The hotspots conjecture can be expressed in simple English as:
Suppose a flat piece of metal, represented by a two-dimensional bounded connected domain R, is given a generic initial heat distribution which then flows throughout the metal. Assuming the metal is insulated (i.e. no heat escapes from the piece of metal), then at all sufficiently large times, the hottest and coldest points on the metal both will lie on its boundary...
Equivalently, the 2nd eigenfunction of the laplacian has its global max and global min on the boundary of the domain.
If the conjecture is to hold for some domain, then all eigenfunctions of the Laplacian must have their extrema on the boundary, because one can impose the initial condition that the temperature distribution is already an eigenfunction. I suppose "generic" means "a little bit of every eigenfunction", so eventually the second eigenfunction dominates. If an eigenvalue is degenerate, then every eigenfunction in the eigenspace must have its extrema on the boundary. Note that the first eigenfunction is the constant distribution. -- Gene