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Consider a bounded domain $S\subset R^2$ and an elliptic system of two first order PDEs, namely a generalization of the Cauchy-Riemann system allowing nonconstant coefficients and lower order terms. Assume enough regularity for the coefficients to ensure Holder bounds. The coefficients depend on the solution itself, rendering the system quasilinear. I'm not imposing boundary conditions and I assume that there exists some solution.

Does the set of all solutions constitute a MANIFOLD in the corresponding Holder space, at least locally around the given solution? Which sort of manifold? Can anyone please suggest literature on this type of questions?

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  • $\begingroup$ Yes, you can consider a differential equation as specifying submanifolds of a manifold. Let $D$ be a domain and you have a set of differential equation for a collection of functions $(f_1,\ldots,f_n):D\to X$. Then solutions to the diffeqs are sections of $X$ fibered over $D$, and the total space of this fiber bundle is the manifold that you are interested in. You can think of coordinates on the total space as $(x_1,\ldots,x_k,f_1,\ldots,f_n)$ where the $x_i$ are coordinates on $D$. $\endgroup$ Mar 22 '13 at 17:22
  • $\begingroup$ In what sense is the total space of the fiber bundle equal to the space of all solutions to the system of PDE's? $\endgroup$
    – Deane Yang
    Mar 22 '13 at 17:52
  • $\begingroup$ The space of solutions is essentially an infinite dimensional vector space. If you assume that the boundary values live in some reasonable topological vector space, then the space of solutions can be given a similar topology and is itself essentially an infinite dimensional topological vector space and therefore an infinite-dimensional manifold. Is this what you're looking for? $\endgroup$
    – Deane Yang
    Mar 22 '13 at 17:57
  • $\begingroup$ Thanks, Deane, you are right. If the coefficients do not depend on the solution the system is linear and then the set of solutions is a linear space. What I meant is a nonlinear equation. I have now removed the confusing sentence. $\endgroup$
    – Mike
    Mar 22 '13 at 18:22
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    $\begingroup$ Mike, what I wrote is still largely true for a quasilinear system. The space of solutions is essentially determined by what boundary conditions are allowed and whether they determine the solution uniquely. But, if you restrict to boundary values of a given regularity, the space of allowable boundary values will still be an open set of a topological vector space and therefore an infinite dimenisional manifold. $\endgroup$
    – Deane Yang
    Mar 22 '13 at 18:28
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Yes. Please look at http://arxiv.org/pdf/math/0101017v3.pdf p. 30; the idea is that functions defined in a disk which satisfy such an equation determine, by an explicit integral expression, a holomorphic function, and this is a Banach space isomorphism. Quite a lot of the standard analysis of holomorphic disks works in that setting. The geometric description of the PDEs might not be clear to you, but the theory works for any smooth elliptic system in the plane, even fully nonlinear.

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  • $\begingroup$ Dear Ben, thanks very much for your answer. This is more or less what I was looking for. $\endgroup$
    – Mike
    Apr 11 '13 at 16:54

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