First boundary-value problem for a certain fourth-order differential equation (Q1062214)

''Let X be a bounded domain in \({\mathbb{R}}^ m\), Y be a bounded domain in \({\mathbb{R}}^ n\), \(\Omega =X\times Y\). Let us consider the following equation in the domain \(\Omega\) : Lu(x,y)\(\equiv MMu(x,y)-Nu(x,y)= f(x,y)\), where \[ Mu(x,y)=k_ 1(y)\sum^{m_ 1}_{i,j=1}[a^{ij}(x,y)u_{x_ i}]_{x_ j}+k_ 2(x)\sum^{n_ 1}_{i,j=1}[b^{ij}(x,y)u_{y_ 1}]_{y_ j}+a(x,y)u, \] \[ Nu(x,y)=\sum^{m_ 1}_{i,j=1}c^{ij}(x,y)u_{x_ ix_ j}+\sum^{n_ 1}_{i,j=1}e^{ij}(x,y)u_{y_ iy_ j}+\sum^{m}_{i=1}c^ i(x,y)u_{x_ i}+\sum^{n}_{i=1}e^ i(x,y)u_{y_ i}+c(x,y)u, \] \(m_ 1\leq m\), \(n_ 1\leq n\), the \(m_ 1\times m_ 1\)-matrices \(\{a^{ij}(x,y)\}\), \(\{b^{ij}(x,y)\}\) and the \(n_ 1\times n_ 1\)-matrices \(\{c^{ij}(x,y)\}\), \(\{e^{ij}(x,y)\}\) are symmetric and non-negative'' (from the author's introduction). The piecewise-smooth boundary of \(\Omega\) is splitted into five parts \(\Gamma_ 1\), \(\Gamma_ 2\), \(\Gamma_ 3\), \(\Gamma^+_ 4\), \(\Gamma^-_ 4\) according to the type of degeneracy of L. Various kinds of data are prescribed on each particular part of \(\partial \Omega:\) On \(\Gamma_ 1\) and \(\Gamma_ 2\), u as well as its conormal derivative (with respect to M) are supposed to vanish whereas on \(\Gamma_ 3\) and \(\Gamma^+_ 4\) only u is prescribed as zero, and nothing is imposed on \(\Gamma^-_ 4\). For the adjoint poblem the role of \(\Gamma^+_ 4\) and \(\Gamma^-_ 4\) are interchanged. The author then introduces various Sobolev spaces of inhomogeneous strength and gives an existence result for weak and semi-weak solutions as well as a uniqueness result for strong solutions [in the sense of \textit{Yu. M. Berezanskij}, ''Expansions in eigenfunctions of selfadjoint operators'', (1965; Zbl 0142.372)].