# Methods for determining domains of influence

Given a hyperbolic PDE, the domain of influence of a spacetime point $x$, say $I_x$ though $x$ could be replaced by any set, can be defined in two ways. Lets call one of them geometric ($I_x^G$) and the other analytical ($I_x^A$). In Lorentzian geometry, the geometric domain of influence consists of the interior of the cone of null geodesics emanating from $x$ (let me not bother about whether to include the boundary in the definition of $I_x^G$ or not). In general, a similar definition can be given using characteristic cones instead of null cones. The analytical domain of influence can be defined as the set of all spacetime points $y$ such that for every neighborhood neighborhood $O$ of $x$ there exist two solutions $u_1$ and $u_2$ satisfying the condition $u_1(x')=u_2(x')$, for all $x'$ on a Cauchy surface passing through $x$ except for $x'\in O$, and also the condition $u_1(y)\ne u_2(y)$. The latter one is the definition used in Lax's book on Hyperbolic PDEs.

Similar defintions can be given for the geometric and analytical domain of dependence, say $D_S^G$ and $D_S^A$. Such a definition should capture the desired equality $D_K = I_{S\setminus K}$, for a Cauchy surface $S$ and $K\subset S$ (once again, being sloppy with boundaries). I know that energy methods can be used to establish that $D^G_K \subseteq D^A_K$ (the analytical domain of dependence is at least as large as the geometric one). Hence, by duality, the same methods establish $I_K^A \subseteq I_K^G$ (that the geometric domain of influence is at least as large as the analytical one).

My question is about the reverse inclusion, $I_K^G \subseteq I_K^A$ or by duality $D_K^A \subseteq D_K^G$. Maybe it's too much to ask for the analytical and geometric definitions to coincide. But when they do, what methods are used to establish that? When they don't what methods can identify the obstruction? Except briefly in Lax's book, I don't know what references discuss this problem explicitly, so those would also be appreciated!

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Just added a bounty to raise the question's profile. –  Igor Khavkine Feb 21 '12 at 13:41
Your definition of analytic domain of dependence needs to be more carefully re-written. I think instead of $\exists O$ you need $\forall O$: otherwise take any Cauchy surface $\Sigma$ through $x$ and any $y\in \Sigma$: you can certainly choose neighbourhood $O$ of $x$ composed of $O_x\cup O_y$, where $O_y$ is a neighbourhood of $y$. Then by your definition $y$ is in the domain of influence, which is nonsensical. –  Willie Wong Feb 24 '12 at 16:15
Willie, you're right. I want the analytical domain of influence to be as small as possible, so I should ask for the relevant conditions to be satisfied for arbitrarily small neighborhoods $O$ of $x$. I'll edit the question accordingly. –  Igor Khavkine Feb 28 '12 at 10:51

I highly doubt the result you actually asked for is true.

Consider the linear wave equation on $(1+3)$ Minkowski space. The analytic domain of influence of a point $x$ as Lax defined it, which morally says that $y$ is in the analytic domain only if one can find perturbations in arbitrary small neighborhoods of $x$ that change $y$ (if I interpret your question statement correctly), actually consists of only the null cone emanating from $x$ and nothing more, since strong Huygen's principle holds.

The same is true for the linear wave equation on $(1+(2k+1))$ Minkowski spaces.

The opposite conclusion can be drawn on $(1+2k)$ dimensional Minkowski spaces, where the Green's function have support inside the cone.

For more general situations, you may want to consult the classical result of Atiyah-Bott-Garding on the existence of Petrowsky lacunae. For any linear hyperbolic equation that admits a lacuna, the analytic domain of influence cannot cover the entirety of the geometric one.

But for the result that you seem to actually want, where you should replace the analytic domain of dependence by a suitable "convex" envelope of it, I don't know if such a result is proven anywhere, but my guess is that, at least for the "local" version one can approach it using some sort of geometric optics construction.

For possible references (I haven't actually finished reading either, so they may not contain what you want), maybe you want to look at Michael Beals' book on propagation of singularities (sorry, the title escapes me at the moment) or Rauch's notes on Hyperbolic PDEs and Geometric Optics which I think you can find floating around on the internet.

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Willie, that's for the answer! I had the feeling that the equality of the geometric and analytical domains of influence/dependence was probably too good to be true. I'm basically happy to learn of the relevant PDE lingo that can be seen as addressing the difference between the two. From your answer and from looking through the literature, I see that the main relevant key words are indeed lacunae and geometric optics. I'll accept this answer. And thanks for the references. –  Igor Khavkine Feb 28 '12 at 11:00

I'm not quite sure if this is really the situation you are interested in, but in the book of Bär, Ginoux, and Pfäffle: Wave Equations on Lorentzian Manifolds and Quantization. ESI Lectures in Mathematics and Physics, European Mathematical Society, 2007, they discuss in quite some detail the Cauchy problem for hyperbolic linear wave equations on, and that is the catch, globally hyperbolic spacetimes. I guess that one should require something like that since otherwise you can at best hope for some local statements. But in their situation, I'm pretty sure to remember correctly, they have statements like the one you are looking for (don't they?). In any case, this is maybe a too special situation for you, but the book is nevertheless very nice. Unlike many other texts on hyperbolic PDE, it emphasizes the geometry very much.

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Hi, Stefan. Thanks for the tip! I know that book and also think highly of it. I know for sure that they have a discussion of the domain of dependence theorem of the kind I described above ($D^G\subseteq D^A$). I'll take a look again to see if they talk about inclusion in the other direction or something related. –  Igor Khavkine Feb 21 '12 at 23:45
Hi Igor, maybe you're right and it is only about this inclusion. So the point would be to construct a particular wave equation which extremizes this support of its solutions. So I fear that my reference does not answer your question :( sorry –  Stefan Waldmann Feb 22 '12 at 8:37