This paper is concerned with the existence of two non-trivial weak solutions for a p(x)-Kirchho type problem by using the mountain pass theorem of Ambrosetti and Rabinowitz and Ekeland's variational principle and the theory of the variable exponent Sobolev spaces.

We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in t...

We describe a recent, one-parameter family of characterizations Sobolev and BV functions on $\mathbb{R}^n$ n, using sizes superlevel sets suitable difference quotients. This provides an alternative point view to the BBM formula by Bourgain, Brezis, Mironescu, complements in case some results Cohen, Dahmen, Daubechies, DeVore about wavelet coefficients such functions. An application towards Gagl...

We prove a regularity result on polyhedral domains P ⊂ R using the weighted Sobolev spaces Ka (P). In particular, we show that there is no loss of Ka –regularity for solutions of strongly elliptic systems with smooth coefficients. In the proof, we identify Ka (P) with the Sobolev spaces on P associated to the metric r P gE , where gE is the Euclidean metric and rP(x) is a smoothing of the Eucli...

We study some basic analytic questions related to differential operators on Lie manifolds, which are manifolds whose large scale geometry can be described by a a Lie algebra of vector fields on a compactification. We extend to Lie manifolds several classical results on Sobolev spaces, elliptic regularity, and mapping properties of pseudodifferential operators. A tubular neighborhood theorem for...

Abstract We obtain a sparse domination principle for an arbitrary family of functions $f(x,Q)$ , where $x\in {\mathbb R}^{n}$ and Q is cube in ${\mathbb . When applied to operators, this result recovers our recent works [37, 39]. On the other hand, can be also non-operator objects. In particular, we show applications generalised Poincaré–Sobolev inequalities, tent spaces general dyadic sums. Mo...

Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hubert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer o...

We study the Sobolev spaces of exponential type associated with the Dunkl-Bessel Laplace operator. Some properties including completeness and the imbedding theorem are proved. We next introduce a class of symbols of exponential type and the associated pseudodifferential-difference operators, which naturally act on the generalized Dunkl-Sobolev spaces of exponential type. Finally, using the theo...

In this paper, metric complexities of certain classes of continuous-time systems are studied, using the time-domain sampling approach and the concepts of Kolmogorov, Gel’fand and sampling n-widths for certain classes of Sobolev space. A sampling theorem is obtained which extends Shannon’s sampling theorem to systems with possibly non-band-limited spectra. The theorem demonstrates that continuou...

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