H-principle

In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations (PDRs). The h-principle is good for underdetermined PDEs or PDRs, such as occur in the immersion problem, isometric immersion problem, and other areas.

The theory was started by Yakov Eliashberg, Mikhail Gromov and Anthony V. Phillips. It was based on earlier results that reduced partial differential relations to homotopy, particularly for immersions. The first evidence of h-principle appeared in the Whitney–Graustein theorem. This was followed by the Nash-Kuiper Isometric ${\displaystyle C^{1}}$ embedding theorem and the Smale-Hirsch Immersion theorem.

Assume we want to find a function ƒ on R^m which satisfies a partial differential equation of degree k, in co-ordinates ${\displaystyle (u_{1},u_{2},\dots ,u_{m})}$. One can rewrite it as

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