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.     

Approximation numbers of Sobolev imbeddings over unbounded domains

Let Ω ? $\text{R}$^N be an open set with $\text{dist}\text{(x, ?Ω) = O(¦ x ¦}{}^{\mathrm{?l}}\text{)}\text{for}\text{x ? Ω}$ and some l > 0 satisfying an additional regularity condition. We give asymptotic estimates for the approximation numbers α_n of Sobolev imbeddings

over these quasibounded domains Ω. Here

denotes the Sobolev space obtained by completing $\text{C}{}_0{}^{\mathrm{staggered\infty }}\text{(Ω)}$ under the usual Sobolev norm. We prove $\text{α}{}_{\mathrm{n}}\text{(I}{}_{\mathrm{p,q}}{}^{\mathrm{m}}\text{)}\text{\$}\text{?}\text{n}{}^{\mathrm{?\gamma }}$, where

. There are quasibounded domains of this type where γ is the exact order of decay, in the case p ? q under the additional assumption that either 1 ? p ? q ? 2 or 2 ? p ? q ? ∞. This generalizes the known results for bounded domains which correspond to l = ∞. Similar results are indicated for the Kolmogorov and Gelfand numbers of I_p,q^m. As an application we give the rate of growth of the eigenvalues of certain elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω is a quasibounded domain of the above type.     

Clifford analysis and the Navier-Stokes equations over unbounded domains

We consider parabolic Dirac operators which do not involve fractional derivatives and use them to show the solvability of the in-stationary Navier-Stokes equations over time-varying domains. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Growth properties of Sobolev space functions over unbounded domains

Summary We study Sobolev space functions with prescribed growth properties on large spheres. In particular, we prove a weighted Poincaré type inequality for such functions. An extension to weighted Sobolev spaces is sketched.

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Sunto Si studiano funzioni che appartengono a certi spazi di Sobolev e che hanno crescita assegnata su sfere di grande raggio. Per tali funzioni si dimostra una disuguaglianza di tipo Poincaré con peso. Si indica inoltre anche una estensione di tale risultato a funzioni in spazi di Sobolev con peso.

     

Elliptic boundary value problems of fluid dynamics over unbounded domains

In this paper we develop a Clifford operator calculus over unbounded domains whose complement contains a non‐empty open set by using add‐on terms to the Cauchy kernel. Using the knowledge about the Poisson equation allows us to prove a direct decomposition of the space , which will be applied to solve the linear Stokes problem in scales of ‐spaces over this kind of unbounded domains. This result will be used to investigate the Navier–Stokes equations by means of a Banach contraction principle. In the end, steady solutions of stream problems with free convection will be studied. Copyright © 2000 John Wiley & Sons, Ltd.     

Minimal graphs

Elementary properties of harmonic maps between Riemannian manifolds are interpreted via their graphs, viewed as nonparametric minimal submanifolds (Proposition 1). Then examples, are given of nonparametric submanifolds of compact Riemannian manifolds which cannot be deformed-through nonparametric submanifolds-to nonparametric minimal submanifolds (Propositions 2 and 4).     

Semilinear problems in unbounded domains

As formulated by Silva [E.A. de B.e. Silva, Linking theorems and applications to semilinear elliptic problems at resonance, Nonlinear Anal. 16 (1991) 455-477] and Schechter [M. Schechter, A generalization of the saddle point method with applications, Ann. Polon. Math. 57 (3) (1992) 269-281; M. Schechter, New saddle point theorems, in: Generalized Functions and Their Applications, Varanasi, 1991, Plenum, New York, 1993, pp. 213-219], the sandwich theorem has become a very useful tool in finding critical points of functionals leading to solutions of partial differential equations. In the present paper, this theorem is strengthened to apply to more general situations. We present some applications.     

A note on unbounded metric temporal logic over dense time domains

We investigate the consequences of removing the infinitary axiom and rules from a previously defined proof system for a fragment of propositional metric temporal logic over dense time (see [1]). (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Minimal functions with unbounded integral

We prove theorems which imply the following results. (1) “Most” almost periodic functionsb(t) with unbounded integral oscillate in a strong sense. (2) IfB is a continuous function on a minimal flow (Ω,R), then either the time averages all converge, or they diverge on a residual set.     

