Weighted Sobolev theorem in Lebesgue spaces with variable exponent

For the Riesz potential operator Iα there are proved weighted estimates

     

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.     

Embedding of Sobolev spaces and properties of the domain

We establish the embedding of the Sobolev space W _p ^s (G) ? L _q (G) for an irregular domain G in the case of a limit exponent under new relations between the parameters depending on the geometric properties of the domain G.     

Embedding constants for periodic Sobolev spaces of fractional order

We obtain an explicit expression for the norms of the embedding operators of the periodic Sobolev spaces into the space of continuous functions (the norms of this type are usually called embedding constants). The corresponding formulas for the embedding constants express them in terms of the values of the well-known Epstein zeta function which depends on the smoothness exponent s of the spaces under study and the dimension n of the space of independent variables. We establish that the embeddings under consideration have the embedding functions coinciding up to an additive constant and a scalar factor with the values of the corresponding Epstein zeta function. We find the asymptotics of the embedding constants as s → n/2.     

Characterisation of zero trace functions in variable exponent Sobolev spaces

It is well known that if u belongs to the Sobolev space , where Ω is an open subset of and , then if belongs to weak , where dist . Results of this type are given here for Sobolev spaces with a variable exponent p , under the conditions that Ω is bounded and satisfies a mild regularity condition, and p is a bounded, log‐Hölder continuous function that is bounded away from 1. The outcome includes theorems that are new even when p is constant. In particular it is shown that if and only if and .     

On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent

We consider the nonlinear eigenvalue problem

in , on , where is a bounded open set in with smooth boundary and , are continuous functions on such that , , and for all . The main result of this paper establishes that any sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland's variational principle.

     

A mass transportation approach for Sobolev inequalities in variable exponent spaces

In this paper we provide a proof of the Sobolev–Poincaré inequality for variable exponent spaces by means of mass transportation methods, in the spirit of Cordero-Erausquin et al. (Adv Math 182(2):307–332, 2004). The importance of this approach is that the method is flexible enough to deal with different inequalities. As an application, we also deduce the Sobolev-trace inequality improving the result of Fan (J Math Anal Appl 339(2):1395–1412, 2008) by obtaining an explicit dependence of the exponent in the constant.     

Sobolev spaces on Lie groups: Embedding theorems and algebra properties

Let G be a noncompact connected Lie group, denote with ρ a right Haar measure and choose a family of linearly independent left-invariant vector fields X on G satisfying Hörmander's condition. Let χ be a positive character of G and consider the measure ${\mu }_{\chi }$ whose density with respect to ρ is χ. In this paper, we introduce Sobolev spaces $L_{\alpha }^p({\mu }_{\chi })$ adapted to X and ${\mu }_{\chi }$ ($1<p<\infty$, $\alpha \ge 0$) and study embedding theorems and algebra properties of these spaces. As an application, we prove local well-posedness and regularity results of solutions of some nonlinear heat and Schrödinger equations on the group.     

The embedding theorem for Sobolev spaces with mixed norm for limit exponents

This article is devoted to the investigation of some properties of Sobolev spaces with mixed norm.Translated fromMatematicheskie Zametki, Vol. 59, No. 1, pp. 62–72, January, 1996.     

On the boundary limits of monotone Sobolev functions in variable exponent Orlicz spaces

Our aim in this note is to deal with boundary limits of monotone Sobolev functions with ▽u∈ Lp(·)logLq(·)(B) for the unit ball BRn. Here p(·) and q(·) are variable exponents satisfyingthe log-Hlder and the log log-Hlder conditions, respectively.     

On the Sobolev embedding theorem for variable exponent spaces in the critical range

In this paper we study the Sobolev embedding theorem for variable exponent spaces with critical exponents. We find conditions on the best constant in order to guaranty the existence of extremals. The proof is based on a suitable refinement of the estimates in the Concentration–Compactness Theorem for variable exponents and an adaptation of a convexity argument due to P.L. Lions, F. Pacella and M. Tricarico.     

On the Sobolev trace Theorem for variable exponent spaces in the critical range

In this paper, we study the Sobolev trace Theorem for variable exponent spaces with critical exponents. We find conditions on the best constant in order to guaranty the existence of extremals. Then, we give local conditions on the exponents and on the domain (in the spirit of Adimurthy and Yadava) in order to satisfy such conditions and therefore to ensure the existence of extremals.     

A fluctuations limit for scaled age distributions and weighted Sobolev spaces

It is shown that under the central-limit scaling, the fluctuations of the space—time renormalized age distributions of particles (whose development is controlled by critical linear birth and death processes) around the law-of-large-numbers limit converge in a Hilbert space (containing the class of signed Radon measures with finite moment generating functionals) to a continuous Gaussian process satisfying a Langevin equation. So far, the space of rapidly decreasing functions has been considered to be the natural state space for the kind of limit theorem considered here. However, the space of rapidly decreasing functions is not suitable in the present context and we are led to define an appropriate family of Sobolev spaces. In fact, we construct a scale of Hilbert spaces based on the eigenfunctions expansions of an elliptic operator defined on a weightedL ²-space.This research was partially supported by an NSERC of Canada grant.     

Sobolev embeddings for Riesz potential spaces of variable exponents near 1 and Sobolev's exponent

Our aim in this paper is to deal with Sobolev embeddings for Riesz potentials of order α for functions f satisfying the Orlicz type condition