Reference no: EM133132541

Problem 1: Let f ∈ C² (B(0, R)) and Δf = 0 in B(0, R). Given k ≥ 0, define

W_k(r) = (1/r^n-2+2k)D(r) - k/(r^n-1+2k).H(r),

where as in the lectures

H(r) = ∫_Sr f²dσ, D(r) = ∫_Br |∇f|²dx.

(a) Prove that for every 0 < r < R one has

W'_k(r) = 2/r^n+2k ∫_Sr (<∇f,x> - kf)²dx

therefore r → Wk(r) is increasing in (0, R).

(b) Prove that if W_k(r) const. for 0 < r < R, then f must be homogeneous of degreer in B(0, R).

Problem 2. Let G(x, y, t) = (4Πt)^-n/2 e-|x-y|²/4t be the Gauss-Weierstrass kernel in Rⁿ, and consider the heat semigroup P_tf(x) = ∫_Rn G(x, y, t) f(y)dy.

1) Prove that for every function f ∈ C²₀ (Rⁿ) one has for every x ∈ Rⁿ and t > 0

P_t(Δf)(x) = Δ(P_tf)(x)

2) Assume that n ≥ 3. Prove that the following equation holds in D''(Rⁿ)

Δ_y(₀∫^∞ G(x, y, t)dt) -δx,

where δx indicates the Dirac delta at z.

Attention: For part 2) you are not allowed to compute explicitly ₀∫^∞ G(x, y, t)dt and use theorems proved in the course! You must produce a direct proof, independent from results proved in the course. You can take for granted that for every x ∈ Rⁿ the function y → ₀∫^∞ G(x, y, t)dt be- longs to L¹_loc(Rⁿ), and therefore it identifies a well-defined element of D'(Rⁿ), i.e. a distribution on Rⁿ

Problem 3. Let E(x, t) = 1/2 1_Ωt(x, t), where 1_Ω denotes the indicator function of the set Ω = {(x, t) ∈ R² | x ∈ R, t > |x|}. Prove that

(i) E ∈ L¹_loc (R)², and therefore E ∈ D'(R²);

(ii) E is a fundamental solution of = ∂²/∂t² - ∂²/ ∂x² in R², i.e., one has in D' (R²)

E = ∂²E/∂t² - ∂²E/∂x² = δ

or equivalently

1/2∫_Ω Ψ(x, t) dxdt = Ψ(0,0)

for any Ψ ∈ C₀^∞ (R²).

Hint: In part (ii) it is useful to use the change of variable

ξ = x - t, η = x + t,

that transforms the forward cone Ω into the second quadrant Ω* = {(ξ, n) ∈ R² | ξ < 0,  η > 0}

Problem 4. Let f be a harmonic function in the unit ball B₁ {x ∈ Rⁿ| |x| < 1}. For any 0 < r < 1, let B_r {x ∈ Rⁿ | |x| < r} and consider the function

J_α(r) = 1/r^α ∫_Br |∇f(x)|²/|x|^n-αdx,  0 < α < n.

Prove that the function r → J_α(r) is monotone nondecreasing in (0, 1).

Hint: Show that the function x → |∇f(x)|² is subharmonic in B₁ and therefore its spherical averages are nondecreasing

Problem 5. Let c > b > a > 1, and consider the ellipsoid

Ω = {(x, y, z) ∈ R³ | x²/a² + y²/b² + z²/c² < 1}

Let f ∈ C²(Ω) ∩ C(∩¯) be the solution to the problem

{ Δf(x, y, z) = - (x²+y²+z²) in Ω,

f = 0, on ∂Ω

Prove that

(a³-1)/12 ≤ f(0, 0, 1) ≤ (c³-1)/12

Problem 6. For a real number a, let a₊ = max{a, 0}, and consider the continuous function with compact support (1 - |x|²)₊, x ∈ Rⁿ. Since such a function is obviously in L¹_loc (Rⁿ) it defines a distribution in D'(Rⁿ).

(a) Compute the distribution T ∈ D'(Rⁿ) such that Δ((1-|x|²)+/2n) = T in D'(Rⁿ). This means that you must find the distribution T such that
for every Ψ ∈ C₀^∞(Rⁿ).

(b) Find the support of the distribution T.

Problem 7. Let n ≥ 2 and u ∈ C²(Rⁿ) be a solution in Rⁿ of the equation Δu = α|x|^k, for some α ≥ 0 and some k ≥ 0. With σ_n-1 being the (n - 1)-dimensional measure of the unit sphere, denote by

M_u(r) = 1/σ_n-1r^n-1∫_{∂Br(0, r)}u(y)dσ(y)

the spherical mean of u over the sphere centred at the origin with radius r.

a) Prove that for every r ≥ 0 one has

M_u(r) = u(0) + α. r^k+2/(n + k)(k + 2)

b) Show that if there exist C ≥ 0 and ε > 0 such that

|u(x)| ≤ C(1 + |x|^k+2-ε) , x ∈ Rⁿ

then it must be Δu ≡ 0 in Rⁿ.