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Singular PDE’s geometry and boundary value problems. (English) Zbl 1171.35006

The geometric theory of PDE’s generally works in some regularity conditions where general theorems can be built in order to obtain local and global solutions existence theorems. However, the great complexity of natural and mathematical phenomena requires to handle singular PDE’s. The authors give a geometric characterization of such mathematical objects and to find general existence theorem of local and global solutions passing through singular points of PDE’s. The characterization of singularities can be obtained by means of algebraic-geometric techniques and algebraic-topological differential techniques. These induce effects on the integral structure of PDE’s, described by means of their Cartan distributions, and formal prolongations properties. Singular points in PDE’s are sources of interesting phenomena, thus one can say that they constitute a more richness and contribute to more versatile behaviors in PDE’s solutions.
The authors obtain three main results: The first result relates singular integral bordism groups of PDE’s to global solutions passing through singular points of PDE’s. The second result characterizes singular ODE’s singular solutions and their stability. The third result completes the second result on the same subject emphasizing the role played by singular points to conserve the smoothness or to create bifurcation in solutions of singular ODE’s.
Some examples of singular PDE’s and ODE’s are considered in some details in order to show how the general theory works. The authors show how by using only the geometric framework one can obtain local and global existence theorems also for singular boundary value problems.

MSC:

35A20 Analyticity in context of PDEs
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
58J22 Exotic index theories on manifolds
20H15 Other geometric groups, including crystallographic groups
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