Monotonicity and best approximation in Orlicz--Sobolev spaces with the Luxemburg norm
Let \(\Omega\) be a bounded and connected open subset of \(\mathbb{R}^{n}\), and let \((\Omega,\Sigma,\mu)\) be a nonatomic finite measure space. The modular of a measurable function \(u\) on \(\Omega\) is defined by \(\rho_{A}(u)=\int_{\Omega }A(u(t))\,dt\), where \(A(u)\) is a given \(N\)-function. The Orlicz space is \(L_{A}(\Omega )=\{u(t):\) there\ exists \(\lambda >0\) such\ that \(\rho (\lambda u)<\infty \}\) and the Orlicz--Sobolev space is defined as \[ W_{m,A}=\{u\in L_{A}(\Omega):\partial^{\alpha }u\in L_{A}(\Omega),\;0\leq\left|\alpha \right|\leq m\}, \] where \(m\in \{0,1,2,\dots\}\), \(\alpha =(\alpha _{1},\dots,\alpha_{n})\), \( \left|\alpha\right|=\alpha_{1}+\cdots+\alpha_{n}\), \(\alpha_{i}\geq 0\) (\(i=1,2,\dots,n\)) are integers and \(\partial^{\alpha}u\) is the \(\alpha\)-th distributional derivative of \(u\). The space \(W_{m,A}\) is equipped with the norm \[ \left\| u\right\|_{m,A}=\left(\sum_{0\leq\left|\alpha\right| \leq m}\left\|\partial^{\alpha }u\right\|^{p}\right)^{1/p} \] (\(p\in(1,\infty)\) fixed), where \(\left\| u\right\| =\inf \{k>0:\rho_{A}(u/k)\leq 1\}\) is the Luxemburg norm in the Orlicz space \(L_{A}(\Omega)\). The space \(W_{m,A}\) is also equipped with the partial order \(u\leq \upsilon \) whenever \(\partial^{\alpha}u(t)\leq \partial^{\alpha }\upsilon(t)\) a.e.\ in \(\Omega\), for all \(\alpha=(\alpha_{1},\dots,\alpha _{n})\) with \(0\leq\left|\alpha\right|\leq m\). Thus \(W_{m,A}\) is a Banach lattice, and the authors study the monotonicity properties in \(W_{m,A}\), namely, strict monotonicity, upper (lower) local uniform monotonicity, uniform monotonicity, etc. The connections of these properties with some geometric properties of the unit ball of \(W_{m,A}\) are also established. Using these properties in the last section of the paper, the problem of best dominated approximation by elements of a convex set in \(W_{m,A}\) is studied (existence and unicity of the best dominated approximant, stability and continuity of the best approximation operator). An example is considered.
upper (lower) local uniform monotonicity
scientific article; zbMATH DE number 5289176
scientific article