H-principle

In mathematics, the homotopy principle (h-principle) is a very general way to solve partial differential equations (PDE), and more generally partial differential relations (PDR). The h-principle is good for underdetermined PDE or PDR such as immersion problem, isometric immersions problem and so on.

Rough idea

Assume we want to find a function f on Rm which satisfies a partial differential equation of degree k, in co-ordinates (u_1,u_2,...,u_m). One can rewrite it as

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:Psi(u_1,u_2,...,u_m, J^k_f)=0!,

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where J^k_f stands for all partial derivatives of f up to order k. Let us exchange every variable in J^k_f for new independent variables

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y_1,y_2,...,y_N.

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Then our original equation can be thought as a system of

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:Psi^{}_{}(u_1,u_2,...u_m,y_1,y_2,...y_N)=0!,

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and some number of equations of the following type

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:y_j={partial y_iover partial u_k}.!,

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A solution for

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:Psi^{}_{}(u_1,u_2,...u_m,y_1,y_2,...y_N)=0!,

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is called a non-holonomic solution, and a solution for the system (which is a solution of our original PDE) is called a holonomic solution.

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In order to check if a solution exists, first check if there is a non-holonomic solution (usually it is quite easy and if not then our original equation did not have any solutions).

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A PDE satisfies the h-principle if any non-holonomic solution can be deformed into a holonomic one in the class of non-holonomic solutions.

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Therefore, once you prove that an equation satisfies h-principle it is really easy to check whether it has solutions. It is surprising that most underdetermined partial differential equations satisfy the h-principle.

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