Minimal non-neighborhood-perfect graphs

Neighborhood-perfect graphs form a subclass of the perfect graphs if the Strong Perfect Graph Conjecture of C. Berge is true. However, they are still not shown to be perfect. Here we propose the characterization of neighborhood-perfect graphs by studying minimal non-neighborhood-perfect graphs (MNNPG). After presenting some properties of MNNPGs, we show that the only MNNPGs with neighborhood independence number one are the 3-sun and 3K₂. Also two further classes of neighborhood-perfect graphs are presented: line-graphs of bipartite graphs and a 3K₂-free cographs. © 1996 John Wiley & Sons, Inc.     

Minimal claw-free graphs

A graph G is a minimal claw-free graph (m.c.f. graph) if it contains no K _1,3 (claw) as an induced subgraph and if, for each edge e of G, G ™ e contains an induced claw. We investigate properties of m.c.f. graphs, establish sharp bounds on their orders and the degrees of their vertices, and characterize graphs which have m.c.f. line graphs. Support by the South African National Research Foundation is gratefully acknowledged.     

Clifford analysis and the Navier‐Stokes equations over unbounded domains

We describe a finite difference scheme based on a discrete function theory for the Navier‐Stokes equations over unbounded domains which has only low regularity demands on the solution for convergence. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two-dimensional Navier-Stokes flow in unbounded domains

Dedicated to Professor Hiroki Tanabe on the occasion of his sixtieth birthday     

Positive integral operators in unbounded domains

We study positive integral operators in with continuous kernel k(x,y). We show that if the operator is compact and Hilbert-Schmidt. If in addition k(x,x)→0 as |x|→∞, k is represented by an absolutely and uniformly convergent bilinear series of uniformly continuous eigenfunctions and is trace class. Replacing the first assumption by the stronger then and the bilinear series converges also in L¹. Sharp norm bounds are obtained and Mercer's theorem is derived as a special case.     

The dam problem in unbounded domains

Summary The time dependent dam problem in a general unbounded domain is studied and an existence theorem is proved by means of a rather general approximation procedure. Moreover the propagation of the support of the solution is studied in a number of geometrical situations of interest. The corresponding stationary problem is also studied and an existence theorem is proved. Finally several one-dimensional examples are given in which the solution can be either written explicitely or found solving an ordinary differential equation.This work has been supported by MURST (Italy), by I.A.N.-C.N.R. of Pavia (Italy), and by SFB 123 of University of Heidelberg (Germany).     

Minimal 2-matching-covered graphs

A perfect 2-matching M of a graph G is a spanning subgraph of G such that each component of M is either an edge or a cycle. A graph G is said to be 2-matching-covered if every edge of G lies in some perfect 2-matching of G. A 2-matching-covered graph is equivalent to a “regularizable” graph, which was introduced and studied by Berge. A Tutte-type characterization for 2-matching-covered graph was given by Berge. A 2-matching-covered graph is minimal if G−e is not 2-matching-covered for all edges e of G. We use Berge’s theorem to prove that the minimum degree of a minimal 2-matching-covered graph other than K₂ and K₄ is 2 and to prove that a minimal 2-matching-covered graph other than K₄ cannot contain a complete subgraph with at least 4 vertices.     

Minimal configuration bicyclic graphs

The nullity η(G) of a graph G is the multiplicity of zero as an eigenvalue of the adjacency matrix of G. If η(G)?=?1, then the core of G is the subgraph induced by the vertices associated with the nonzero entries of the kernel eigenvector. The set of vertices which are not in the core is the periphery of G. A graph G with nullity one is minimal configuration if no two vertices in the periphery are adjacent and deletion of any vertex in the periphery increases the nullity. An ∞-graph ∞(p,?l,?q) is a graph obtained by joining two vertex-disjoint cycles C _p and C _q by a path of length l?≥?0. Let ?* be the class of bicyclic graphs with an ∞-graph as an induced subgraph. In this article, we characterize the graphs in ?* which are minimal configurations.     

Higher-order parabolic variational inequality in unbounded domains

We prove the existence and uniqueness of a solution of a nonlinear parabolic variational inequality in an unbounded domain without conditions at infinity. In particular, the initial data may infinitely increase at infinity, and a solution of the inequality is unique without any restrictions on its behavior at infinity. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 7, pp. 949–968, July, 2008.     

Optimal global regularity for minimal graphs over convex domains in hyperbolic space

We use the concept of the inside-(a, η, h) domain to construct a subsolution to the Dirichlet problem for minimal graphs over convex domains in hyperbolic space. As an application, we prove that the Hölder exponent \max \{1/a,1/(n+1)\} for the problem is optimal for any a\in [2,+\infty ].