Operator properties of Sobolev imbeddings over unbounded domains

For an open set Ω ? $\text{R}$^N, 1 ? p ? ∞ and λ ∈ $\text{R}$⁺, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators $\text{U}$, 1 ? p, q ? ∞ and a quasibounded domain Ω ? $\text{R}$^N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the given Banach ideal $\text{U}$: Assume the quasibounded domain fulfills condition C_k^l for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any $\text{x ? Ω}$ to the boundary ?Ω tends to zero as $\text{O(¦ x ¦}{}^{\mathrm{?l}}\text{)}$ for $\text{¦ x ¦ → ∞}$, and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ $\text{N}$, μ > λ _S($\text{U}$; p,q:N) and v > N/l · λ_D($\text{U}$;p,q), one has that $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ belongs to the Banach ideal $\text{U}$. Here λ_D($\text{U}$;p,q;N)∈$\text{R}$⁺ and λ_S($\text{U}$;p,q;N)∈$\text{R}$⁺ are the D-limit order and S-limit order of the ideal $\text{U}$, introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings l_pⁿ → l_qⁿ for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω fulfills condition C₁^l.For an open set Ω in $\text{R}$^N, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in $\text{R}$^N and give sufficient conditions on λ such that the Sobolev imbedding operator $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω is a quasibounded open set in $\text{R}$^N.     

Extreme value distribution for normalized sums from stationary Gaussian sequences

Let {X_i, i?0} be a sequence of independent identically distributed random variables with finite absolute third moment. Then Darling and Erdös have shown that

for -∞<t<∞ where and $\text{X}{}_{\mathrm{n}}\text{= (2 ln ln n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. The result is extended to dependent sequences but assuming that {X_i} is a standard stationary Gaussian sequence with covariance function {r_i}. When {X_i} is moderately dependent (e.g. when $\text{v(∑}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{X}{}_{\mathrm{i}}\text{) ? n}{}_{\mathrm{a}}\text{, 0 < a < 2)}$ we get

where H_a is a constant. In the strongly dependent case (e.g. when $\text{v(∑}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{X}{}_{\mathrm{i}}\text{) ? n}{}^2\text{r(n))}$ we get

for-∞<t<∞.     

The completion of a B-convex normed Riesz space is reflexive

If

is a B-convex normed Riesz space, then the topological completion of

is a closed subspace of

⁷, the second Banach dual of

. If $\text{N=}\textcolor{red}{}{}^7$ or $\text{N=}\textcolor{red}{}{}^{\mathrm{7x}}$, then $\text{N}$ is a B-convex σ-Dedekind complete normed Riesz space which is the Banach dual of a normed Riesz space. In such a $\text{N}$, if u₁ ? u₂ ? … ? 0 and inf_n{u_n} = 0, then lim_n∥u_n∥ = 0. This is the key step in verifying that Ogasawara's criteria that a normed Riesz space be reflexive are satisfied by

⁷. Thus the topological completion of

as a closed subspace of

⁷ is also reflexive.     

The minimal number of basic elements in a multiset antichain

Let M be a finite set consisting of k_i elements of type i, i = 1, 2,…, n and let S denote the set of subsets of M or, equivalently, the set of all vectors x = (x₁, x₂,…,x_n) with integral coefficients x_i satisfying 0 ? x_i ? k_i, i = 1, 2,…, n. An antichain

is a subset of S in which there is no pair of distinct vectors x and y such that x is contained in y (that is, there is no pair of distinct vectors x and y such that the inequalities x_i ? y_i, i = 1, 2,…, n all hold). Let $\text{∥Y}\textcolor{red}{}\text{∥}$ denote the number of vectors in S which are contained in at least one vector in

and let $\text{∥B}\textcolor{red}{}\text{∥=∑}{}_{\mathrm{x\in }}\text{(X}{}_1\text{+X}{}_2\text{+?+X}{}_{\mathrm{n}}\text{)}$, the number of basic elements in

. For given m we give procedures for calculating min $\text{∥Y}\textcolor{red}{}\text{∥}$ and min $\text{∥B}\textcolor{red}{}\text{∥}$, where the minima are taken over all m-element antichains

in S.     

Generic periodic solutions of functional differential equations

Consider the class of retarded functional differential equations

$\text{x(t) = f(x}{}_{\mathrm{t}}\text{)}$

, (1) where x_t(θ) = x(t + θ), ?1 ? θ ? 0, so x_t?C = C([?1, 0], Rⁿ), and $\text{f∈}\textcolor{red}{}\text{=C}{}^{\mathrm{r}}\text{(C,R}{}^{\mathrm{n}}\text{)}$. Let 2 ? r ? ∞ and give $\text{X}$ the appropriate (Whitney) topology. Then the set of $\text{f∈}\textcolor{red}{}$ such that all fixed points and all periodic solutions of (¹) are hyperbolic is residual in

.     

On the nonexistence of global solutions to a nonlinear Euler-Poisson-Darboux equation

In this paper, we show that the initial boundary value problem for the (singular) nonlinear EPD (Euler-Poisson-Darboux) equation

does not possess global solutions for arbitrary choices of $\text{u(x, 0). (x ? Ω ? R}{}^{\mathrm{n}}\text{, Ω}\text{bounded}\text{, Δ}{}_{\mathrm{n}}\text{= n}\text{dimensional Laplacian}$) when 0 < k ? 1 for a wide class of nonlinearities $\text{T}$, which includes all the even powers of u and the functions u^{2n + 1}, n = 1, 2,…. The solutions are assumed to vanish on the “walls” of the spacetime cylinder and satisfy $\text{?u}\text{?t(x, 0)}\text{= 0, x ? Ω}$. The result is independent of the space dimension.     

