Alcuni risultati di teoria spettrale per l'operatore di Laplace in regioni non limitate con frontiera non regolare
Soit un ensemble quelconque d’opérateurs différentiels en deux variables à coefficients complexes constants. Soit l’espace des fonctions continues complexes tendant vers zéro à l’infini dans le plan euclidien. Soit l’espace , tout . Classifier ces espaces équivaut à trouver des conditions nécessaires et suffisantes sur des opérateurs différentiels pour que . Il paraît que ce problème général est bien difficile. Nous présentons ici la solution complète dans le cas spécial des stables...
1. Introduction. It is well known that methods of algebraic geometry and, in particular, Riemann surface techniques are well suited for the solution of nonlinear integrable equations. For instance, for nonlinear evolution equations, so called 'finite gap' solutions have been found by the help of these methods. In 1989 Korotkin [9] succeeded in applying these techniques to the Ernst equation, which is equivalent to Einstein's vacuum equation for axisymmetric stationary fields. But, the Ernst equation...
We provide a detailed treatment of the Camassa-Holm (CH) hierarchy with special emphasis on its algebro-geometric solutions. In analogy to other completely integrable hierarchies of soliton equations such as the KdV or AKNS hierarchies, the CH hierarchy is recursively constructed by means of a basic polynomial formalism invoking a spectral parameter. Moreover, we study Dubrovin-type equations for auxiliary divisors and associated trace formulas, consider the corresponding algebro-geometric initial...
We construct Almansi decompositions for a class of differential operators, which include powers of the classical Laplace operator, heat operator, and wave operator. The decomposition is given in a constructive way.