This is the Laplace equation. The best place to start would be to look at posts about this equation here and on stackexchange such as:
http://community.wolfram.com/groups/-/m/t/315261
http://mathematica.stackexchange.com/questions/60663/solving-laplaces-equation-in-rectangle-domain-by-separation-of-variables
DSolve is useful for finding closed-form solutions to equations. Here is some documentation on what it does with this equation (http://reference.wolfram.com/language/tutorial/DSolveSecondOrderPDEs.html).The solution above is more of a statement that the solution T can be represented as a Fourier Series.
