# Uniqueness of Solutions to First-Order, Linear, Homogeneous, Boundary-Value PDE

Consider a homogeneous, linear, first-order PDE

$$L u \equiv \left( \sum_{i = 1}^d f^i(x) \frac{\partial}{\partial x^i} + c(x) \right) u(x) = 0$$

on some compact domain $$\Omega \subset \mathbb{R}^d$$. Obviously this system always has $$u = 0$$ as a solution; my question is what sorts of conditions on the coefficients $$f^i(x)$$ and $$c(x)$$ are sufficient in order to guarantee that the zero solution is unique subject to the boundary condition $$u|_{\partial \Omega} = 0$$.

I know that well-posedness of first-order PDEs is usually studied via the method of characteristics, but as I understand that's typically useful in thinking of the PDE as an initial value problem in which boundary conditions are specified on an initial-value surface and evolved from there. Because here I'm treating the system as a Dirichlet problem, the inhomogenous problem $$Lu = g$$, $$u|_{\partial \Omega} = h$$ may not in general be well-posed; but that's OK because I just care about uniqueness of the zero solution to the homogeneous problem.

I have one partial result from Oleinik and Radkevic (https://www.springer.com/gp/book/9781468489675), which consider second-order linear PDEs with nonnegative characteristic form, of which the equation I gave above is a special case (since its characteristic form is identically zero). Then from e.g. Theorem 1.6.2 of this book I can conclude that the zero solution is unique if $$c^* < 0$$ in $$\Omega \cup \partial \Omega$$, where $$c^* \equiv c - \sum_{i = 1}^d \partial_i f^i$$ is the zero-derivative term of the adjoint $$L^*$$ of $$L$$. But because the operator $$L$$ I care about is genuinely a first-order operator, while the condition $$c^* < 0$$ comes from considering second-order operators, I imagine there must be much more general sufficient conditions for the uniqueness of the zero solution than just $$c^* < 0$$.

• I don't know how useful this is (but I have nothing smarter to say - this seems like a tough problem), but uniqueness of solutions for boundary value problems can be sometimes proven using the method of energies. Maybe you can think of a method of constructing an energy functional that would fit a large family of differential operators $L$ which could be used to prove uniqueness? Perhaps you could find a sufficient criterion for a differential operator to satisfy in order to 'fit' the energy functional you constructed. Commented Aug 21, 2020 at 21:52

The method of characteristics looks like the right way to solve this. Along paths that satisfy $${\rm d}x_i/{\rm d}t = f_i(\vec{x})$$, one finds $$u(\vec{x}(t))$$ evolves according to $${\rm d}u/{\rm d}t = -c u$$. If the path terminates at $$\partial\Omega$$, then $$u(x) = 0$$ along the whole path. This leads to our first necessary condition for the existence of a nonzero solution:

(1) $$\exists$$ path $$\vec{x}(t)$$ satisfying $${\rm d}x_i/{\rm d}t = f_i(\vec{x})$$ with origin and terminus (limits as $$t \rightarrow \pm\infty$$) in the interior of $$\Omega$$.

For a continuous $$u(\vec{x})$$, the value of $$u(\vec{x}(t))$$ cannot diverge when $$t \rightarrow \pm\infty$$. Excepting a set of measure zero, all paths $$\vec{x}(t)$$ start at a repulsor and end at an attractor (rather than, say, a saddle point). Two more necessary conditions for the existence of a nonzero solution are therefore:

(2) $$c < 0$$ at $$\vec{x}(-\infty)$$

(3) $$c > 0$$ at $$\vec{x}(+\infty)$$

Except for a set of measure zero, we can probably assume these inequalities are strict, i.e. $$c < 0$$ and $$c > 0$$, respectively (convergence is possible for $$c = 0$$ but not guaranteed, depending on derivative terms). With the strict inequalities, conditions (1-3) are also sufficient for nonzero solutions $$u(\vec{x})$$ to exist. That can be seen as follows:

Starting with a point $$\vec{x}_0$$ along the path $$\vec{x}(t)$$, define a size-$$\epsilon$$ cross section (orthogonal to the streamlines of $${\rm d}x_i/{\rm d}t = f_i(\vec{x})$$) and posit that $$u(\vec{x})$$ varies smoothly from $$u(x_0) = 1$$ to $$u = 0$$ at the boundaries of the cross section. The value of $$u(\vec{x})$$ along the "past" and "future" of this cross section is obtained by propagating along the characteristics using $${\rm d}u/{\rm d}t = -c u$$. All these characteristics originate from the same repulsor (where $$u = 0$$) and terminate at the same attractor (also where $$u = 0$$). Fill in the rest of $$\Omega$$ with the null solution $$u = 0$$. Thus we have constructed a nonzero, continuous-valued solution to the PDE.

