An elliptic boundary value problem occuring in magnetohydrodynamics
Let \(\alpha, \beta \in \mathbb{R}\), \(\alpha < \beta\), \(I_{\alpha, \beta} : = (\alpha, \beta) \times \mathbb{R}\). The authors consider the elliptic operator \[ Lu : = - \sum^ 2_{i=1} D_ i \left( \sum^ 2_{j = 1} a_{ij} (x) D_ ju + a_ i (x)u \right) + \sum^ 2_{i = 1} \overline a_ i (x)D_ i u + a_ 0(x)u \] in \(I_{\alpha, \beta}\). Here \((a_{ij} (x))\) is a positive definite Hermitian matrix, \(a_ 0\) is a bounded real function. The coefficients \(a_{ij}\), \(a_ i\) are \(2 \pi\)-periodic in \(x_ 2\). Aditionally \(a_{ij}\) and \(a_ 1\), \(a_ 2\) are assumed to be uniformly Lipschitz-continuous. \(L\) is considered under Dirichlet boundary conditions on \(x_ 1 = \alpha\) and \(x_ 1 = \beta\) and periodic boundary conditions with respect to \(x_ 2\). The authors mention, that this boundary value problem arises in magnetohydrodynamics. First they precisely describe the domain of definition \(D(L) \subset L^ 2 ((\alpha, \beta) \times (0,2 \pi))\) in order that \(L\) be selfadjoint. Then they show, that the spectrum of \(L\) is purely discrete, bounded from below and has no finite accumulation point. Depending on additional smoothness assumptions on the coefficients of \(L\), higher regularity of the eigenfunctions of \(L\) is proven. The authors study further the distribution of eigenvalues. Let \(N(t)\) denote the number of eigenvalues of \(L\) in the interval \((-\infty,t)\), counted according to their multiplicity. They find an explicit number \(\kappa\), such that for any \(\varepsilon \in (0,1/8)\) there holds: \[ N(t) = \kappa t + {\mathcal O} \biggl( t^{(7/8) + \varepsilon} \biggr) \quad \text{as} \quad t \to \infty. \] Finally for some appropriate real \(\sigma\) Green's function for \((L + \sigma) u = f\) under Dirichlet/periodic boundary conditions is constructed and investigated.
scientific article; zbMATH DE number 554392
scientific article