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2 fixed a typo.; deleted 4 characters in body

Dear Pawel,

The H-principle is a technique by which you solve differential equations and inequalities by first defining a space of pre-solutions and then showing that every pre-solution can be deformed to a solution. The existence of pre-solutions is usually a topological problem and to show that the differential equation/inequality satisfies de h-principle is another matter (Gromov's book introduces something like four general techniques to do this).

To define pre-solutions (I'm just using this term to explain things) one needs the language of jet spaces. For example, suppose you want to look at the laplacian in the plane. Consider the space of $2$ jets where each point is classically denoted by $(x,y,z,q,p,r,s,t)$. The $2$-jet of a function $u$ is $$(x,y) \mapsto (x,y,u(x,y),\partial_xu(x,y),\partial_yu(x,y),\partial^2_{xx}u(x,y), \partial^2_{xy}u(x,y), \partial^2_{yy}u(x,y))$$

Harmonic functions are functions whose $2$-jets lie inside the submanifold $r^2 r + t^2 t = 0$ inside the space of $2$-jets. In this case, the pre-solutions I've been talking about would be given by maps of the form $$(x,y) \mapsto (x,y,u(x,y),q(x,y),p(x,y),r(x,y),s(x,y),t(x,y))$$ such that $r(x,y)^2 r(x,y) + t(x,y)^2 t(x,y) = 0$. What is missing is that $q(x,y)$ is not the partial derivative of $u$ with respect to $x$ and so on: the map (more precisely, this section of the $2$-jet bundle) is not necessarily "holonomic" in Gromov's terminology.

Gromov's incredible idea is that for lots and lots of differential equations and inequalities in geometry any pre-solution can be deformed to a solution. Alas, the method does not seem to be too useful in mathematical physics, although I think some people in elasticity have used it to construct weak solutions to some variational problems.

I've always wanted to use this method, but no luck yet ...

1

Dear Pawel,

The H-principle is a technique by which you solve differential equations and inequalities by first defining a space of pre-solutions and then showing that every pre-solution can be deformed to a solution. The existence of pre-solutions is usually a topological problem and to show that the differential equation/inequality satisfies de h-principle is another matter (Gromov's book introduces something like four general techniques to do this).

To define pre-solutions (I'm just using this term to explain things) one needs the language of jet spaces. For example, suppose you want to look at the laplacian in the plane. Consider the space of $2$ jets where each point is classically denoted by $(x,y,z,q,p,r,s,t)$. The $2$-jet of a function $u$ is $$(x,y) \mapsto (x,y,u(x,y),\partial_xu(x,y),\partial_yu(x,y),\partial^2_{xx}u(x,y), \partial^2_{xy}u(x,y), \partial^2_{yy}u(x,y))$$

Harmonic functions are functions whose $2$-jets lie inside the submanifold $r^2 + t^2 = 0$ inside the space of $2$-jets. In this case, the pre-solutions I've been talking about would be given by maps of the form $$(x,y) \mapsto (x,y,u(x,y),q(x,y),p(x,y),r(x,y),s(x,y),t(x,y))$$ such that $r(x,y)^2 + t(x,y)^2 = 0$. What is missing is that $q(x,y)$ is not the partial derivative of $u$ with respect to $x$ and so on: the map (more precisely, this section of the $2$-jet bundle) is not necessarily "holonomic" in Gromov's terminology.

Gromov's incredible idea is that for lots and lots of differential equations and inequalities in geometry any pre-solution can be deformed to a solution. Alas, the method does not seem to be too useful in mathematical physics, although I think some people in elasticity have used it to construct weak solutions to some variational problems.

I've always wanted to use this method, but no luck yet ...