Approximation in Weighted Sobolev Spaces

Abstract

For p≥1 we define L ^m,p(ℝⁿ) as the set of distributions u on ℝⁿ such that

$$\|u\|_{m, p}=\Biggl(\, \sum^m_{k=1} \int\bigl|\nabla_k u(x)\bigr |^p\, \mathrm{d}x\Biggr)^{1/p}<\infty.$$

Here ∇ _k u denotes the vector (D ^α u)_|α|=k , $\mathaccent"7017{L}^{m, p}(\mathbb{R}^{n})$ is the completion of $C^{\infty}_{0}(\mathbb{R}^{n})$ with respect to the L ^m,p seminorm. Now let Ω⊂ℝⁿ be an open set and let μ be a nontrivial positive Radon measure on Ω. We will study the space $H^{m,p}_{\mu}(\varOmega)$, defined as the completion of $L_{p}(\mu) \cap L^{m,p}(\mathbb{R}^{n}) \cap C^{\infty}_{0}(\varOmega)$ with respect to the norm

$$\| u\|_{m, p; \mu}=\|u\|_{L_p(\mu)}+\| u\|_{m, p}.$$

The closure of $C^{\infty}_{0}(\varOmega)$ in $H^{m, p}_{\mu}(\varOmega)$ is denoted $\mathaccent"7017{H}^{m, p}_{\mu}(\varOmega)$. Note that if p<n then by Sobolev’s inequality the elements in $\mathaccent"7017{L}^{m, p}$ can be identified with functions in $L_{p^{*}}$, where $p^{*}=\frac{np}{n-p}$. If a domain in the notations of a space is not indicated it is assumed to be ℝⁿ. The elements in $\mathaccent"7017{H}^{m, p}_{\mu}(\varOmega)$ are naturally identified with elements in $\mathaccent"7017{L}^{m, p}$. Note also that L ^m,p⊂L _p (ℝⁿ,loc).