥ 2. Thus, for almost every x ∈ Ω, we can speak of the linear transformation D f(x) : Rn → Rn, called differential of f at x. Its norm is defined by |D f(x)| = sup{|D f(x)h| : h ∈ Sn−1}. We shall often identify D f(x) with its matrix, and denote by J(x, f ) = det D f(x) the Jacobian determinant. Thus, using the language of differential forms, we can write

A counterexample to “An extension of the Vitali-Hahn-Saks theorem” and a compactness result

We give a counterexample to An extension of the Vitali-HahnSaks theorem and from that highlight the sharp frame within which any attempt to change the version of such an extension should be possible. Lastly a sequential compactness criterion for Radon measures absolutely continuous with respect to a prescribed Radon measure defined on a locally compact separable metric space (taking into account the ideas of Hernandez-Lerma and Lasserre) is proved. The results deal with Radon measures but yield obvious corollaries on real (or vector-valued) Radon measures and so on functions with bounded variation on open subsets of Rn.

Th�or�me de division avec des conditions au bord

Let Ω be an open subset of ℝn and\(\mathcal{O}(\Omega )\) be a subalgebra of the algebra of analytic functions on Ω. We suppose that\(\mathcal{O}(\Omega )\) satisfies some weak conditions of noetherianity such that we can construct a finite stratification for each ideal of\(\mathcal{O}(\Omega )\). We also suppose that\(\mathcal{O}(\Omega )\) satifies global £ojasiewicz's inequalities. We prove the following: Let\(f_{1, \ldots ,} f_p \in \mathcal{O}(\Omega )\) andf C∞ on\(\overline \Omega \) flat on ∂Ω; if for eacha ∈ Ω the Taylor's serie off ata, Taf, is in the ideal generated byTaf1,...,Tafp in the ring of formal power series, then there existα1,...,αp,C∞ on\(\overline \Omega \) flat on ∂Ω such that\(f = \sum\nolimits_{j = 1}^p {\alpha _j f_j } \). This result extends the classic Hormander's theorem of division (for a polynomial) or the £ojasiewicz-Malgrange theorem in the local analytic case.

Integration and Microlocal Analysis in Colombeau Algebras of Generalized Functions

We study integration and Fourier transform in the Colombeau algebra Gτ of tempered generalized functions using a general damping factor. This unifies different settings described earlier by J. F. Colombeau, M. Nedeljikov, S. Pilipović, and M. Damsma (for a simplified version). Further we prove characterizations of regularity for generalized functions in two situations: compactly supported or in the image of S′ inside Gτ. Finally we investigate the notion of wave front set in the Colombeau algebra G(Ω), Ω an open subset of Rn, and show that it is in fact independent of the damping measure used for Fourier transform.

THE OSCILLATION PATTERN OF SOLUTIONS TO PARABOLIC EQUATIONS AS TIME GOES TO INFINITY

We consider the semilinear parabolic equation [Formula: see text] where Ω be a bounded, connected open subset of ℝN with a Lipschitz continuous boundary and f is a locally Lipschitz continuous function: ℝ→ℝ. If u is a bounded solution de (1.1) for which the ω-limit set satisfies [Formula: see text] the number of connected components of {x∈Ω; φ(x)≠=0}) is equal to a constant n∞ on ω(u) and there exists T>0 such that for all t≥T, the set {x∈Ω; u(t,x)≠0} has a finite number of connected components precisely equal to n∞.

Sublinear functions chich generate the group of normal forms of piecewise smooth morse Functions*

For Piecewise Smooth Morse functions Defined on an Open subset of Rn the normal forms around a nondegenerate critical point have been determined in [2] and [7]. The non- smooth part of these normal forms is given by a max-min combination of linear functions with integer coefficients. The additive group generated by these functions is a combina- torial Picard group (see [4]) corresponding to a certain fan in Rn In this paper we show that the generators of this group are the support functions of the face complex of a polytope and give an explicit representation for the two-dimensional case.

On an analogue of the runge theorem for harmonic differential forms

A variant of the classical theorem of Runge is established for harmonic differenrial forms on an open subset of ℝn. It generalizes the case of analytic functions for n=2. Harmonic forms with point singularities are introduced, and a theorem of displacement of poles is proved. An integral representation analogous to the Cauchy formula is constructed. Bibliography: 5 titles.

