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The two-dimensional Laplace equation is defined as the sum of the second partial derivatives of a scalar function $u$ with respect to the spatial coordinates $x$ and $y$ equaling zero. It is presented on a rectangular domain defined by $0 < x < a$ and $0 < y < b$.

Conditions: The function $u$ is a scalar field depending on two independent spatial variables $x$ and $y$.; The domain is an open rectangle in the $xy$-plane.; The equation is homogeneous (equals zero).

The two-dimensional Laplace equation is defined as the sum of the second partial derivatives of a scalar function $u$ with respect to the spatial coordinates $x$ and $y$ equaling zero. It is presented on a rectangular domain defined by $0 < x < a$ and $0 < y < b$.

Conditions: The function $u$ is a scalar field depending on two independent spatial variables $x$ and $y$.; The domain is an open rectangle in the $xy$-plane.; The equation is homogeneous (equals zero).

Boundary conditions are mentioned because the Laplace equation alone has infinitely many solutions. To determine the specific temperature distribution $u(x,y)$ for a given physical plate, one must specify the values of $u$ (or its derivatives) on the boundary of the domain.

Conditions: The problem is a boundary-value problem for the Laplace equation.; The domain is bounded (e.g., a rectangle).; The general solution contains arbitrary constants or functions.

Boundary conditions are mentioned because the Laplace equation alone has infinitely many solutions. To determine the specific temperature distribution $u(x,y)$ for a given physical plate, one must specify the values of $u$ (or its derivatives) on the boundary of the domain.

Conditions: The problem is a boundary-value problem for the Laplace equation.; The domain is bounded (e.g., a rectangle).; The general solution contains arbitrary constants or functions.