I am studying Artin's Algebra. In Chapter 1 Exercise M11, he asks to show that every discrete Dirichlet problem on a finite discrete set in plane has a unique solution. This is essentially the question:
Let $R$ be a finite subset of $\mathbb{ Z\times Z}$. Let $\partial R$ be the set of all the points, not in $R$, which are unit distance away from some point in $R$. Let $\beta$ be a function on $\partial R$. Then we need to show that there exists a unique function $f$ on $R\cup \partial R$ such that $f(u, v) = \beta_{u, v}$ for all $(u, v)\in\partial R$ and that $f$ satisfies the discrete Laplace equation$^1$ for all points in $R$.
The full question can be found here.
My attempt at solving this:
After arbitrarily ordering $R$, we get a linear system $LX = B$ to solve for $X$ where $B$ contains combinations of $\beta_{u, v}$'s. We just need to show that any such $L$ is invertible, or equivalently, has a nonzero determinant.
I abstracted out the following properties that any coefficient matrix $L$ must satisfy (possibly after a rearrangement of equations, provided that we write the equations as mentioned in the footnote):
- The diagonal entries are all $4$.
- Nondiagonal entries can be either $0$ or $-1$.
- Any row can have at most four $-1$ entries (since any point in $R$ can have at most four points in $\partial R$ that are a unit distance away from it).
- Any column in $L$ can also have a maximum of four $-1$ entries (since any point in $R$ can be a unit distance away from at most four points in $R$).
Now I consider an arbitrary matrix $A$ satisfying the above properties. I rearrange the columns and rows (by elementary row and column operations) to get the following matrix: $$ \begin{bmatrix} B & \ast\\ \ast & B' \end{bmatrix}. $$ Here $B$ and $B'$ are matrices satisfying the above properties, with the additional condition on $B'$: its first column is entirely nonzero, and contains all the $-1$'s of the first column of the original $A$. Thus $b'$ has size at most $5\times 5$ and at least $1\times 1$. By induction, we can row reduce $B'$ to $I$, and thus we can row reduce $A$ to $$ \begin{bmatrix} B' & 0\\ \ast & I \end{bmatrix}. $$
Now all I need to show is that $B'$ is also invertible. The brute force method contains $2^{20}$ cases with the general matrix to be analyzed for invertibility being $$ \begin{bmatrix} 4 & &\ast\\ &\ddots&\\ \ast&&4 \end{bmatrix}_{5\times 5} $$ with the nondiagonal entries being $-1$ or $0$. (This case will also suffice for the cases when $B$ has a smaller size.)
But I am unable to prove this base case without brute force.
Can you help?
$^1$$f$ satisfies the Laplace equation at $(u, v)\in R$ iff $4f(u, v) = f(u+1, v) + f(u-1, v) + f(u, v+1) + f(u, v-1)$.