A long cylinder of material with a with a rectangular cross-section occupies the region 0 < x < a, 0 < y < b in space. The faces at x = 0 and x = a are held at temperature u=0 , and the face at y = b is thermally insulated. The face at y = 0 is held at temperature (const.) until a steady temperature u(x,y) is reached in the cylinder. Show that
for 0 < x < a, 0 < y < b.