# Uniqueness of the solution to systems of first-order linear PDEs

Context:

Let $$\Omega \subset \mathbb{R}^p$$ be an domain. For functions $$A_{jk}^i : \Omega \to \mathbb{R}$$ and $$B_k^i : \Omega \to \mathbb{R}$$ with some regularity, I am interested in the following system of PDEs: $$\sum_{j = 1}^n \sum_{k = 1}^m A_{jk}^i (x) \frac{\partial f_k}{\partial x_j} (x) + B_{k}^i (x) f_k (x) = 0 \ \ \text{for} \ \ i = 1, \dots, r$$ where $$f = (f_1, \dots, f_m)$$ are variables. Assume that the number of equations is no less than the number of variables, i.e., $$r \geq m.$$ Also, assume the Dirichlet boundary condition: $$f_k = 0$$ on $$\partial \Omega.$$

Question:

It is obvious that $$f_k = 0$$ is a solution. I am wondering under what conditions on $$A_{jk}^i$$ and $$B_k^i$$ the uniqueness is guaranteed.

What I have tried:

I found some potentially useful books/papers/notes.

(1) Wavefronts and Rays as Characteristics and Asymptotics by Andrej Bóna and Michael A Slawinski.

In section 1.7 (page 24), they briefly explain how to solve the system. Their setup is almost the same as what I have in mind, but their method relies on the higher-order differentiability of $$A_{jk}^i,$$ which is not ideal for me.

(2) METHODS OF MATHEMATICAL PHYSICS VOLUME II by R. COUHANT and D. HILBERT.

In section 2.2 (pages 14-15), they address the system saying that by Cramer's rule, one can transform the system into independent PDEs. But I am not pretty sure how I should apply Cramer's rule to differential operators.

(3) Lecture note by Evy Kersalé.

In section 2.4, he discusses interesting methods to solve the system for $$p = 2.$$ I would like to know more general treatments.

What I want: I conjecture that a sort of full rankness of $$A_{jk}^i$$ and $$B_k^i$$ is sufficient for the uniqueness, but I have not found such a result. Also, I do not want to put too strong differentiability assumptions on $$A_{jk}^i$$ and $$B_k^i.$$ Ideally, they should be up to $$C^1.$$ Any thoughts and references are helpful.

Thanks!

• Are you making any assumptions about the number of unknown functions versus the number of equations? If there are fewer equations than functions, uniqueness is highly unlikely. If you are not assuming any boundary conditions, uniqueness probably doesn’t hold when the number of equations equals the number of functions. Things get much more complicated if the number of equations is greater than the number of functions. Nov 18 at 1:28
• Yes, I am implicitly assuming the number of equations is equal to or more than the number of variables. For the boundary condition, please assume the Cauchy condition. I'll add these. Nov 18 at 1:33
• Rather strong assumptions on the differential operator (i.e., the coefficients of the operator) are needed. The most general result I know is for symmetric positive systems by K. O. Friederichs. dept.math.lsa.umich.edu/~rauch/pisabvp/friedrichs1958.pdf Nov 19 at 5:33
• The subscript $k$ is not really relevant, is it? The components $f_k$ are determined by seperate sets of coupled equations, or do I miss something? Nov 27 at 15:54
• You write that $f$ are variables. But they are a function of $x$, right? Nov 27 at 15:56