به انگلیسی

Solving equations too rough to differentiate

Mathematics · Differential Equations · ages 22-24

Name ______________________ Date ____________

Let u(x) equal the absolute value of (x minus 1) on (negative 2, 3). What is the weak derivative of u on the interval (negative 2, 1)?
Answer: ______________
A tent is 0 at x = -1 and x = 1 and peaks at 1 at x = 0. Someone claims it solves u'' = 0 classically on (-1, 1). Is that right?
Circle one: True False
Let u(x) = |x + 2| on (-5, 5). What is the weak derivative of u on the interval (-5, -2)?
Answer: ______________
Let phi be a test function supported inside (negative 1, 1). Moving both derivatives off u, what does the integral of u double prime times phi equal?
- Minus the integral of u times phi double prime
- The integral of u times phi double prime
- The integral of u prime times phi prime
A student says every H1 function on an interval has a continuous representative. Is that true in one dimension?
Circle one: True False
Moving a derivative off u(x) = sin x with a test function phi on (0, π), what is the boundary term [sin x · phi] from 0 to π?
Answer: ______________
A tent function is 0 at x = 0 and x = 4 and peaks at 2 at x = 2. What is its weak derivative on (2, 4)?
- 1
- -1
- 0
- 2
What does a Sobolev space collect?
- Only smooth functions that vanish near the boundary
- Every integrable function, with no condition on slopes
- Functions in L2 whose weak derivatives are in L2 too
The function equal to x for x below 0 and x plus 1 for x at least 0 belongs to H1 on (negative 1, 1).
Circle one: True False
Let u(x) = |x - 1| on (0, 2). What is its weak first derivative?
- The sign of (x - 1): -1 left of 1, 1 right of 1
- 1 everywhere
- 0 everywhere
- It has none

Answer key

For grown-ups. Fold this page away before handing over the rest.

Solving equations too rough to differentiate W1-mt_AKlC3Ol70p-s1

-1 · Left of the kink at 1, u(x) = 1 - x, whose slope is -1.
False · The tent has no classical second derivative at 0, where its slope jumps from 1 to -1, so it is not a classical solution there.
-1 · Left of the kink at -2, u(x) = -(x + 2), whose slope is -1.
The integral of u times phi double prime · Each move flips the sign and the bracket dies: the integral of u double prime phi = minus the integral of u prime phi prime = the integral of u phi double prime.
True · In one dimension every H1 function agrees almost everywhere with a continuous function, so the tent has no jump to hide.
0 · phi vanishes near both ends, so the bracket is 0.
-1 · On (2, 4) the tent falls from 2 to 0 over a run of 2, so its slope is -1.
Functions in L2 whose weak derivatives are in L2 too · Sobolev membership controls both the function and its weak derivatives in L2.
False · The function jumps by 1 at 0. Every H1 function on an interval has a continuous representative, so a jump rules it out; its weak derivative would contain a point mass.
The sign of (x - 1): -1 left of 1, 1 right of 1 · Left of 1 the slope is -1 and right of 1 it is 1, so the weak derivative is the sign of (x - 1); its value at 1 does not matter.