2.5. PROBLEMS 85 1. Write down an explicit formula for a function u solving the initial- value problem ut + b · Du + cu = 0 in Rn × (0, ∞) u = g on Rn × {t = 0}. Here c R and b Rn are constants. 2. Prove that Laplace’s equation Δu = 0 is rotation invariant that is, if O is an orthogonal n × n matrix and we define v(x) := u(Ox) (x Rn), then Δv = 0. 3. Modify the proof of the mean-value formulas to show for n 3 that u(0) = ∂B(0,r) g dS + 1 n(n 2)α(n) B(0,r) 1 |x|n−2 1 rn−2 f dx, provided −Δu = f in B0(0,r) u = g on ∂B(0,r). 4. Give a direct proof that if u C2(U) C( ¯ U ) is harmonic within a bounded open set U, then max ¯ U u = max ∂U u. (Hint: Define := u + ε|x|2 for ε 0, and show cannot attain its maximum over ¯ U at an interior point.) 5. We say v C2( ¯ U ) is subharmonic if −Δv 0 in U. (a) Prove for subharmonic v that v(x) B(x,r) v dy for all B(x, r) U. (b) Prove that therefore max ¯ U v = max∂U v. (c) Let φ : R R be smooth and convex. Assume u is harmonic and v := φ(u). Prove v is subharmonic. (d) Prove v := |Du|2 is subharmonic, whenever u is harmonic.
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