Math 173: Homework 2 (Due: April 24th) Solve 5 problems for full credit. Do 7 for extra. 1. Prove that if |u(x)| ≤ C(|x|m + 1) for all x ∈ Rn , where m ∈ N, and u is harmonic in Rn , then u is a polynomial of degree less than or equal to m. 2. Let Ω ⊂ Rn be a bounded open set. Consider a sequence un of harmonic functions such that un → u uniformly on U . Prove that u is an harmonic function. 3. Let u : Rn → R be a harmonic function. Prove that Z u2 (x) dx < +∞, then u ≡ 0. (Hint: Note that u2 must be subharmonic and use (a) If Rn problem 5 from Homework 1). Z (b) If |∇u2 (x)| dx < +∞, then u is constant. Rn 4. Suppose Ω is a bounded domain in Rn and consider the operator: X X Lv = aij (x)∂i ∂j v + bi (x)∂i v i,j i where aij and bi are continuous functions on Ω and the matrix (aij )(x) is positive definite for each x ∈ Ω. (a) Suppose v is a function in C 2 (Ω) satisfying Lv > 0. Prove that v cannot have a local maximum in Ω. (Hint: Use a change of coordinates to make aij diagonal at a point if necessary.) (b) Show that if x0 ∈ / Ω and M > 0 is a sufficiently large constant, then w(x) = exp(−M |x− x0 |2 ) satisfies Lw > 0 in Ω. (c) Use (a) and (b) to prove: Suppose u ∈ C 2 (Ω) ∩ C(Ω) and Lu = 0 in Ω, then max u = max u. ∂Ω Ω 5. Solve problem 9 of Evans, page 87 (1st edition). 6. Suppose u is a continuous function on Ω. If for each x ∈ Ω there is a sequence of positive numbers rj → 0 such that Z u(x) = − u(y) dS(y), ∂Brj (x) for all j, then u is harmonic on Ω. (Hint: try to use the fact that we can solve the Dirichlet problem in a ball.) 1 7. Let u be a harmonic function in Ω\{x0 } and assume that u = o(Φ(x − x0 )) as x → x0 , where Φ is the fundamental solution of the Laplacian equation (with singularity at 0), that is: lim u(x)|x − x0 |n−2 = 0, if n ≥ 3; or lim x→x0 x→x0 u(x) = 0, if n = 2. log |x − x0 | Prove that x0 is a removable singularity for u. 8. Consider an increasing sequence un : B(0, R) → R of harmonic functions, that is un (x) ≤ un+1 (x) for every x ∈ B(0, R). Assume that un (0) is a Cauchy sequence. Show that there exists a harmonic function u such that un → u uniformly in B(0, r) for every r < R. 2
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