List of top Partial Differential Equations Questions on Laplace Equation and Dirichlet Problem

Let \( \Omega = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 < 1\} \) be the open unit disc and \( \partial\Omega \) be its boundary. If \( u(x, y) \) is the solution of the following Dirichlet problem
\[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \quad \text{in } \Omega \]
\[ u(x, y) = x^2 - y^2 \quad \text{on } \partial\Omega, \]
then the value of \( 4\left(u\left(\frac{1}{2}, 0\right) - u\left(0, \frac{1}{2}\right)\right) \) is ______.

Consider the Laplace equation
\[ \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0, \qquad 0 < x < 1,\; 0 < y < 1, \] with the boundary conditions
\[ T(x,0) = x, \qquad T(0,y) = y \] \[ T(x,1) = 1+x, \qquad T(1,y) = 1+y. \] Then the value of \(T\left(\dfrac{1}{2}, \dfrac{1}{3}\right)\) is equal to