Differential operators preserving the space of harmonic functions (aka higher symmetries of the Laplacian) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T19:32:15Z http://mathoverflow.net/feeds/question/28277 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/28277/differential-operators-preserving-the-space-of-harmonic-functions-aka-higher-sym Differential operators preserving the space of harmonic functions (aka higher symmetries of the Laplacian) robot 2010-06-15T16:58:27Z 2010-07-13T23:22:16Z <p>The article <a href="http://arxiv.org/abs/hep-th/0206233" rel="nofollow">http://arxiv.org/abs/hep-th/0206233</a> (published in Ann. of Math. (2) 161 (2005), no. 3) deals with linear differential operators $D$ for which there exists another linear differential operator $\delta$ such that $\Delta D = \delta \Delta$. Obviously these operators preserve the kernel of $\Delta$, i.e. the space of harmonic functions. The mentioned article finds essentially all such operators $D$. The result is that up to trivial operators $D = P\Delta$ all the operators $D$ have polynomial coefficients and are generated by sums of compositions of first order operators of this kind.</p> <p>First question: Let $D$ be any differential operator preserving the space of harmonic functions. It is easy to see that the operator $\delta = \Delta D (\Delta)^{-1}$ is well defined and satisfies $\Delta D = \delta \Delta$. Is $\delta$ also a differential operator?</p> <p>Second question: Is it true that all differential operators, which preserve the space of harmonic functions, are generated by first order ones with this property?</p> <p>One can also ask these questions only for linear differential operators or for operators from the Weyl algebra (i.e. linear differential operators with polynomial coefficients). For example, by a theorem of Peetre, the answer to the first question is affirmative if the operator $\delta = \Delta D (\Delta)^{-1}$ is local (i.e. the support of $\delta u$ is contained in the support of $u$).</p> <p>Third question: What makes the linked article so interesting that it was published in Annals?</p> http://mathoverflow.net/questions/28277/differential-operators-preserving-the-space-of-harmonic-functions-aka-higher-sym/28311#28311 Answer by mathphysicist for Differential operators preserving the space of harmonic functions (aka higher symmetries of the Laplacian) mathphysicist 2010-06-15T20:29:52Z 2010-06-15T21:05:05Z <p>The answer to your second question (unless I somehow misread it) is <strong>yes</strong> precisely because of the result of the paper you refer to (you may also wish to look at <a href="http://www.springerlink.com/content/r1540864nn27rt97/" rel="nofollow">this paper</a> and the preprint <a href="http://arxiv.org/abs/math-ph/0506002" rel="nofollow">math-ph/0506002</a> which address the same subject). This is the case because if $D$ is a differential operator that preserves the space of harmonic functions then there indeed exists a <strong>differential</strong> operator $\delta$ such that $\Delta D = \delta \Delta$. The latter holds (see e.g. the discussion at p.290 near Eq.(5.5) of the book <em><a href="http://books.google.com/books?id=sI2bAxgLMXYC" rel="nofollow">Applications of Lie groups to Differential Equations</a></em> by P.J. Olver) because the equation $\Delta f=0$ is totally nondegenerate in the sense of Definition 2.83 of the same book. In spite of the rather technical language the idea behind all this is very simple: if you have a submanifold $N$ of an manifold $M$ defined by the equations $F_1=0, \dots, F_k=0$ with smooth $F$'s and $k&lt;\mathrm{dim}\ M$, then a smooth function $h$ vanishes on $N$ iff there exist smooth functions <i>h<sub>j</sub></i> on $M$ such that $$h=h_{1} F_1+\cdots+h_k F_k$$ provided $dF_1\wedge \dots \wedge dF_k\neq 0$ on $N$ (see Proposition 2.10 of the same book). In a sense, this is a smooth counterpart of the famous Hilbert's Nullstellensatz in the form stated e.g. <a href="http://mathworld.wolfram.com/HilbertsNullstellensatz.html" rel="nofollow">here</a>. This result is then applied to the case when $M$ is a <a href="http://en.wikipedia.org/wiki/Jet_bundle" rel="nofollow">jet bundle</a> and $N$ is a submanifold thereof defined by a system of differential equations and all its differential consequences (more precisely, one should rather consider the consequences only up to a certain order, to avoid dealing with infinitely many equations), et voila.</p>