There are a bunch of singular edge cases where the necessary and sufficient conditions don't coincide, i.e. if $$\lVert f \rVert = u = 0$$ at the same point (fixable by rescaling $$f$$ and $$u$$), if $$\lVert f\rVert = 0$$ over an open subset of $$\Omega$$, if $$\lVert f\rVert = 0$$ on the boundary $$\partial\Omega$$, if $$c = 0$$ at $$\vec{x}(\pm\infty)$$. In the space of possible functions $$(\vec{f}, u)$$, these singular cases only occur in a set of measure zero, so are not very interesting. Almost everywhere, conditions (1-3) are both necessary and sufficient.

Putting this another way, we can say (almost everywhere) that the zero solution is unique if:

$$\forall$$ paths $$\vec{x}(t)$$ satisfying $${\rm d}x_i/{\rm d}t = f_i(\vec{x})$$ with origin and terminus in the interior of $$\Omega$$,

$$c > 0$$ at $$\vec{x}(-\infty)$$ or $$c < 0$$ at $$\vec{x}(+\infty)$$.

Coming back to your condition $$c^* < 0$$: Note that $$\partial_i f^i < 0$$ at attractors (this always holds, regardless of whether it's a node, limit cycle, toroid, chaotic attractor, etc.). Therefore, if $$c^* < 0$$ on $$\Omega$$, it follows that $$c = c^* + \partial_i f^i < 0$$ at all of the attractors. Therefore, the second condition above is always satisfied when $$c^* < 0$$. The condition above is the more general sufficient (and necessary) condition for uniqueness (with the caveats noted above).

Since any dynamical system can be represented by $${\rm d}x_i/{\rm d}t = f_i(\vec{x})$$ and dynamical systems can be really, really complicated, the general condition can be hard to work with, so more specific conditions like $$c^* < 0$$ might be more useful.

Also, defining the value of $$c$$ is tricky when the attractor / repulsor isn't a point. Taking the average over limit cycles is straightforward, chaotic attractors less so (ergodic theory).

• Thanks! This is extremely clear, and gave me a much better understanding of the method of characteristics than I'd been able to get from quickly skimming the literature. I notice you treat the $c = 0$ case as an exception; do you know what can be said when $c = 0$ everywhere in $\Omega$? (Besides requiring that there simply be no paths $\vec{x}(t)$ satisfying $dx_i/dt = f_i(\vec{x})$ with origin and terminus in the interior of $\Omega$.) Commented Aug 24, 2020 at 15:55
• Naively going off of your answer, I would think that continuity of $u$ in the $c = 0$ case would require $u$ to vanish at any origins and/or termini in the interior of $\Omega$, so that the zero solution must always be the unique one. But I might be being too fast, especially if there are attractors or repulsors that aren't points. Commented Aug 24, 2020 at 15:57
• More precisely, since $u = 0$ along any path that reaches $\partial \Omega$, I would think that continuity of $u$ in the $c = 0$ case would require $u$ to also vanish along any path whose origin and terminus can both be reached by paths that hit $\partial \Omega$. But I don't know how to deal with paths whose origin and terminus cannot be reached in such a way. Commented Aug 24, 2020 at 16:12
• For $c = 0$ everywhere, the PDE tells us that $u(\vec{x})$ is constant along characteristics. Unless $\partial_i f^i = 0$ for any open subset of $\Omega$, $\Omega$ can be divided into (I think) a countable number of basins of attraction. If $u = 0$ at the attractor, it is zero in the entire basin. If $u = 0$ in one basin, it is zero in all neighboring basins (by continuity). Since the basins are connected, and $u = 0$ on the boundary, it must be zero anywhere. Commented Aug 24, 2020 at 21:31
• But there is one pathological case where this doesn't work: if $f^i$ is Hamiltonian ($\partial_i f^i = 0$) over any finite subset of $\Omega$. Then there are regions of $\Omega$ that are not basins of attraction. Consider a rotational flow $\vec{f} = r \hat{\phi}$ on a unit disc. The PDE tells us $u(r, \phi)$ is independent of $\phi$, but it can depend on $r$, so nonzero solutions exist. Commented Aug 24, 2020 at 21:36