let $M$ be a Riemannian manifold and $\alpha$ be any some unknown form on $M$. I am interested in solutions or some references of the equation of type $(d + \delta) \alpha = 0$ where $\delta$ is the Hodge operator. And also with some boundary conditions. Is there any literature (books) on this, or papers? Tanks in advance.
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$\begingroup$ Please see the FAQ, and also mathoverflow.net/howtoask This is an overly broad reference request. $\endgroup$– David Roberts ♦Commented May 22, 2012 at 6:27
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$\begingroup$ I added a tag {} $\endgroup$– David Roberts ♦Commented May 22, 2012 at 6:27
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1$\begingroup$ This equation is a bit problematic. If $\alpha$ is a $k$-form, then $d\alpha$ is a $(k+1)$-form and $\delta\alpha$ is a $(k-1)$-form. I suppose you could interpret this equation as living on the entire space of forms, i.e., $\alpha$ is of mixed type. Did you mean to solve $(d+\delta)^2\alpha=0$? Those are the harmonic forms. $\endgroup$– Brendan ForemanCommented May 22, 2012 at 16:38
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$\begingroup$ Is $M$ compact? Is it perhaps not compact but complete? $\endgroup$– Ben McKayCommented May 28, 2012 at 12:10
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3 Answers
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On a compact manifold, the conditions $(d{+}\delta)\alpha=0$ and $\Delta\alpha=0$ are equivalent, but this is not true on noncompact manifolds or manifolds with boundary, and I suspect that this is the situation in the original question.
For example, take the flat metric on $\mathbb{R}^n$ and consider the case of a $1$-form $\alpha$. The (local and global) solutions of $(d{+}\delta)\alpha=0$ are given by $\alpha = da$ where $a$ is any solution to the (determined elliptic) equation $\Delta a = 0$, while the (local and global) solutions of $\Delta \alpha = 0$ are $\alpha = \alpha_1\ dx^1 + \cdots + \alpha_n\ dx^n$, where $\Delta \alpha_i = 0$ for all $i$. Obviously, when $n>1$, there are a lot more solutions of the latter equation than the former.
Moreover, if one wants to specify boundary data that will make the solution unique, then it's clear that one will have to specify boundary data completely differently in the two cases. For example, for solutions of the equation $(d{+}\delta)\alpha=0$ on a domain $D\subset\mathbb{R}^n$ with smooth boundary, specifying the pullback of $\alpha$ to $\partial D$, say, $(\partial D)^*\alpha = \phi$, clearly won't work unless $\phi\in \Omega^1(\partial D)$ satisfies $d\phi=0$ and $[\phi]\in H^1_{dR}(\partial D)$ lies in the image of the restriction mapping $H^1_{dR}(D)\to H^1_{dR}(\partial D)$. On the other hand, for the second equation $\Delta\alpha=0$, the appropriate boundary problem would be to specify the restriction (not the pullback) of $\alpha$ to $\partial D$.
Similar remarks apply for $k$-forms with $k>1$ on a general Riemannian manifold with boundary. This problem is not immediately reducible to the Hodge theorem for compact orientable manifolds without boundary.
answered May 27, 2012 at 16:02
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Warner, ''Foundations of differentiable manifolds and Lie groups'' Jost, ''Riemannian geometry and geometric analysis'', Wells: ''Differential analysis on complex manifolds''. Each of these books contains a chapter concerning this PDE.
answered May 22, 2012 at 7:20
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$\begingroup$ where exactly in Jost "Riemannian geometry and geometric analysis" do I find this? $\endgroup$– pascalCommented May 22, 2012 at 12:17
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1$\begingroup$ 2nd edition, Theorem 2.2.1, page 88ff $\endgroup$ Commented May 22, 2012 at 14:03
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$\begingroup$ I do not see from this theorem how to solve a equation like $(\delta + d)\alpha = 0$? what do you mean ? $\endgroup$– pascalCommented May 22, 2012 at 14:31
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2$\begingroup$ The theorem is called Hodge theorem and it has several equivalent formulations. You can read in, say Warners book, how the different formulations relate to each other. $\endgroup$ Commented May 22, 2012 at 17:07
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For manifolds with or without boundary, I can think of two references that discuss your question in great detail. The first is Ch. Morrey's classic,
Multiple integrals in the calculus of variations, Springer Verlag
The second reference is the first volume of M. Taylor's three volume monograph on PDE's. The language in this reference may be more familiar to the modern reader.
For manifolds without boundary you can also try my lectures.
answered Jun 1, 2012 at 9:19