Some properties of anisotropic Sobolev spaces
Motivated by the study of the equations of ideal Magneto-hydrodynamics with perfectly conducting wall boundary condition, the author introduces and investigates three types of anisotropic Sobolev spaces of integer order \(m\) on \(\Omega\), where is either \(\mathbb R^n_+\) or a bounded subset of \(\mathbb R^n\). Assume first \(\Omega =\mathbb R^n_+\) and let \(\sigma\) be a smooth one variable function which is increasing, \(\sigma (x_1)=x_1\) in a neighbourhood of the origin, and \(\sigma (x_1)=1\) for \(x_1\) large enough. For a multi-index \(\alpha =(\alpha _1,\dots,\alpha _n)\) let \[ \partial^{\alpha}_*= (\sigma (x_1)\partial _1)^{\alpha _1}\partial _2^{\alpha _2}\cdots \partial _n^{\alpha _n}. \] Given \(m\geq 1\), the space \(H^m_*(\Omega)\) is defined as the set of functions \(u\in L_2(\Omega)\) such that \(\partial^{\alpha}_*\partial ^k_1u\in L_2(\Omega)\) if \(|\alpha |+2k\leq m\) and it is normed by \[ \|u\|_{m,*}^2=\sum _{|\alpha |+2k\leq m} \|\partial^{\alpha}_*\partial ^k_1u\|^2. \] The space \(H^m_{**}(\Omega)\) consists of the functions \(u\in L_2(\Omega)\) such that \(\partial^{\alpha}_*\partial ^k_1u\in L_2(\Omega)\) if \(|\alpha |+2k\leq m+1\), \(|\alpha |\leq m\), and it is normed by \[ \|u\|_{m,**}^2=\sum _{\alpha , k} \|\partial^{\alpha}_*\partial ^k_1 u \|^2 \] where the sum is taken over all multi-indices \(\alpha\) and indices \(k\) such that \(|\alpha |+2k\leq m+1\), \(|\alpha |\leq m\). The space \(H^m_{***}(\Omega)\) consists of the functions \(u\in L_2(\Omega)\) such that \(\partial^{\alpha}_*\partial ^k_1u\in L_2(\Omega)\) if \(|\alpha |\leq m\) and if \(|\alpha |+2k\leq m+2\) for \(k\geq 2\), \(|\alpha |+2k\leq m+1\) for \(k=1\), and it is normed by \[ \|u\|_{m,***}^2=\sum _{\alpha , k} \|\partial^{\alpha}_*\partial ^k_1u\|^2 \] where the sum is taken over all multi-indices \(\alpha\) and indices \(k\) such that \(|\alpha |\leq m\) and \(|\alpha |+2k\leq m+2\) for \(k\geq 2\), \(|\alpha |+2k\leq m+1\) for \(k=1\). If \(\Omega\) is a bounded open subset of \(\mathbb R^n\) with \(C^{\infty}\) boundary then the above spaces are defined via tangential vector fields. The aim of the paper is to present some properties of these function spaces such as: multiplication, embeddings and compact embeddings, trace results. It is announced that these properties, which provide basic tools for calculations, will be used in two forthcoming papers of the author.
scientific article; zbMATH DE number 1547969