HOMEWORK 1, DUE JANUARY 26 IN CLASS
Problem 1: (This problem uses facts from real analysis) Let u ∈ W1,p(0,1) for 1 ≤ p ≤ ∞.
Show that u equals almost everywhere an absolutely continuous function v and its weak
derivative u′ equals the pointwise derivative v′ almost everywhere. (Hint: Pick b > a ∈ (0, 1)
and arbitrary. Consider the function ψε(x) = φε(x − b) − φε(x − a), set ηε = x ψε(z)dz, use 0
the definition of the weak derivative and let ε go to zero. Here φε is a non-negative ‘bump’ function, i.e., smooth with support in (−ε, ε) and φε(x)dx = 1)
Problem 2: a) Prove the inequality
∥f∥2∞ ≤ ∥f∥L2(R)∥f′∥L2(R)
for all functions in C1(R). (Hint: Write f(x)2 = 2 x ff′dx and also f(x)2 = −2 ∞ ff′dx c −∞ x
and use Schwarz’s inequality.)
b) Is there a function, not necessarily in Cc1(R), that yields equality?
Problem 3: Fix any point x0 ∈ R and consider the linear functional l(φ) = φ(x0) where φ ∈ Cc∞(R.
a) Show that l can be uniquely extended to a bounded linear functional on H1(R).
b) Show that there exists a unique u0 ∈ H1(R) such that (u0, v)H1(R) = l(v) for all v ∈ H1(R) and check that u0(x) = e−|x−x0|.
Problem 4: A function u : Rn → R is H ̈older continuous of order 0 < α < 1 if ∥u∥Cα :=∥u∥∞+sup|u(x)−u(y)|<∞.
x̸=y |x − y|α
This space Cα(R) is a Banach space. Show that any function u ∈ W1,p(R) for some 1 ≤ p < ∞
is almost everywhere equal to a function that is Ho ̈lder continuous of order α = 1 − 1 . (Hint: p
Prove the estimate
for functions u ∈ Cc1(R) and then use the fact that these functions are dense in W1,p(R).)
∥u∥Cα ≤ C∥u∥W1,p(R)
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