Sobolev embeddings involving symmetry

Given a bounded regular domain with cylindrical symmetry, then functions having such symmetry and belonging to the first Sobolev space can be embedded compactly into some weighted Lp spaces, with p superior to the critical Sobolev exponent. A simple application to elliptic boundary value problem is also considered.     

Boundary trace embedding theorems for variable exponent Sobolev spaces

We study boundary trace embedding theorems for variable exponent Sobolev space W1,p(⋅)(Ω). Let Ω be an open (bounded or unbounded) domain in RN satisfying strong local Lipschitz condition. Under the hypotheses that p∈L^∞(Ω), 1?infp(x)?supp(x)<N, |∇p|∈Lγ(⋅)(Ω), where γ∈L^∞(Ω) and infγ(x)>N, we prove that there is a continuous boundary trace embedding W1,p(⋅)(Ω)→Lq(⋅)(∂Ω) provided q(⋅), a measurable function on ∂Ω, satisfies condition for x∈∂Ω.     

Sobolev embedding theorems for a class of anisotropic irregular domains

Sufficient conditions for the embedding of a Sobolev space in Lebesgue spaces on a domain depend on the integrability and smoothness parameters of the spaces and on the geometric features of the domain. In the present paper, Sobolev embedding theorems are obtained for a class of domains with irregular boundary; this class includes the well-known classes of σ-John domains, domains with the flexible cone condition, and their anisotropic analogs.     

A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems

We make a contribution to the theory of embeddings of anisotropic Sobolev spaces into L ^p -spaces (Sobolev case) and spaces of H?lder continuous functions (Morrey case). In the case of bounded domains the generalized embedding theorems published so far pose quite restrictive conditions on the domain’s geometry (in fact, the domain must be “almost rectangular”). Motivated by the study of some evolutionary PDEs, we introduce the so-called “semirectangular setting”, where the geometry of the domain is compatible with the vector of integrability exponents of the various partial derivatives, and show that the validity of the embedding theorems can be extended to this case. Second, we discuss the a priori integrability requirement of the Sobolev anisotropic embedding theorem and show that under a purely algebraic condition on the vector of exponents, this requirement can be weakened. Lastly, we present a counterexample showing that for domains with general shapes the embeddings indeed do not hold.     

A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems

We make a contribution to the theory of embeddings of anisotropic Sobolev spaces into L ^p -spaces (Sobolev case) and spaces of Hölder continuous functions (Morrey case). In the case of bounded domains the generalized embedding theorems published so far pose quite restrictive conditions on the domain’s geometry (in fact, the domain must be “almost rectangular”). Motivated by the study of some evolutionary PDEs, we introduce the so-called “semirectangular setting”, where the geometry of the domain is compatible with the vector of integrability exponents of the various partial derivatives, and show that the validity of the embedding theorems can be extended to this case. Second, we discuss the a priori integrability requirement of the Sobolev anisotropic embedding theorem and show that under a purely algebraic condition on the vector of exponents, this requirement can be weakened. Lastly, we present a counterexample showing that for domains with general shapes the embeddings indeed do not hold.     

Compact embedding results of Sobolev spaces and existence of positive solutions to quasilinear equations

In this paper, we study mainly the existence of multiple positive solutions for a quasilinear elliptic equation of the following form on ${\mathbf{R}}^N$, when $N\ge 2$,

(0.1)$?{\mathrm{\Delta }}_{N}u+V(x){\left|u\right|}^{N?2}u=\lambda {\left|u\right|}^{r?2}u+f(x,u).$

Here, $V(x)>0:{\mathbf{R}}^N\to \mathbf{R}$ is a suitable potential function, $r\in (1,N)$, $f(x,u)$ is a continuous function of N-superlinear and subcritical exponential growth without having the Ambrosetti–Rabinowitz condition, while $\lambda >0$ is a constant. A suitable Moser–Trudinger inequality and the compact embedding $W_{\mathrm{V}}^{1,N}\left({\mathbf{R}}^N\right)?L^r\left({\mathbf{R}}^N\right)$ are proved to study problem (0.1). Moreover, the compact embedding $H_{\mathrm{V}}^1\left({\mathbf{R}}^N\right)?L_{\mathrm{K}}^t\left({\mathbf{R}}^N\right)$ is also analyzed to investigate the existence of a positive ground state to the following nonlinear Schrödinger equation

(0.2)$?\mathrm{\Delta }u+V(x)u=K(x)g(u)$

with potentials vanishing at infinity in a measure-theoretic sense when $N\ge 3$.     