The p-harmonic system with measure-valued right hand side

For 2-12<p<n we prove existence of a distributional solution u of the p-harmonic system -div(|∇u|p-2∇u)=μinΩu=0on∂Ω, where Ω is an open subset of ℝn (bounded or unbounded), u : Ω → ℝm, and μ is an ℝm-valued Radon measure of finite mass. For the solution u we establish the Lorentz space estimate ‖Du‖Lq,∞+‖u‖Lq*,∞≤C‖μ‖M1p-1with q=nn-1(p-1) and q*=nn-p(p-1). The main step in the proof is to show that for suitable approximations the gradients Duk converge a.e. This is achieved by a choice of regularized test functions and a localization argument to compensate for the fact that in general u ∉ W1,p.

Existence and regularity results for relaxed Dirichlet problems with measure data

We study the following relaxed Dirichlet problem $$\left\{ \begin{gathered} Lu + \mu u = vin\Omega , \hfill \\ u = 0on\partial \Omega , \hfill \\ \end{gathered} \right.$$ where Ω is a bounded open subset ofRN,Lu=−div(A∇u) is an elliptic operator, μ is a positive Borel measure on Ω not charging polar sets, and v is a measure with bounded variation on Ω. We give a definition of solution for such a problem, and then prove existence and regularity results. As a consequence, the Green function for relaxed Dirichlet problems can be defined, and some of its properties are proved, including the standard representation formula for solutions.

Submanifolds of constant sectional curvature in Pseudo-Riemannian manifolds

The generalized equation and the intrinsic generalized equation are considered. The solutions of the first one are shown to correspond to Riemannian submanifolds Mn(K) of constant sectional curvature of psedo-Riemannian manifolds % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGnbaaamaaDaaaleaaieGacaWFZbaabaGaaGOmaiaad6gacqGHsisl% caaIXaaaaOGaaiikamaanaaabaGaam4saaaacaGGPaaaaa!3D97!\[\overline M _s^{2n - 1} (\overline K )\] of index s, with % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabgc% Mi5oaanaaabaGaam4saaaaaaa!3965!\[K \ne \overline K \], flat normal bundle and such that the normal principal curvatures are different from % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabgk% HiTmaanaaabaGaam4saaaaaaa!388B!\[K - \overline K \]. The solutions of the intrinsic generalized equation correspond to Riemannian metrics defined on open subsets of Rn which have constant sectional curvature. The relation between solutions of those equations is given. Moreover, it is proven that the submanifolds M under consideration are determined, up to a rigid motion of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGnbaaaaaa!36D0!\[\overline M \], by their first fundamental forms, as solutions of the intrinsic generalized equation. The geometric properties of the submanifolds M associated to the solutions of the intrinsic generalized equation, which are invariant under an (n − 1)-dimensional group of translations, are given. Among other results, it is shown that such submanifolds are foliated by (n − 1)-dimensional flat submanifolds which have constant mean curvature in M. Moreover, each leaf of the foliation is itself foliated by curves of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGnbaaaaaa!36D0!\[\overline M \] which have constant curvatures.

Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data

We consider the differential problem(*){A(u)=μinΩ,u=0on∂Ω,where Ω is a bounded, open subset of RN, N ≥ 2, A is a monotone operator acting on W01,p(Ω), p > 1, and μ is a Radon measure on Ω that does not charge the sets of zero p-capacity. We prove a decomposition theorem for these measures (more precisely, as the sum of a function in L1(Ω) and of a measure in W−1,p′(Ω)), and an existence and uniqueness result for the so-called entropy solutions of (*).

Behavior at infinity of solutions of second-order nonlinear equations of a particular class

Let Ω be an arbitrary, possibly unbounded, open subset of ℝn, and letL be an elliptic operator of the form $$L = \sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial x_i }}\left( {a_{ij} (x)\frac{\partial }{{\partial x_j }}} \right)} $$ . The behavior at infinity of the solutions of the equationLu=ƒ(¦u¦)signu in Ω is studied, whereƒ is a measurable function. In particular, given certain conditions at infinity, the uniqueness theorem for the solution of the first boundary value problem is proved.

Critical points for multiple integrals of the calculus of variations

In this paper we deal with the existence of critical points of functional defined on the Sobolev space W01,p(Ω), p>1, by $$J(u) = \int\limits_\Omega {\vartheta (x,u,Du)dx,} {\text{ }}$$ where Ω is a bounded, open subset of ℝN. Even for very simple examples in ℝN the differentiability of J(u) can fail. To overcome this difficulty we prove a suitable version of the Ambrosetti-Rabinowitz Mountain Pass Theorem applicable to functionals which are not differentiable in all directions. Existence and multiplicity of nonnegative critical points are also studied through the use of this theorem.