Solving equations too rough to differentiate

Mathematics · Differential Equations · ages 22-24

Name ______________________ Date ____________

Let u(x) equal the absolute value of x on (negative 3, 3). What is the weak derivative of u at x equals 2?
Answer: ______________
Someone claims u(x) = |x| is classically differentiable at 0 with derivative 0. Is that claim right?
Circle one: True False
Let u(x) = |x| on (-3, 3). What is the weak derivative of u at x = 2?
Answer: ______________
Let phi be a test function supported inside (negative 1, 1). What does the integral of u prime times phi over the interval equal?
- The integral of u times phi prime
- Minus the integral of u times phi prime
- The bracket of u phi at the endpoints
A student says every H1 function on an interval has a continuous representative. Is that true in one dimension?
Circle one: True False
When moving a derivative off u(x) = x squared with a test function phi on [-1, 1], the boundary term is [x squared times phi] from -1 to 1. What is its value?
Answer: ______________
A tent function is 0 at x = -2 and x = 2 and peaks at 2 at x = 0, joined by straight lines. What is its weak derivative on (0, 2)?
- -1
- 1
- 0
- It has no weak derivative there
What does a Sobolev space collect?
- Square-integrable functions whose weak derivatives are also square-integrable
- Only smooth functions with compact support
- Every integrable function with no further condition
Every L2 function on (negative 1, 1) has a weak first derivative in L2.
Circle one: True False
Let u(x) = |x| on (-1, 1). Which statement about its weak second derivative is correct?
- 0
- No weak second derivative in L2
- 2
- The sign function

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 · Away from the kink it matches the line y equals x, whose slope is 1.
False · The left slope -1 and right slope 1 never meet, so no tangent exists at 0.
1 · Away from the kink, |x| is the line y = x with slope 1, so its weak derivative at x = 2 is 1.
Minus the integral of u times phi prime · Integration by parts moves the derivative over, and the bracket dies on test functions.
True · In one dimension every H1 function agrees almost everywhere with a continuous function, so the tent has no jump to hide.
0 · Both evaluations hit phi at an endpoint where it is zero, so the bracket is 0.
-1 · On (0, 2) the tent falls from 2 to 0, so its slope and its weak derivative there are -1.
Square-integrable functions whose weak derivatives are also square-integrable · Sobolev membership controls both the function and its weak derivatives in L2.
False · The step function is bounded hence in L2, but its derivative is a point mass.
No weak second derivative in L2 · Differentiating twice concentrates a point mass at the kink, which is not an L2 function, so there is no weak second derivative in L2.