Liouville–type theorems for semilinear elliptic equations involving the Sobolev exponent

In this article, we prove there are no positive smooth solutions of where satisfies one of the following conditions: (i) K is a subharmonic function in with , and the derivative satisfies where for , for and for . (ii) is nondecreasing along each ray for any unit vector in and . (iii) is nondecreasing in , for and for , where and b are constants. Various generalizations to a more general class of nonlinearities are also considered. Received 7 May 1996; in final form 6 January 1997     

Certain embedding theorems

We obtain necessary and sufficient conditions such that, for f(x) from LP(0, 1), the integral ₀ ¹ ¦f (x)¦qdx (0<p<1,p<q<p(1 –p)^–1) is convergent, or for f LP[0, 1] for all p 1, the integral ₀ ¹ e¦f(x)¦dx is convergent.Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 187–200, February, 1976.     

Sobolev imbedding theorems in borderline cases

An imbedding theorem is given for functions whose gradient belongs to a class slightly larger than , .

     

Equivariant embedding theorems and topological index maps

The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a factorisation. These factorisations are models for the topological index map. Under some assumptions concerning the existence of equivariant vector bundles, any smooth map admits a normal factorisation, and two such factorisations are unique up to a certain notion of equivalence. To prove this, we generalise the Mostow Embedding Theorem to spaces equipped with proper groupoid actions. We also discuss orientations of normally non-singular maps with respect to a cohomology theory and show that oriented normally non-singular maps induce wrong-way maps on the chosen cohomology theory. For K-oriented normally non-singular maps, we also get a functor to Kasparov's equivariant KK-theory. We interpret this functor as a topological index map.     

Existentially closed structures and some embedding theorems

Mathematical Notes - Using the notion of an existentially closed structure, we obtain embedding theorems for groups and Lie algebras. We also prove the existence of certain groups and Lie algebras...     

Sobolev embedding theorem for anisotropically irregular domains

Doklady Mathematics -     

Dimension and embedding theorems for geometric lattices

Let G be an n-dimensional geometric lattice. Suppose that 1 ? e, f ? n ? 1, e + f ? n, but e and f are not both n ? 1. Then, in general, there are E, F? G with dim E = e, dim F = f, E ? F = 1, and dim E ∧ F = e + f ? n ? 1; any exception can be embedded in an n-dimensional modular geometric lattice M in such a way that joins and dimensions agree in G and M, as do intersections of modular pairs, while each point and line of M is the intersection (in M) of the elements of G containing it.     

Removability theorems for Sobolev functions and quasiconformal maps

We establish several conditions, sufficient for a set to be (quasi)conformally removable, a property important in holomorphic dynamics. This is accomplished by proving removability theorems for Sobolev spaces inR ⁿ . The resulting conditions are close to optimal. The first author is supported by N.S.F. Grant No. DMS-9423746. The second author is supported by N.S.F. Grants No. DMS-9304580 and DMS-9706875.     

Weighted Sobolev embedding with unbounded and decaying radial potentials

We prove embedding results of weighted W1,p(RN) spaces of radially symmetric functions. The results then are used to obtain ground and bound state solutions of quasilinear equations with unbounded or decaying radial potentials.     

Compact Sobolev imbeddings on finite measure spaces

We provide conditions on a finite measure μ on $\text{R}$ⁿ which insure that the imbeddings W^{k, p}($\text{R}$ⁿdμ)?L^p($\text{R}$ⁿdμ) are compact, where 1 ? p < ∞ and k is a positive integer. The conditions involve uniform decay of the measure μ for large ¦x¦ and are satisfied, for example, by $\text{dμ = e}{}^{\mathrm{?¦x¦\alpha }}\text{dx,}\text{where}\text{α > 1}$.