The overdetermined Cauchy problem in some classes of ultradifferentiable functions
The authors continue their study of the Cauchy problem for overdetermined systems of linear partial differential equations with constant coefficients; here they are concerned with the solvability in spaces of ultradifferentiable functions. They begin by reviewing the algebraic theory of such systems: the system is represented by a finitely generated polynomial module and the considered function space is a further polynomial module. The existence and uniqueness of solutions for the inhomogeneous system may then be formulated in terms of the vanishing of certain Ext modules. For the Cauchy problem on a given hypersurface, the question whether the surface is characteristic is studied via a change of the base ring of the module of the system. The next section discusses some spaces of ultradifferentiable functions: first the functions of class \((M_p)\), then the corresponding Whitney functions. In order to accommodate for functions with different regularity properties with respect to different variables, topological tensor products are treated. After these preliminaries, the abstract setting for the Cauchy problem for a pair of convex sets \((K_1,K_2)\) with \(K_1\) closed in \(K_2\) is introduced. Given two function spaces \(E(K_1)\), \(F(K_2)\) on these sets (the first one for the inhomogeneity of the problem, the second one for the Cauchy data), the pair \(E(K_1)\), \(F(K_2)\) is said to be of evolution, if the Cauchy problem has at least one solution for all allowed data; it is causal, if the Cauchy problem has at most one solution, and it is hyperbolic, if the Cauchy problem is uniquely solvable for all allowed data. Then the authors show how this classification may be performed in the case of spaces of ultradifferentiable functions via Ext modules and maps between them and also develop a dual formulation of these conditions via associated primes. The next topic is the relation of the results obtained so far with a Phragmén-Lindelöf principle. The authors study in particular some generalizations of the classical Petrovsky conditions for evolution that suffice to guarantee the validity of a Phragmén-Lindelöf principle. Finally, the question of well-posedness is investigated for several hyperbolic problems.
scientific article; zbMATH DE number 2035094