Densidade de funções contínuas em espaços de Sobolev com expoentes variáveis

Abstract

The approach analyzed the density of continuous functions in the integrable Riemann function and the space of square integrable Riemann function. The analysis shows that the Cc(X) space is dense in the space of the integrable Lebesgue functions Lpµ(X), where 0 ≤ p < ∞, X Hausdor , locally compact and µ as in Riesz's representation theorem. We explore the density of Cc(Ω), Ω ⊂ Rn, in the generalized Lebesgue spaces Lp(.)(Ω), with p(.) measurable and essentially limited function. Considering Sobolev spaces with variable exponent W1,p(.)(Ω), we discuss conditions about the exponent p(.) that guarantee the density of continuous functions in W1,p(.)(Ω). One result merges a monotonicity condition and a continuous log-Hölder condition. Another result discusses the density using two corollaries, p(.) exponent depends only on the nth coordinate of each Ω point and another where the p(.) exponent depends only on the distance from the point to the origin.