I'm interested in a criterion that determines whether a linear scalar PDE (arbitrary order) has a unique solution given vanishing boundary conditions at spatial infinity. I'll try to formulate the question more precisely below.
Consider a PDE of the form $L[u]=0$ where $u(t,x,y,z)$ is a scalar function of one time $(t)$ and three spatial variables $(x,y,z)$, though this choice of dimensionality is not central to the question. The function $u$ is required to vanish "sufficiently" fast if the $(x,y,z)$ variables are taken to infinity, keeping $t$ fixed. [If that's not enough, it can also be required to vanish at infinity along any hyperplane that is space-like with respect to the Lorentzian metric $\mathrm{diag}(-1,1,1,1)$.] However, no requirements are put on the behavior of $u$ as $t\to\pm\infty$ for fixed $(x,y,z)$. The linear differential operator $L$ can be assumed to have constant coefficients, but could by of any order. Though, I'd also like to know how the answer generalizes to the case when the coefficients and the background Lorentzian metric are no longer constant.
So, my question is this: for which operators $L$ does the equation $L[u]=0$ have a unique solution?
Let me give some examples.
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Equation $\partial_z u=0$ has a unique solution. An arbitrary solution comes from integrating the rhs wrt to $z$ and adding any function that's constant wrt to $z$. From the boundary conditions, it is easy to see that both pieces must be zero. Hence, $u=0$ is the unique solution.
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The same argument does not work for $\partial_t u=0$. For any given solution, I can get another solution by adding a function of $(x,y,z)$ only that vanishes at infinity, and there are plenty of those.
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The equation $(\partial_x^2+\partial_y^2+\partial_z^2)u=0$ is uniquely solvable: ignore $t$ dependence and invert the Laplacian, with uniqueness given by the same argument as in the first example.
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The equation $(-\partial_t^2+\partial_x^2+\partial_y^2+\partial_z^2)u=0$ is not uniqely solvable: solutions are parametrized by Cauchy data on, say, the $t=0$ hyperplane, and so are definitely not unique.
These examples make me think that the answer is some version of an ellipticity condition. Unfortunately, I'm only aware of how to formulate this condition for second order systems. Any help appreciated!
Status Update: Willie Wong provided some good, relevant information below. Let me summarize my understanding of it here and then sharpen my question in light of it.
If the partial differential operator (PDO) $L$ contains no time derivatives, then the Fourier transform $\hat{u}$ of a nontrivial solution $u$ must be supported on the zero set of $P(\xi)$, the symbol of $L$. If this set is of measure zero, then $\hat{u}$ can only be a distribution. On the other hand, local regularity of $\hat{u}$ is controlled by the decay of $u$ at infinity. In particular, if $u$ is in the Schwarz space, then $\hat{u}$ cannot be sufficiently singular to be a distribution. Hence, $L[u]=0$ would admit only the trivial solution, and hence be uniquely solvable.
In principle, I can use the same argument for any $L$ that is expressible only in terms of derivatives parallel to a given space-like hyperplane, by appealing to the fact that I've imposed the same boundary conditions at infinity (say Schwarz) for each such hyperplane.
In principle, the above reasoning gives a nice large space of PDOs that satisfy my criteria. But are there any more? I think there are (see below).
Now, suppose that I cannot ignore time derivatives. Willie's suggestion is to write the equation $L[u]=0$ in evolution form $\partial_t v + Av=0$ and exclude $L$'s for which the evolution equation is well posed as an initial value problem. But not all such $L$ can be excluded, since not all initial data generates solutions that satisfy the boundary conditions (decay at infinity along any space-like direction). I'm thinking that there should be a geometric condition involving the background null cone and the characteristics of $L$ or the zero set of $P(\xi)$. For instance, if $L=-c^{-2}\partial_t^2+\partial_x^2$ with $c>1$, then the corresponding equation has infinitely many solutions parametrized by Cauchy data on $t=0$. However, these solutions would correspond to waves pulses propagating along the $x$-axis at a speed faster than the background speed of light. On the other hand, the solution $u(t,ct)$ must vanish for large $t$ as $(1,c)$ is a space-like vector. I believe this is enough to show that this $L$ also admits only trivial solutions. My reasoning here is based on the exact representation of solutions via the D'Alambert formula, but that doesn't generalize easily. Any idea what kind of geometric condition could be used for more general operators?
Ultimately, I'd like to know something about the geometry of the space of operators that satisfy my criteria. Say I fix the maximal order to make things easier. The space is clearly not linear, but could it be convex or the complement of a convex set? (These last guesses are probably not right. I'm just throwing out ideas.) I'd be happy if I could understand this space for just first and second order operators, preferably with hints of how this understanding could generalize to higher orders.
Best Answer
Hi, I am adding another answer because this suggests a rather different approach then what I have outlined before, and this is targeted at the fact you are willing to grant smooth with compact support on any space-like hyperplane.
If you are willing to let your solutions vanish in such a large set, then the proper tool for the analysis is the theory of unique continuation for solutions to linear differential equations. For example, solutions to elliptic equations tend to have the property of strong unique continuation: if the solution vanish on any open set it must vanish everywhere. Using the geometry you can weaken ellipticity to have cases where no derivatives are taken in certain directions ($\partial_x^2 + \partial_y^2$, for example).
Similarly you can address parabolic type operators. (They have infinite speeds of propagation as well.)
For hyperbolic type, if the coefficients of the operator are real analytic, then you can use Holmgren's Uniqueness theorem if the inner-most characteristic cone of the operator remains outside of the Minkowskian null cone: the compact-support criterion on space-like (to back ground metric) slices will guarantee that there will be a time-like surface (relative to the differential operator, not to the background metric) to one side of which the solution vanishes, then Holmgren kicks in and tells you that the solution must vanish everywhere.
(A version of Holmgren's also applies to ultrahyperbolic operators, with suitable changes in the geometry.)
Outside of these trivial cases, you need to actually consider the geometry of the characteristic cones and compare them against the geometry of the support set of your function. The keywords you will be looking for are Carleman estimates, unique continuation, and pseudo-convexity. There's a nice, comprehensive treatment in Lars Hormander's Analysis of Linear Partial Differential Operators (I think volume III, may be IV?); there are also some nice notes by Daniel Tataru on his website.