Understanding meaning of notation $u(x, y) = g(x, y)$ on $\partial D$

Question

I've a numerical problem which involves the Poisson's equation. I'm clearly not an expert in differential equations at all, so I'm trying to understand here the notation.

The problem starts like this:

The version of Poisson's equation being solved here is

$$\left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right) u(x, y) = f(x, y)$$

$$u(x, y) = g(x, y) \text{ on } \partial D$$

What's the meaning here of $u(x, y) = g(x, y)$? From what I remember having studied about ODEs, usually there's just one equation, not two, relating a function to its derivatives with respect to one independent variable. Here we also have a second equation $u(x, y) = g(x, y)$. Why?

Second question, why on $\partial D$? I know that $D$ is the $2D$ domain, but I'm not understanding the meaning of this $u(x, y) = g(x, y) \text{ on } \partial D$.

Answer 1

$\partial D$ is the boundary of the region $D.$ So $g(x,y)$ is a known function such that $u(x,y) = g(x,y)$ for $(x,y)\in \partial D,$ i.e. points on the boundary. A typical boundary condition would be for $u(x,y)=0$ on a circle. Different boundary conditions will lead to different solutions to the equation.

This looks different from the boundary conditions you're used to for one-dimensional problems, since in that case the boundary consisted of two points $a$ and b, so you just had $u(a) = C$ and $u(b) = D,$ or something like that. Here, the boundary is a one-dimensional object (like a circle).

If you remember that sometimes in ODEs the boundary conditions could on deriviatives, that's true here also. For instance somestimes a boundary condition will be of the form $$\nabla u(x,y)\cdot \hat n(x,y) = h(x,y)$$ where $\hat n$ is the normal vector to the boundary.