Stabilitätsprobleme für Eigenwertaufgaben bei Beschränkungen für die Variationen mit einer Anwendung auf die Platte

We give a criterion that a solution to an eigenvalue variational inequality defines a strong local minimum of an associated energy functional. Furthermore we study the bifurcation problem. As an application of these, we consider the compressed plate where the admissible deflections v satisfy the inequality |v| d, 0<d = const.     

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.