Solution to Laplace equation
Question:
Let $u(x, y)$ be the solution to Laplace’s equation in the unit square, with boundary conditions $$u(x, 0) = u(0, y) = u(x, 1) = u(1, y) = 1, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1.$$
Without using separation of variables, find an explicit expression for $u(x,y)$.
The question gives the following hint: what is the solution to $f ''(x) = 0$ on $0 < x < 1$ with $f(0) = f(1) = 1$?)
From the hint, I see that $f(x)=1$ but I don't see how this helps with the question.
$\endgroup$1 Answer
$\begingroup$Putting $u(x,y)=1$ works: a constant satisfies Laplace's equation, and the boundary conditions work too. The point is that this is analogous to the one-dimensional case, where linear functions are solutions to $f''=0$. In higher dimensions there are many more functions that satisfy Laplace's equation, but linear functions still work.
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