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We consider the existence of solutions of variational inequality form. Find u∈D(J):〈A(u),v-u〉+〈F(u),v-u〉+J(v)-J(u)≥0, ∀v∈W1LM(Ω), whose principal part is having a growth not necessarily of polynomial type, where A is a second-order elliptic operator of Leray-Lions type, F is a multivalued lower order term, and J is a convex functional. We use subsupersolution methods to study the existence and enclosure of solutions in Orlicz-Sobolev spaces. [ABSTRACT FROM AUTHOR]

This paper is concerned with the functional J defined by J(u) = ∫ΩxΩ W(x, y, ∇u(x), ∇u(y))dx dy, where Ω ⊂ ℝN is a regular open bounded set and W is a real-valued function with variable growth. After discussing the theory of Young measures in variable exponent Sobolev spaces, we study the weak lower semicontinuity and relaxation of J. [ABSTRACT FROM AUTHOR]

We study the regularity of the solutions of the surface quasi-geostrophic equation with subcritical exponent 1/2 < α ≤ 1. We prove that if the initial data is small enough in the critical space Ḣ2-2α(ℝ²), then the regularity of the solution is of exponential growth type with respect to time and its Ḣ2-2α(ℝ²) norm decays exponentially fast. It becomes then infinitely differentiable with respect to time and has value in all homogeneous Sobolev spaces Ḣs(ℝ²) for s ≥ 2 - 2α. Moreover, we give some general properties of the global solutions. [ABSTRACT FROM AUTHOR]

The oscillatory hyper-Hilbert transform along curves is of the following form: Hn,α,βf(x) = ∫ 0¹f(x - ⌈(t))e it-β t-1-α dt, where α ≥ 0, β ≥ 0, and ⌈(t) = {tP1, tP2,..., tPn). The study on this operator is motivated by the hyper-Hilbert transform and the strongly singular integráis. The LP bounds for Hn,α,β have been given by Chen et al. (2008 and 2010). In this paper, for some α, β, and p, the boundedness of Hn, α, β on Sobolev spaces LsP(ℝn) and the boundedness of this operator from Ls²(ℝn) to L²(ℝn) are obtained. [ABSTRACT FROM AUTHOR]

We continue our research on Sobolev spaces on locally compact abelian (LCA) groups motivated by our work on equations with infinitely many derivatives of interest for string theory and cosmology. In this paper, we focus on compact embedding results and we prove an analog for LCA groups of the classical Rellich lemma and of the Rellich-Kondrachov compactness theorem. Furthermore, we introduce Sobolev spaces on subsets of LCA groups and study its main properties, including the existence of compact embeddings into Lℙ-spaces. [ABSTRACT FROM AUTHOR]

This paper gives a characterization of Sobolev functions on the real line by means of pointwise inequalities involving finite differences. This is also shown to apply to more general Orlicz-Sobolev, Lorentz-Sobolev, and Lorentz-Karamata-Sobolev spaces. [ABSTRACT FROM AUTHOR]

Spaces with variable exponents Lp(x) H, μ) and Lp(x) (H, μ H) are introduced. After discussing some approximation results of Lp(x) (H, μ), Sobolev spaces on H with variable exponents are introduced. At last, we define Malliavin derivatives in Lp(x) (H, μ) and discuss some properties of Malliavin derivatives in Lp(x)(H, μ). [ABSTRACT FROM AUTHOR]

We introduce a new space QK(∂ ⅅ) of Lebesgue measurable functions on the unit circle connecting closely with the Sobolev space. We obtain a necessary and sufficient condition on K such that QK(∂ ⅅ) = BMO(∂ ⅅ), as well as a general criterion on weight functions K1 and K2, K1 ≤ K2, such that QK (∂ ⅅ) ... QK (∂ ⅅ). We also prove that a measurable function belongs to QK(∂ ⅅ) if and only if it is Mobius bounded in the Sobolev space L²K(∂ ⅅ). Finally, we obtain a dyadic characterization of functions in QK spaces in terms of dyadic arcs on the unit circle. [ABSTRACT FROM AUTHOR]

We introduce a class of Fourier multiplier operators Mb on n-complex unit sphere, where the symbol b ∈ Hs(Sω).We obtained the Sobolev boundedness of Mb. Our result implies that the operators Mb take a role of fractional diferential operators on ∂픹 [ABSTRACT FROM AUTHOR]

We prove optimal embeddings of the generalized Sobolev spaces Wk E, where E is a rearrangement invariant function space, into the generalized Hölder-Zygmund space CH generated by a function space H. [ABSTRACT FROM AUTHOR]

This paper is concerned with the fractional quasigeostrophic equation with modified dissipativity. We prove the local existence of solutions in Sobolev spaces for the general initial data and the global existence for the small initial data when 1/2 ≤ ɑ < 1. [ABSTRACT FROM AUTHOR]

The authors study the mapping properties of Fourier multipliers, with symbols satisfying some generalized Hörmander's condition, on Triebel- Lizorkin-type spaces and Triebel-Lizorkin-Hausdorff spaces. To this end, the authors first establish a new characterization of these spaces via some generalized (weighted) g*λ functions, which essentially improves the known result for Triebel-Lizorkin spaces even when t = 0. Applying this new characterization, the authors then obtain the boundedness of Fourier multipliers on Triebel-Lizorkin-type spaces and Triebel-Lizorkin-Hausdorff spaces, which also give a new proof of the Sobolev embedding theorems for these spaces. [ABSTRACT FROM AUTHOR]

The pseudoparabolic regularization technique is employed to study the local well-posedness of strong solutions for a nonlinear dispersive model, which includes the famous Camassa-Holm equation. The local well-posedness is established in the Sobolev space Hs(R) with s > 3/2 via a limiting procedure. [ABSTRACT FROM AUTHOR]

A high-order iterative scheme is established in order to get a convergent sequence at a rate of order N (N ≥ 1) to a local unique weak solution of a nonlinear Kirchhoff wave equation in the unit membrane. This extends a recent result in (EJDE, 2005, No. 138) where a recurrent sequence converges at a rate of order 2. [ABSTRACT FROM AUTHOR]

We study in this paper a class of second-order linear elliptic equations in weighted Sobolev spaces on unbounded domains of ℝn, n ≥ 3. We obtain an a priori bound, and a regularity result from which we deduce a uniqueness theorem. [ABSTRACT FROM AUTHOR]

We consider the elliptic problem -Δu + u = b(x)∣u∣p-2u + h(x) in Ω, u ε H0¹ (Ω), where 2 < p < (2N/(N - 2)) (N ≥ 3), 2 < p < ∞ (N = 2), Ω is a smooth unbounded domain in ℝN, b(x) ε C(Ω), and h(x) ε H-1(Ω). We use the shape of domain Ω to prove that the above elliptic problem has a ground-state solution if the coefficient b(x) satisfies b(x) → b ∞ > 0 as ∣x∣ → ∞and b(x) ≥ c for some suitable constants c ε (0,b∞), and h(x) ≡ 0. Furthermore, we prove that the above elliptic problem has multiple positive solutions if the coefficient b(x) also satisfies the above conditions, h(x) ≥ 0 and 0 < ∥h∥H-1 < (p-2)(1/(p - 1))(p-1)/(p-2)[bsup Sp(Ω]1/(2-p), where S(Ω) is the best Sobolev constant of subcritical operator in H0¹ (Ω) and bsup = supxεΩb(x). [ABSTRACT FROM AUTHOR]