[SOLVED] : HOMEWORK 1

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 (x) = (x b) (x a), set = x (z)dz, use 0

the definition of the weak derivative and let go to zero. Here is a non-negative bump function, i.e., smooth with support in (, ) and (x)dx = 1)

Problem 2: a) Prove the inequality
f2 fL2(R)fL2(R)

for all functions in C1(R). (Hint: Write f(x)2 = 2 x ffdx and also f(x)2 = 2 ffdx c x

and use Schwarzs 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() = (x0) where 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|xx0|.

Problem 4: A function u : Rn R is H older continuous of order 0 < < 1 if uC :=u+sup|u(x)u(y)|<.

x=y |x y|
This space C(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 = 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).)

uC CuW1,p(R)

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