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Natural Focusing

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Elliptic–Hyperbolic Partial Differential Equations

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Abstract

In this chapter we suggest that one natural context for equations of mixed elliptic–hyperbolic type is the apparently spontaneous focusing of energy. In some cases this process depends on how a body is embedded in a higher-dimensional space. A brief historical review of the isometric embedding problem is included; remarks on the Hard Implicit Function Theorem, used in much of the associated literature, are given. We review the related matter of the reduction of the Darboux equation to a quasilinear system; the technical discussions are based mainly on work by Han, Hong and Lin and by Chen, Slemrod, and Wang. The focusing of elastic energy provides a potential area of physical application for the mathematics of isometric embedding. The well-studied case of crumpling flat sheets is a special case of a larger and still emerging theory for indentations of thin shells. The concentration of energy that characterizes the failure of elastic structures is mathematically analogous to the focusing of electromagnetic energy at a caustic.

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    Thomas H. Otway

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Otway, T.H. (2015). Natural Focusing. In: Elliptic–Hyperbolic Partial Differential Equations. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-19761-6_6

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