9. Suppose that V is an open subset of R2, H = [a, b] x [0, c] C V, u: V -+ R is C2 on V, and u(xo, to) ~ ° for all (xo, to) E 8H.

(a) Show that given E > 0, there is a compact set K c HO such that u(x, t) ~ -E for all (x, t) E H \ K.

(b) Suppose that U(X1,tl) = -£ < ° for some (Xl,tl) E HO, and choose r > ° so small that 2rt1 < £. Apply part

(a) to E := £/2 - rh to choose the compact set K, and prove that the minimum of w(x, t) := u(x, t) + r(t - tl)

on H occurs at some (X2' t2) E K.

(c) Prove that if u satisfies the heat equation, i.e., Uxx - Ut = ° on V, and if u(xo, to) ~ ° for all (xo, to) E 8H, then u(x, t) ~ ° for all (x, t) E H.