MATH4202/6202 Partial Differential Equations Assignment 1 2022
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MATH4202/6202 Partial Differential Equations
Assignment 1
2022
1. (i) (3pts) An integrable function o e L1(U) is called weakly harmonic in U if
o∆φ dz = 0
U
for all φ > 0 in C2(U) with compact support. Show that a weakly harmonic function is harmonic.
(ii) (3pts) Prove that if ∆o = 0 in a connected open set U and o = θo/θ”= 0 on a smooth, open portion of θU where ”is the normal vector field to θU , then o 三 0 in U .
2. (3pts) Let o be a nonnegative harmonic function in a ball B(0﹐ R). Use the Poisson integral formula to prove the following version of Harnack’s inequality:
RI塞2(R - IzI) RI塞2(R + IzI)
(R + IzI)I塞1 (R - IzI)I塞1
3. (4pts) A function u e C2(U) n C(
) is called subharmonic (resp. super- harmonic) if
∆u > 0 (resp. < 0) in U．
(i) Prove that if u is subharmonic in U , then
u(z) < 1 u(y) dy for all B(z﹐ t) c U．
α(n)tI B(t尸r)
(ii) Prove that
max u = max u．
尸U
(iii) Let φ : R → R be smooth and convex. Prove that u = φ(o) is subhar- monic if o is harmonic.
(iv) Prove that u = IDoI2 is subharmonic if o is harmonic.
4. (4pts) Let o e C2(RI × [0 ﹐ o)) solve
ot - ∆o = 0 in RI × (0 ﹐ o)﹐
o = f on RI × {u = 0}﹐
where f is continuous with compact support.
(i) Show that limt去o o(z﹐ u) = 0 uniformly for z e RI and obtain an explicit decay estimate for o.
(ii) Show that all the derivatives of o decay to zero uniformly as u → o.
5. (4pts) [Evans, Q17, page 88] Let o e C2(R × [0 ﹐ o)) solve the initial value problem for the wave equation in one space dimension
ott - o北北 = 0 in R × (0 ﹐ o)﹐
o = g﹐ ot = h on R × {u = 0}﹐
where g﹐ h are smooth with compact support.
The kinetic energy is k(u) = 
塞o(o) ot(2)(z﹐ u) dz and the potential energy is r(u) = 
塞o(o) o北(2)(z﹐ u) dz.
Prove that
(i) k(u) + r(u) is constant in u.
(ii) k(u) = r(u) for all large enough times u.
6. (4pts) [Evans, Q18, page 88] Let o solve
ott - ∆o = 0 in R3 × (0 ﹐ o)﹐
o = g﹐ ot = h on R3 × {u = 0}﹐
where g﹐ h are smooth and have compact support.
(i) Show that there exists a constant C such that
Io(z﹐ u)I < 
for (z﹐ u) e R3 × (0 ﹐ o)．
(ii) Show by giving an example that the decay estimate need not hold if g﹐ h are smooth bounded functions that do not decay to zero sufficiently fast as z → o. (in your example, g, h must decay but not too fast)
2022-09-14