Density of smooth functions in weighted Sobolev spaces with variable exponent
In a bounded Lipschitz domain \(\Omega\subset\mathbb{R}^{n}\), \(n\geq 2\), we are given a weight \(\rho \) that is a nonnegative measurable function summable on \(\Omega.\) Additionally, let \(p=p(x) \) be a given exponent that is measurable in \(\Omega\) and satisfies the condition \[ 1<\alpha <p( x) <\beta <\infty . \tag{1} \] The Luxemburg norm is defined as \[ \| f\| _{p( .) ,\rho dx}=\inf \left\{ \lambda>0:\int | \lambda ^{-1}f| ^{p( x) }\rho dx\leq 1\right\} . \] The classes \(L^{p( .) }( \Omega;\rho dx) \) with this norm are reflexive separable Banach spaces with dual \(L^{p'(.)}(\Omega;\rho dx)\), where \(p'(x)=\frac{p(x)}{p(x)-1}.\) Define the weighted Sobolev-Orlicz space by \[ W=\left\{ u\in W_{0}^{1,1}( \Omega) :\int_{\Omega}| \nabla u| ^{p( x) }\rho dx<\infty \right\} \] with the norm \[ \| u\| _{W}=\| \nabla u\| _{p(.) ,\rho dx}. \] Assume that \(\rho ^{-\frac{1}{p}}\in L^{p'}( \Omega;dx) .\) Then \(W\) is complete. The space \(H\) is defined as\ the closure of \(C_{0}^{\infty }( \Omega) \) with respect to the norm \(\| .\| _{W}.\) The weight \(\rho \) is said to be regular if \(H=W.\) A non-negative locally summable function (weight) \(\omega\) is said to belong to \(A_{p(.)}(\Omega)\) if \[ \| \omega\| _{A_{p( .) }( \Omega)}:=\sup_{x\in Q\subset \Omega}\left( \int_{Q}\omega dy\right) ^{\frac{1}{p(x)}}\frac{\| \omega^{-\frac{1}{p}}\| _{L^{p'}}( Q) }{| Q| }<\infty , \] where the supremum is taken over the cubes \(Q\subset \Omega\) whose faces are parallel to the coordinate hyperplanes. We define the logarithmic condition by \[ | p( x) -p( y) | \leq \frac{k_{0}}{\ln \frac{1}{| x-y| }},\quad | x-y| <1,\tag{2} \] where \(k_{0}\) does not depend on \(x,y.\) The main result of this paper is the following. Theorem. Let an exponent \(p( .) \) satisfy conditions (1) and (2), let \(\rho\) be a weight that can be represented as \(\rho=\omega\omega_{0}\), where \(\omega_{0}\in A_{p( .) }(\Omega)\), and let \[ \lim\inf_{t\rightarrow \infty } \left( \int_{\Omega}\omega^{-t}\omega_{0}dx\right) ^{\frac{1}{t}}.\left( \int_{\Omega}(t^{-p( x) }\omega) ^{t}\omega_{0}dx\right) ^{\frac{1}{t}}<\infty . \] Then the weight \(\rho \) is regular.
weighted Sobolev spaces with variable exponent
scientific article; zbMATH DE number 6355635