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• W1891117966 abstract "We discuss the dimensional characterization of the solutions space of a formally integrable system of partial differential equations and provide certain formulas for calculations of these dimensional quantities. 1. Introduction: what is the solutions space? Let E be a system of partial differential equations (PDEs). We would like to discuss the dimensional characterization of its solutions space. However it is not agreed upon what should be called a solution. We can choose between global or local and even formal solutions or jet-solutions to a certain order. Hyperbolic systems hint us about shock waves as multiple- valued solutions and elliptic PDEs suggest generalized functions or sections. A choice of category, i.e. finitely differentiable C k , smooth C ∞ or analytic C ω together with many others, plays a crucial role. For instance there are systems of PDEs that have solutions in one category, but lacks them in another (we can name the famous Lewy's example of a formally integrable PDE without smooth or analytic solutions, (L)). In this paper we restrict to local or even formal solutions. The reason is lack of reasonable existence and uniqueness theorems (in the case of global solutions even for ODEs). In addition this helps to overcome difficulties with blow-ups and multi-values. If the category is analytic, then Cartan-Kahler theorem (Ka) guarantees local solutions of formally integrable equations (Go) and even predicts their quantity. We then measure it by certain dimension characteristics. If the category is smooth, formal integrability yields existence of solutions only if coupled with certain additional conditions (see for instance (Ho)). Thus it is easier in this case to turn to formal solutions, which in regular situations give the same dimension characteristics. With this vague idea let us call the space of solutions Sol(E). With this approach it is easy to impose a topology on the solutions space. However we shall encounter the situations, when the topological structure is non-uniform. To illustrate the above discussion, let's consider some model ODEs (in which case we possess existence and uniqueness theorem). The space of local 2000 Mathematics Subject Classification. Primary: 35N10, 58A20, 58H10; Secondary: 35A30." @default.
• W1891117966 created "2016-06-24" @default.
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• W1891117966 date "2006-10-26" @default.
• W1891117966 modified "2023-09-27" @default.
• W1891117966 title "Dimension of the solutions space of PDEs" @default.
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