Das Maximum-Prinzip vom Miranda-Agmon-Typ für Lösungen der biharmonischen Gleichung in einem Rechteck

Utilizing the quadratic functional criteria and positive functionals, the author develops a comparison theorem for oscillation of partial differential equations. This result parallels a comparison of the oscillation of scalar ordinary differential equations to oscillation of vector-matrix differential equations established by LEWIS and WRIGHT . We want to look at the matrix partial differential equation.     

Über die existenz und eindeutigkeit einer lösung für ein randwertproblem für eine gleichung vom gemischten typ in einem rechteck

In the paper a boundary value problem is studied for the equation of mixed typek(y)u_xx + u_yy + r(x, y)u = f(x, y) in the rectangular domain {(x, y)| ?1 < x < +1, y_c < y < y_H} with y_c < 0, y_H > 0, k(y) = sign y|y|^m, m > 0 (and more generally for a function k = k(y) with k(O) = 0, k(y)y > O for y ≠ O). Specific for the stated problem is that no data are prescribed on the line {(x, y_c), ?1 < x < +1}. It is proved that the formulated problem is well-posed in the sense that there is at most one quasi-regular solution and that a generalized solution exists. The energy-integral-(abc-)method is used to show uniqueness and to obtain an apriori estimate for the solution of the adjoint problem whence the existence statement follows.     

Ein Hodge-Satz für Mannigfaltigkeiten mit nicht-glattem Rand

In this paper the question of determining the dimension of the space of harmonic Dirichlet and Neumann differential forms on a Riemannian manifold with non-smooth boundary is answered for a wide class of boundaries. The admissible boundaries can be characterized using a generalized “global segment property”. The well-known relation between the Betti numbers and the dimension of these spaces is established in this more general case, too. Bounded and non-bounded manifolds are treated (“exterior and interior domains”).     

Ein Existenzsatz für die Vlasov-Gleichung mit selbstkonsistentem Magnetfeld

The Vlasov equation with electric and magnetic fields generated by the distribution of the plasma is considered. For a simplified model with one space and two velocity coordinates and a mollification of the spatial density and the mean velocity in the Maxwell equations an existence and uniqueness theorem is proved.