Minimax estimation of a multivariate normal mean under polynomial loss

Let X be an observation from a p-variate (p ≥ 3) normal random vector with unknown mean vector θ and known covariance matrix

. The problem of improving upon the usual estimator of θ, δ⁰(X) = X, is considered. An approach is developed which can lead to improved estimators, δ, for loss functions which are polynomials in the coordinates of (δ ? θ). As an example of this approach, the loss L(δ, θ) = |δ ? θ|⁴ is considered, and estimators are developed which are significantly better than δ⁰. When

is the identity matrix, these estimators are of the form $\text{δ(X) = (1 ? (}\text{b}\text{(d + |X|}{}^2\text{)}\text{))X}$.     

Cohérence des faisceaux d'idéaux multiplicateurs avec estimations

Let $\text{Ω?}\text{C}{}^{\mathrm{n}}$ be a bounded pseudoconvex open set and let ? be a plurisubharmonic function on $\text{Ω.}$ For every positive integer m, we consider the multiplier ideal sheaf $\text{I}\text{(m?)}$ and the Hilbert space $\text{H}{}_{\mathrm{\Omega }}\text{(m?)}$ of holomorphic functions f on $\text{Ω}$ such that $\text{|f|}{}^2\text{e}{}^{\mathrm{?2m?}}$ is integrable on $\text{Ω.}$ We give an effective version, with estimates, of Nadel's result stating that the sheaf $\text{I}\text{(m?)}$ is coherent and generated by an arbitrary orthonormal basis of $\text{H}{}_{\mathrm{\Omega }}\text{(m?).}$ This result is expected to play a major part in the context of current regularizations with estimates of the Monge–Ampère masses. To cite this article: D. Popovici, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A variant of Poincaré's inequality

We show that if $\text{Ω?}\text{R}{}^{\mathrm{N}}\text{,}\text{N?2}$, is a bounded Lipschitz domain and $\text{(ρ}{}_{\mathrm{n}}\text{)?L}{}^1\text{(}\text{R}{}^{\mathrm{N}}\text{)}$ is a sequence of nonnegative radial functions weakly converging to δ₀ then there exist C>0 and n₀?1 such that

$\text{∫}\text{Ω}\text{f??}\text{∫}\text{Ω}\text{f}{}^{\mathrm{p}}\text{?C}\text{∫}\text{Ω}\text{∫}\text{Ω}\text{|f(x)?f(y)|}{}^{\mathrm{p}}\text{|x?y|}{}^{\mathrm{p}}\text{ρ}{}_{\mathrm{n}}\text{(|x?y|)}\text{d}\text{x}\text{d}\text{y}\text{?f∈L}{}^{\mathrm{p}}\text{(Ω)}\text{?n?n}{}_0\text{.}$

The above estimate was suggested by some recent work of Bourgain, Brezis and Mironescu (in: Optimal Control and Partial Differential Equations, IOS Press, 2001, pp. 439–455). As n→∞ in (1) we recover Poincaré's inequality. We also extend a compactness result of Bourgain, Brezis and Mironescu. To cite this article: A.C. Ponce, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Continuous dynamic programming approach to inequalities

We present in this paper some results concerning the following nonlinear system of P.D.E.

where $\text{z′ = z}{}_{\mathrm{t}}\text{, D}{}_{\mathrm{z}}\text{= z}{}_{\mathrm{x}}\text{and}\text{a(u) = ∝}{}_0{}^1\text{¦Du¦}{}^2\text{dx}$.The above system is a mathematical model which describes coupled flexural and torsional oscillations of an open cross-section beam. In Part I we consider the abstract initial value problem associated with the above system, prove the existence and uniqueness of solutions in a weak sense and mention two applications. In Part II we obtain regular solutions when adequate conditions on the data are assumed.     

Homogenization in open sets with holes

Let Q^r be a cylindrical bar with r cylindrical cavities having generators parallel to those of Q^r. Let Ω be the cross-section of the bar, Ω1 the cross-section of the domain occupied by the material and $\text{Ω}{}^{\mathrm{i}}\text{(i = 1,…, r)}$ the cross- sections of the cavities:

$\text{Ω}\text{?}{}^{\mathrm{i}}\text{? Ω}\text{Ω}\text{?}{}^{\mathrm{i}}\text{∩}\text{Ω}\text{?}{}^{\mathrm{k}}\text{= φ, i ≠ k}$

. The study of the elastic torsion of this bar leads to the following problem [see 2., 3., 267–320)]:

$\text{Δ?}{}_{\mathrm{r}}\text{+ 2μα = 0 in Ω1}$

$\text{?}{}_{\mathrm{r¦?\Omega }}\text{= 0}$

(1)

$\text{?}{}_{\mathrm{r}}\text{=}\text{constant on}\text{?Ω}{}^{\mathrm{i}}\text{; i = 1,…, r}$

where μ is the shear modulus of the material, α is the angle of twist and $\text{?}{}_{\mathrm{r}}$ represents the stress function. In this paper the problem (1) with an increasing number of holes which are distributed periodically is considered. One would like to know if $\text{?}{}_{\mathrm{r}}$ has a limit $\text{?}{}_{\mathrm{\infty }}\text{as}\text{r → + ∞}$, and if so, the equation satisfied by this limit. This is an “homogenization” problem — the heterogeneous bar Q^r is replaced by a homogeneous one, the response of which under torsion approximates as closely as possible that of Q^r. A more general problem will be studied and the case of elastic torsion will be obtained as an application. The proof uses the energy method [see Lions (Collège de France, 1975–1977), Tartar (Collège de France, 1977)] and extension theorems. A related problem is the homogenization of a perforated plate [cf. Duvaut (to appear)].